First Edition, 2012
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[email protected] Table of Contents Chapter 1 - Euclidean Group Chapter 2 - Dihedral Group Chapter 3 - Euclidean Plane Isometry Chapter 4 - Euler Angles Chapter 5 - Euler's Rotation Theorem Chapter 6 - Rotation Representation (Mathematics) Chapter 7 - Orthogonal Group Chapter 8 - Point Groups in Three Dimensions
Chapter 1
Euclidean Group
In mathematics, the Euclidean group E(n), sometimes called ISO(n) or similar, is the symmetry group of n-dimensional Euclidean space. Its elements, the isometries associated with the Euclidean metric, are called Euclidean moves. These groups are among the oldest and most studied, at least in the cases of dimension 2 and 3 — implicitly, long before the concept of group was known.
Overview Dimensionality The number of degrees of freedom for E(n) is n(n + 1)/2, which gives 3 in case n = 2, and 6 for n = 3. Of these, n can be attributed to available translational symmetry, and the remaining n(n − 1)/2 to rotational symmetry.
Direct and indirect isometries There is a subgroup E+(n) of the direct isometries, i.e., isometries preserving orientation, also called rigid motions; they are the rigid body moves. These include the translations, and the rotations, which together generate E+(n). E+(n) is also called a special Euclidean group, and denoted SE(n). The others are the indirect isometries. The subgroup E+(n) is of index 2. In other words, the indirect isometries form a single coset of E+(n). Given any indirect isometry, for example a given reflection R that reverses orientation , all indirect isometries are given as DR, where D is a direct isometry. The Euclidean group for n = 3 is used for the kinematics of a rigid body, in classical mechanics. A rigid body motion is in effect the same as a curve in the Euclidean group. Starting with a body B oriented in a certain way at time t = 0, its orientation at any other
time is related to the starting orientation by a Euclidean motion, say f(t). Setting t = 0, we have f(0) = I, the identity transformation. This means that the curve will always lie inside E+(3), in fact: starting at the identity transformation I, such a continuous curve can certainly never reach anything other than a direct isometry. This is for simple topological reasons: the determinant of the transformation cannot jump from +1 to −1. The Euclidean groups are not only topological groups, they are Lie groups, so that calculus notions can be adapted immediately to this setting.
Relation to the affine group The Euclidean group E(n) is a subgroup of the affine group for n dimensions, and in such a way as to respect the semidirect product structure of both groups. This gives, a fortiori, two ways of writing down elements in an explicit notation. These are: 1. by a pair (A, b), with A an n×n orthogonal matrix, and b a real column vector of size n; or 2. by a single square matrix of size n + 1, as explained for the affine group. Details for the first representation are given in the next section. In the terms of Felix Klein's Erlangen programme, we read off from this that Euclidean geometry, the geometry of the Euclidean group of symmetries, is therefore a specialisation of affine geometry. All affine theorems apply. The extra factor in Euclidean geometry is the notion of distance, from which angle can then be deduced.
Detailed discussion Subgroup structure, matrix and vector representation The Euclidean group is a subgroup of the group of affine transformations. It has as subgroups the translational group T, and the orthogonal group O(n). Any element of E(n) is a translation followed by an orthogonal transformation (the linear part of the isometry), in a unique way:
where A is an orthogonal matrix or an orthogonal transformation followed by a translation: . T is a normal subgroup of E(n): for any translation t and any isometry u, we have
u−1tu again a translation (one can say, through a displacement that is u acting on the displacement of t; a translation does not affect a displacement, so equivalently, the displacement is the result of the linear part of the isometry acting on t). Together, these facts imply that E(n) is the semidirect product of O(n) extended by T. In other words O(n) is (in the natural way) also the quotient group of E(n) by T: O(n)
E(n) / T.
Now SO(n), the special orthogonal group, is a subgroup of O(n), of index two. Therefore E(n) has a subgroup E+(n), also of index two, consisting of direct isometries. In these cases the determinant of A is 1. They are represented as a translation followed by a rotation, rather than a translation followed by some kind of reflection (in dimensions 2 and 3, these are the familiar reflections in a mirror line or plane, which may be taken to include the origin, or in 3D, a rotoreflection). We have: SO(n)
E+(n) / T.
Subgroups Types of subgroups of E(n):
Finite groups. They always have a fixed point. In 3D, for every point there are for every orientation two which are maximal (with respect to inclusion) among the finite groups: Oh and Ih. The groups Ih are even maximal among the groups including the next category. Countably infinite groups without arbitrarily small translations, rotations, or combinations, i.e., for every point the set of images under the isometries is topologically discrete. E.g. for 1 ≤ m ≤ n a group generated by m translations in independent directions, and possibly a finite point group. This includes lattices. Examples more general than those are the discrete space groups. Countably infinite groups with arbitrarily small translations, rotations, or combinations. In this case there are points for which the set of images under the isometries is not closed. Examples of such groups are, in 1D, the group generated by a translation of 1 and one of √2, and, in 2D, the group generated by a rotation about the origin by 1 radian. Non-countable groups, where there are points for which the set of images under the isometries is not closed. E.g. in 2D all translations in one direction, and all translations by rational distances in another direction.
Non-countable groups, where for all points the set of images under the isometries is closed. E.g. o all direct isometries that keep the origin fixed, or more generally, some point (in 3D called the rotation group) o all isometries that keep the origin fixed, or more generally, some point (the orthogonal group) + o all direct isometries E (n) o the whole Euclidean group E(n) o one of these groups in an m-dimensional subspace combined with a discrete group of isometries in the orthogonal n-m-dimensional space o one of these groups in an m-dimensional subspace combined with another one in the orthogonal n-m-dimensional space
Examples in 3D of combinations:
all rotations about one fixed axis ditto combined with reflection in planes through the axis and/or a plane perpendicular to the axis ditto combined with discrete translation along the axis or with all isometries along the axis a discrete point group, frieze group, or wallpaper group in a plane, combined with any symmetry group in the perpendicular direction all isometries which are a combination of a rotation about some axis and a proportional translation along the axis; in general this is combined with k-fold rotational isometries about the same axis (k ≥ 1); the set of images of a point under the isometries is a k-fold helix; in addition there may be a 2-fold rotation about a perpendicularly intersecting axis, and hence a k-fold helix of such axes. for any point group: the group of all isometries which are a combination of an isometry in the point group and a translation; for example, in the case of the group generated by inversion in the origin: the group of all translations and inversion in all points; this is the generalized dihedral group of R3, Dih(R3).
Overview of isometries in up to three dimensions E(1), E(2), and E(3) can be categorized as follows, with degrees of freedom: E(1) - 1:
E+(1): identity - 0 translation - 1 those not preserving orientation: o reflection in a point - 1 o o
E(2) - 3:
E+(2):
identity - 0 translation - 2 rotation about a point - 3 those not preserving orientation: o reflection in a line - 2 o reflection in a line combined with translation along that line (glide reflection) - 3 o o o
E(3) - 6:
E+(3): identity - 0 translation - 3 rotation about an axis - 5 rotation about an axis combined with translation along that axis (screw operation) - 6 those not preserving orientation: o reflection in a plane - 3 o reflection in a plane combined with translation in that plane (glide plane operation) - 5 o rotation about an axis by an angle not equal to 180°, combined with reflection in a plane perpendicular to that axis (roto-reflection) - 6 o inversion in a point - 3 o o o o
Commuting isometries For some isometry pairs composition does not depend on order:
two translations two rotations or screws about the same axis reflection with respect to a plane, and a translation in that plane, a rotation about an axis perpendicular to the plane, or a reflection with respect to a perpendicular plane glide reflection with respect to a plane, and a translation in that plane inversion in a point and any isometry keeping the point fixed rotation by 180° about an axis and reflection in a plane through that axis rotation by 180° about an axis and rotation by 180° about a perpendicular axis (results in rotation by 180° about the axis perpendicular to both) two rotoreflections about the same axis, with respect to the same plane two glide reflections with respect to the same plane
Conjugacy classes The translations by a given distance in any direction form a conjugacy class; the translation group is the union of those for all distances.
In 1D, all reflections are in the same class. In 2D, rotations by the same angle in either direction are in the same class. Glide reflections with translation by the same distance are in the same class. In 3D:
Inversions with respect to all points are in the same class. Rotations by the same angle are in the same class. Rotations about an axis combined with translation along that axis are in the same class if the angle is the same and the translation distance is the same, and in corresponding direction (right-hand or left-hand screw). Reflections in a plane are in the same class Reflections in a plane combined with translation in that plane by the same distance are in the same class. Rotations about an axis by the same angle not equal to 180°, combined with reflection in a plane perpendicular to that axis, are in the same class.
Chapter 2
Dihedral Group
In mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.
Notation There are two competing notations for the dihedral group associated to a polygon with n sides. In geometry the group is denoted Dn, while in algebra the same group is denoted by D2n to indicate the number of elements. Here, Dn (and sometimes Dihn) refers to the symmetries of a regular polygon with n sides.
Definition Elements
The six reflection symmetries of a regular hexagon A regular polygon with n sides has 2n different symmetries: n rotational symmetries and n reflection symmetries. The associated rotations and reflections make up the dihedral group Dn. If n is odd each axis of symmetry connects the mid-point of one side to the opposite vertex. If n is even there are n/2 axes of symmetry connecting the mid-points of opposite sides and n/2 axes of symmetry connecting opposite vertices. In either case, there are n axes of symmetry altogether and 2n elements in the symmetry group. Reflecting in one axis of symmetry followed by reflecting in another axis of symmetry produces a rotation through twice the angle between the axes. The following picture shows the effect of the sixteen elements of D8 on a stop sign:
The first row shows the effect of the eight rotations, and the second row shows the effect of the eight reflections.
Group structure As with any geometric object, the composition of two symmetries of a regular polygon is again a symmetry. This operation gives the symmetries of a polygon the algebraic structure of a finite group.
The composition of these two reflections is a rotation. The following Cayley table shows the effect of composition in the group D3 (the symmetries of an equilateral triangle). R0 denotes the identity; R1 and R2 denote counterclockwise rotations by 120 and 240 degrees; and S0, S1, and S2 denote reflections across the three lines shown in the picture to the right.
R0 R1 R2 S0 S1 S2 R0 R0 R1 R2 S0 S1 S2 R1 R1 R2 R0 S1 S2 S0 R2 R2 R0 R1 S2 S0 S1 S0 S0 S2 S1 R0 R2 R1 S1 S1 S0 S2 R1 R0 R2 S2 S2 S1 S0 R2 R1 R0
For example, S2S1 = R1 because the reflection S1 followed by the reflection S2 results in a 120-degree rotation. (This is the normal backwards order for composition.) Note that the composition operation is not commutative. In general, the group Dn has elements R0,...,Rn−1 and S0,...,Sn−1, with composition given by the following formulae:
In all cases, addition and subtraction of subscripts should be performed using modular arithmetic with modulus n.
Matrix representation
The symmetries of this pentagon are linear transformations. If we center the regular polygon at the origin, then elements of the dihedral group act as linear transformations of the plane. This lets us represent elements of Dn as matrices, with composition being matrix multiplication. This is an example of a (2-dimensional) group representation. For example, the elements of the group D4 can be represented by the following eight matrices:
In general, the matrices for elements of Dn have the following form:
Rk is a rotation matrix, expressing a counterclockwise rotation through an angle of 2πk ⁄ n. Sk is a reflection across a line that makes an angle of πk ⁄ n with the x-axis.
Small dihedral groups For n = 1 we have Dih1. This notation is rarely used except in the framework of the series, because it is equal to Z2. For n = 2 we have Dih2, the Klein four-group. Both are exceptional within the series:
They are abelian; for all other values of n the group Dihn is not abelian. They are not subgroups of the symmetric group Sn, corresponding to the fact that 2n > n ! for these n.
The cycle graphs of dihedral groups consist of an n-element cycle and n 2-element cycles. The dark vertex in the cycle graphs below of various dihedral groups stand for the identity element, and the other vertices are the other elements of the group. A cycle consists of successive powers of either of the elements connected to the identity element.
Dih1 Dih2 Dih3 Dih4 Dih5
Dih6
Dih7
The dihedral group as symmetry group in 2D and rotation group in 3D An example of abstract group Dihn, and a common way to visualize it, is the group Dn of Euclidean plane isometries which keep the origin fixed. These groups form one of the two series of discrete point groups in two dimensions. Dn consists of n rotations of multiples of 360°/n about the origin, and reflections across n lines through the origin, making angles of multiples of 180°/n with each other. This is the symmetry group of a regular polygon with n sides (for n ≥3, and also for the degenerate case n = 2, where we have a line segment in the plane). Dihedral group Dn is generated by a rotation r of order n and a reflection s of order 2 such that
(in geometric terms: in the mirror a rotation looks like an inverse rotation). In matrix form, an anti-clockwise rotation and a reflection in the x-axis are given by
(in terms of complex numbers: multiplication by
and complex conjugation).
By setting
and defining and product rules for Dn as
for
we can write the
(Compare coordinate rotations and reflections.) The dihedral group D2 is generated by the rotation r of 180 degrees, and the reflection s across the x-axis. The elements of D2 can then be represented as {e, r, s, rs}, where e is the identity or null transformation and rs is the reflection across the y-axis.
The four elements of D2 (x-axis is vertical here) D2 is isomorphic to the Klein four-group.
If the order of Dn is greater than 4, the operations of rotation and reflection in general do not commute and Dn is not abelian; for example, in D4, a rotation of 90 degrees followed by a reflection yields a different result from a reflection followed by a rotation of 90 degrees:
D4 is nonabelian (x-axis is vertical here). Thus, beyond their obvious application to problems of symmetry in the plane, these groups are among the simplest examples of non-abelian groups, and as such arise frequently as easy counterexamples to theorems which are restricted to abelian groups. The 2n elements of Dn can be written as e, r, r2, ..., rn−1, s, r s, r2 s, ..., rn−1 s. The first n listed elements are rotations and the remaining n elements are axis-reflections (all of which have order 2). The product of two rotations or two reflections is a rotation; the product of a rotation and a reflection is a reflection. So far, we have considered Dn to be a subgroup of O(2), i.e. the group of rotations (about the origin) and reflections (across axes through the origin) of the plane. However, notation Dn is also used for a subgroup of SO(3) which is also of abstract group type Dihn: the proper symmetry group of a regular polygon embedded in three-dimensional space (if n ≥ 3). Such a figure may be considered as a degenerate regular solid with its face counted twice. Therefore it is also called a dihedron (Greek: solid with two faces), which explains the name dihedral group (in analogy to tetrahedral, octahedral and icosahedral group, referring to the proper symmetry groups of a regular tetrahedron, octahedron, and icosahedron respectively).
Examples of 2D dihedral symmetry
2D D6 symmetry – The Red Star of David
2D D24 symmetry – Ashoka Chakra, as depicted on the National flag of the Republic of India.
Equivalent definitions Further equivalent definitions of Dihn are:
The automorphism group of the graph consisting only of a cycle with n vertices (if n ≥ 3). The group with presentation
or (The only finite groups that can be generated by two elements of order 2 are the dihedral groups and the cyclic groups. If the two elements of order 2 are distinct, then the group generated is dihedral.) From the second presentation follows that Dihn belongs to the class of Coxeter groups.
The semidirect product of cyclic groups Zn and Z2, with Z2 acting on Zn by inversion (thus, Dihn always has a normal subgroup isomorphic to the group Zn
is isomorphic to Dihn if φ(0) is the identity and φ(1) is inversion.
Properties If we consider Dihn (n ≥ 3) as the symmetry group of a regular n-gon and number the polygon's vertices, we see that Dihn is a subgroup of the symmetric group Sn via this permutation representation. The properties of the dihedral groups Dihn with n ≥ 3 depend on whether n is even or odd. For example, the center of Dihn consists only of the identity if n is odd, but if n is even the center has two elements, namely the identity and the element rn / 2 (with Dn as a subgroup of O(2), this is inversion; since it is scalar multiplication by −1, it is clear that it commutes with any linear transformation). For odd n, abstract group Dih2n is isomorphic with the direct product of Dihn and Z2. In the case of 2D isometries, this corresponds to adding inversion, giving rotations and mirrors in between the existing ones. If m divides n, then Dihn has n / m subgroups of type Dihm, and one subgroup Zm. Therefore the total number of subgroups of Dihn (n ≥ 1), is equal to d(n) + σ(n), where d(n) is the number of positive divisors of n and σ(n) is the sum of the positive divisors of n.
Conjugacy classes of reflections All the reflections are conjugate to each other in case n is odd, but they fall into two conjugacy classes if n is even. If we think of the isometries of a regular n-gon: for odd n there are rotations in the group between every pair of mirrors, while for even n only half of the mirrors can be reached from one by these rotations. Geometrically, in an odd polygon every axis of symmetry passes through a vertex and a side, while in an even polygon half the axes pass through two vertices, and half pass through two sides. Algebraically, this is an instance of the conjugate Sylow theorem (for n odd): for n odd, each reflection, together with the identity, form a subgroup of order 2, which is a Sylow 2-subgroup (2 = 21 is the maximum power of 2 dividing 2n = 2(2k + 1)), while for n even, these order 2 subgroups are not Sylow subgroups because 4 (a higher power of 2) divides the order of the group. For n even there is instead an outer automorphism interchanging the two types of reflections (properly, a class of outer automorphisms, which are all conjugate by an inner automorphism).
Automorphism group The automorphism group of Dihn is isomorphic to the affine group Aff(Z/nZ) number of k in
and has order nφ(n), where φ is Euler's totient function, the coprime to n.
It can be understood in terms of the generators of a reflection and an elementary rotation (rotation by k(2π / n), for k coprime to n); which automorphisms are inner and outer depends on the parity of n.
For n odd, the dihedral group is centerless, so any element defines a non-trivial inner automorphism; for n even, the rotation by 180° (reflection through the origin) is the non-trivial element of the center. Thus for n odd, the inner automorphism group has order 2n, and for n even the inner automorphism group has order n. For n odd, all reflections are conjugate; for n even, they fall into two classes (those through two vertices and those through two faces), related by an outer automorphism, which can be represented by rotation by π / n (half the minimal rotation). The rotations are a normal subgroup; conjugation by a reflection changes the sign (direction) of the rotation, but otherwise leaves them unchanged. Thus automorphisms that multiply angles by k (coprime to n) are outer unless
Examples of automorphism groups Dih9 has 18 inner automorphisms. As 2D isometry group D9, the group has mirrors at 20° intervals. The 18 inner automorphisms provide rotation of the mirrors by multiples of 20°, and reflections. As isometry group these are all automorphisms. As abstract group there are in addition to these, 36 outer automorphisms, e.g. multiplying angles of rotation by 2. Dih10 has 10 inner automorphisms. As 2D isometry group D10, the group has mirrors at 18° intervals. The 10 inner automorphisms provide rotation of the mirrors by multiples of 36°, and reflections. As isometry group there are 10 more automorphisms; they are conjugates by isometries outside the group, rotating the mirrors 18° with respect to the inner automorphisms. As abstract group there are in addition to these 10 inner and 10 outer automorphisms, 20 more outer automorphisms, e.g. multiplying rotations by 3. Compare the values 6 and 4 for Euler's totient function, the multiplicative group of integers modulo n for n = 9 and 10, respectively. This triples and doubles the number of automorphisms compared with the two automorphisms as isometries (keeping the order of the rotations the same or reversing the order).
Generalizations There are several important generalizations of the dihedral groups:
The infinite dihedral group is an infinite group with algebraic structure similar to the finite dihedral groups. It can be viewed as the group of symmetries of the integers. The orthogonal group O(2), i.e. the symmetry group of the circle, also has similar properties to the dihedral groups. The family of generalized dihedral groups includes both of the examples above, as well as many other groups. The quasidihedral groups are family of finite groups with similar properties to the dihedral groups.
Chapter 3
Euclidean Plane Isometry
In geometry, a Euclidean plane isometry is an isometry of the Euclidean plane, or more informally, a way of transforming the plane that preserves geometrical properties such as length. There are four types: translations, rotations, reflections, and glide reflections. The set of Euclidean plane isometries forms a group under composition: the Euclidean group in two dimensions. It is generated by reflections in lines, and every element of the Euclidean group is the composite of at most three distinct reflections.
Informal discussion Informally, a Euclidean plane isometry is any way of transforming the plane without "deforming" it. For example, suppose that the Euclidean plane is represented by a sheet of transparent plastic sitting on a desk. Examples of isometries include:
Shifting the sheet one inch to the right. Rotating the sheet by ten degrees around some marked point (which remains motionless). Turning the sheet upside down. Notice that if a picture is drawn on one side of the sheet, then after turning the sheet upside down, we see the mirror image of the picture.
These are examples of translations, rotations, and reflections respectively. There is one further type of isometry, called a glide reflection. However, folding, cutting, or melting the sheet are not considered isometries. Neither are less drastic alterations like bending, stretching, or twisting.
Formal definition An isometry of the Euclidean plane is a distance-preserving transformation of the plane. That is, it is a map
such that for any points p and q in the plane,
where d(p, q) is the usual Euclidean distance between p and q.
Classification of Euclidean plane isometries It can be shown that there are four types of Euclidean plane isometries (five if we include the identity). (Note: the notations for the types of isometries listed below are not completely standardised.)
Translation Translations, denoted by Tv, where v is a vector in R2. This has the effect of shifting the plane in the direction of v. That is, for any point p in the plane,
or in terms of (x, y) coordinates,
Rotations, denoted by Rc,θ, where c is a point in the plane (the centre of rotation), and θ is the angle of rotation. In terms of coordinates, rotations are most easily expressed by breaking them up into two operations. First, a rotation around the origin is given by
Rotation
These matrices are the orthogonal matrices (i.e. each is a square matrix G whose transpose is its inverse, i.e. GGT = GTG = I2.), with determinant 1 (the other possibility for orthogonal matrices is -1, which gives a mirror image). They form the special orthogonal group SO(2). A rotation around c can be accomplished by first translating c to the origin, then performing the rotation around the origin, and finally translating the origin back to c. That is, or in other words, Alternatively, a rotation around the origin is performed, followed by a translation:
Reflection Reflections, or mirror isometries, denoted by Fc,v, where c is a point in the plane and v is a unit vector in R2. (F is for "flip".) This has the effect of reflecting the point p in the line L that is perpendicular to v and that passes through c. The line L is called the reflection axis or the associated mirror. To find a formula for Fc,v, we first use the dot product to find the component t of p − c in the v direction,
and then we obtain the reflection of p by subtraction,
The combination of rotations about the origin and reflections about a line through the origin is obtained with all orthogonal matrices (i.e. with determinant 1 and -1) forming orthogonal group O(2). In the case of a determinant of -1 we have:
which is a reflection in the x-axis followed by a rotation by an angle θ, or equivalently, a reflection in a line making an angle of θ/2 with the x-axis. Reflection in a parallel line corresponds to adding a vector perpendicular to it.
Glide reflection Glide reflections, denoted by Gc,v,w, where c is a point in the plane, v is a unit vector in R2, and w is a vector perpendicular to v. This is a combination of a reflection in the line described by c and v, followed by a translation along w. That is,
or in other words, (It is also true that that is, we obtain the same result if we do the translation and the reflection in the opposite order.) Alternatively we multiply by an orthogonal matrix with determinant -1 (corresponding to a reflection in a line through the origin), followed by a translation. This is a glide reflection, except in the special case that the translation is perpendicular to the line of reflection, in which case the combination is itself just a reflection in a parallel line. The identity isometry, defined by I(p) = p for all points p, can be considered a fifth kind. Thus there are five mutually exclusive categories. Alternatively, we can consider the identity a special case of a translation, and also a special case of a rotation. Similarly we can consider every reflection to be a special case of a glide reflection. In that case we have only three categories: rotations, translations, and glide reflections, which are mutually exclusive except for the identity. In all cases we multiply the position vector by an orthogonal matrix and add a vector; if the determinant is 1 we have a rotation, a translation, or the identity, and if it is -1 we have a glide reflection or a reflection. A "random" isometry, like taking a sheet of paper from a table and randomly laying it back, "almost surely" is a rotation or a glide reflection (they have three degrees of freedom). This applies regardless of the details of the probability distribution, as long as θ and the direction of the added vector are independent and uniformly distributed and the length of the added vector has a continuous distribution. A pure translation and a pure
reflection are special cases with only two degrees of freedom, while the identity is even more special, with no degrees of freedom.
Isometries as reflection group Reflections, or mirror isometries, can be combined to produce any isometry. Thus isometries are an example of a reflection group.
Mirror combinations In the Euclidean plane, we have the following possibilities.
[d ] Identity Two reflections in the same mirror restore each point to its original position. All points are left fixed. Any pair of identical mirrors has the same effect. [db] Reflection As Alice found through the looking-glass, a single mirror causes left and right hands to switch. (In formal terms, topological orientation is reversed.) Points on the mirror are left fixed. Each mirror has a unique effect. [dp] Rotation Two distinct intersecting mirrors have a single point in common, which remains fixed. All other points rotate around it by twice the angle between the mirrors.
Any two mirrors with the same fixed point and same angle give the same rotation, so long as they are used in the correct order. [dd] Translation Two distinct mirrors that do not intersect must be parallel. Every point moves the same amount, twice the distance between the mirrors, and in the same direction. No points are left fixed. Any two mirrors with the same parallel direction and the same distance apart give the same translation, so long as they are used in the correct order. [dq] Glide reflection Three mirrors. If they are all parallel, the effect is the same as a single mirror (slide a pair to cancel the third). Otherwise we can find an equivalent arrangement where two are parallel and the third is perpendicular to them. The effect is a reflection combined with a translation parallel to the mirror. No points are left fixed.
Three mirrors suffice Adding more mirrors does not add more possibilities (in the plane), because they can always be rearranged to cause cancellation. Proof. An isometry is completely determined by its effect on three independent (not collinear) points. So suppose p1, p2, p3 map to q1, q2, q3; we can generate a sequence of mirrors to achieve this as follows. If p1 and q1 are distinct, choose their perpendicular bisector as mirror. Now p1 maps to q1; and we will pass all further mirrors through q1, leaving it fixed. Call the images of p2 and p3 under this reflection p2′ and p3′. If q2 is distinct from p2′, bisect the angle at q1 with a new mirror. With p1 and p2 now in place, p3 is at p3′′; and if it is not in place, a final mirror through q1 and q2 will flip it to q3. Thus at most three reflections suffice to reproduce any plane isometry. ∎
Recognition We can recognize which of these isometries we have according to whether it preserves hands or swaps them, and whether it has at least one fixed point or not, as shown in the following table (omitting the identity). Preserves hands? Yes No Reflection Yes Rotation Fixed point? No Translation Glide reflection
Group structure Isometries requiring an odd number of mirrors — reflection and glide reflection — always reverse left and right. The even isometries — identity, rotation, and translation —
never do; they correspond to rigid motions, and form a normal subgroup of the full Euclidean group of isometries. Neither the full group nor the even subgroup are abelian; for example, reversing the order of composition of two parallel mirrors reverses the direction of the translation they produce. Proof. The identity is an isometry; nothing changes, so distance cannot change. And if one isometry cannot change distance, neither can two (or three, or more) in succession; thus the composition of two isometries is again an isometry, and the set of isometries is closed under composition. The identity isometry is also an identity for composition, and composition is associative; therefore isometries satisfy the axioms for a semigroup. For a group, we must also have an inverse for every element. To cancel a reflection, we merely compose it with itself. (Reflections are involutions.) And since every isometry can be expressed as a sequence of reflections, its inverse can be expressed as that sequence reversed. Notice that the cancellation of a pair of identical reflections reduces the number of reflections by an even number, preserving the parity of the sequence; also notice that the identity has even parity. Therefore all isometries form a group, and even isometries a subgroup. (Odd isometries do not include the identity, so are not a subgroup.) This subgroup is a normal subgroup, because sandwiching an even isometry between two odd ones yields an even isometry. ∎ Since the even subgroup is normal, it is the kernel of a homomorphism to a quotient group, where the quotient is isomorphic to a group consisting of a reflection and the identity. However the full group is not a direct product, but only a semidirect product, of the even subgroup and the quotient group.
Composition Composition of isometries mixes kinds in assorted ways. We can think of the identity as either two mirrors or none; either way, it has no effect in composition. And two reflections give either a translation or a rotation, or the identity (which is both, in a trivial way). Reflection composed with either of these could cancel down to a single reflection; otherwise it gives the only available three-mirror isometry, a glide reflection. A pair of translations always reduces to a single translation; so the challenging cases involve rotations. We know a rotation composed with either a rotation or a translation must produce an even isometry. Composition with translation produces another rotation (by the same amount, with shifted fixed point), but composition with rotation can yield either translation or rotation. It is often said that composition of two rotations produces a rotation, and Euler proved a theorem to that effect in 3D; however, this is only true for rotations sharing a fixed point.
Translation, rotation, and orthogonal subgroups We thus have two new kinds of isometry subgroups: all translations, and rotations sharing a fixed point. Both are subgroups of the even subgroup, within which translations are normal. Because translations are a normal subgroup, we can factor them out leaving the subgroup of isometries with a fixed point, the orthogonal group.
Proof. If two rotations share a fixed point, then we can swivel the mirror pair of the second rotation to cancel the inner mirrors of the sequence of four (two and two), leaving just the outer pair. Thus the composition of two rotations with a common fixed point produces a rotation by the sum of the angles about the same fixed point. If two translations are parallel, we can slide the mirror pair of the second translation to cancel the inner mirror of the sequence of four, much as in the rotation case. Thus the composition of two parallel translations produces a translation by the sum of the distances in the same direction. Now suppose the translations are not parallel, and that the mirror sequence is A1, A2 (the first translation) followed by B1, B2 (the second). Then A2 and B1 must cross, say at c; and, reassociating, we are free to pivot this inner pair around c. If we pivot 90°, an interesting thing happens: now A1 and A2′ intersect at a 90° angle, say at p, and so do B1′ and B2, say at q. Again reassociating, we pivot the first pair around p to make B2″ pass through q, and pivot the second pair around q to make A1″ pass through p. The inner mirrors now coincide and cancel, and the outer mirrors are left parallel. Thus the composition of two non-parallel translations also produces a translation. Also, the three pivot points form a triangle whose edges give the head-to-tail rule of vector addition: 2(p c) + 2(c q) = 2(p q). ∎
Nested group construction The subgroup structure suggests another way to compose an arbitrary isometry: Pick a fixed point, and a mirror through it.
1. If the isometry is odd, use the mirror; otherwise do not. 2. If necessary, rotate around the fixed point. 3. If necessary, translate. This works because translations are a normal subgroup of the full group of isometries, with quotient the orthogonal group; and rotations about a fixed point are a normal subgroup of the orthogonal group, with quotient a single reflection.
Discrete subgroups
The subgroups discussed so far are not only infinite, they are also continuous (Lie groups). Any subgroup containing at least one non-zero translation must be infinite, but subgroups of the orthogonal group can be finite. For example, the symmetries of a regular pentagon consist of rotations by integer multiples of 72° (360° / 5), along with reflections in the five mirrors which perpendicularly bisect the edges. This is a group, D5, with 10
elements. It has a subgroup, C5, of half the size, omitting the reflections. These two groups are members of two families, Dn and Cn, for any n > 1. Together, these families constitute the rosette groups. Translations do not fold back on themselves, but we can take integer multiples of any finite translation, or sums of multiples of two such independent translations, as a subgroup. These generate the lattice of a periodic tiling of the plane. We can also combine these two kinds of discrete groups — the discrete rotations and reflections around a fixed point and the discrete translations — to generate the frieze groups and wallpaper groups. Curiously, only a few of the fixed-point groups are found to be compatible with discrete translations. In fact, lattice compatibility imposes such a severe restriction that, up to isomorphism, we have only 7 distinct frieze groups and 17 distinct wallpaper groups. For example, the pentagon symmetries, D5, are incompatible with a discrete lattice of translations. (Each higher dimension also has only a finite number of such crystallographic groups, but the number grows rapidly; for example, 3D has 320 groups and 4D has 4783.)
Isometries in the complex plane In terms of complex numbers, the isometries of the plane are addition of a complex constant (translation), multiplication by a complex constant with modulus 1 (rotation), complex conjugation (reflection in the real axis), and combinations.
Chapter 4
Euler Angles
The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body. To describe such an orientation in 3-dimensional Euclidean space three parameters are required. Euler angles also represent three composed rotations that move a reference frame to a given referred frame. This is equivalent to saying that any orientation can be achieved by composing three elemental rotations (rotations around a single axis of a basis), and also equivalent to saying that any rotation matrix can be decomposed as a product of three elemental rotation matrices. Without considering the possibilities of different signs for the angles or moving the reference frame, there are twelve different conventions divided in two groups. One of them is called "proper" Euler angles and the other Tait–Bryan angles. Sometimes "Euler angles" is used for all of them.
Definition
Euler angles - The xyz (fixed) system is shown in blue, the XYZ (rotated) system is shown in red. The line of nodes, labeled N, is shown in green. Euler angles are a means of representing the spatial orientation of any frame (coordinate system) as a composition of rotations from a frame of reference (coordinate system). In the following the fixed system is denoted in lower case (x,y,z) and the rotated system is denoted in upper case letters (X,Y,Z). The definition is Static. Given a reference frame and the one whose orientation we want to describe, first we define the line of nodes (N) as the intersection of the xy and the XY coordinate planes (in other words, line of nodes is the line perpendicular to both z and Z axis). Then we define its Euler angles as:
α (or ψ) is the angle between the x-axis and the line of nodes. β (or θ) is the angle between the z-axis and the Z-axis. γ (or φ) is the angle between the line of nodes and the X-axis.
Euler angles between two frames are defined only if both frames have the same handedness. Euler angles are just one of the several ways of specifying the relative orientation of two such coordinate systems. Different authors may use different sets of angles to describe these orientations, or different names for the same angles, leading to different conventions. Therefore any discussion employing Euler angles should always be preceded by their definition.
Angles signs and ranges Normally, angles are defined in such a way that they are positive when they rotate counter-clock-wise (how they rotate depends on which side of the rotation plane we observe them from. The positive side will be the one of the positive axis of rotation) About the ranges:
α and γ range are defined modulo 2π radians. A valid range could be (-π, π]. β range covers π radians (but can't be said to be modulo π). For example could be [0, π] or [-π/2, π/2].
The angles α, β and γ are uniquely determined except for the singular case that the xy and the XY planes are identical, the z axis and the Z axis having the same or opposite directions. Indeed, if the z-axis and the Z-axis are the same, β = 0 and only (α+γ) is uniquely defined (not the individual values), and, similarly, if the z-axis and the Z-axis are opposite, β = π and only (α-γ) is uniquely defined (not the individual values). These ambiguities are known as gimbal lock in applications.
Conventions
Tait-Bryan angles. ZYX (intrinsic composition) convention There are two main types of conventions called "proper" Euler angles and Tait–Bryan angles, after Peter Guthrie Tait and George H. Bryan, also known as Nautical or Cardan angles, after Cardan. Their static difference is the definition for the line of nodes. In the first case two homologous planes (planes overlapping when the angles are zero) are used. In the second one, they are replaced by non-homologous planes (perpendicular when angles are zero). Nevertheless, it is unusual to use static conventions when speaking about Euler angles. The intrinsic rotations equivalence or the extrinsic rotations equivalence are used instead. According with these equivalences, proper Euler angles are equivalent to three combined
rotations repeating exactly one axis. Tait-Bryan angles are equivalent to three composed rotations in different axes. No information is lost when using the rotation equivalence because the static parameters can be calculated from the name of the convention. For example, given the convention XY’-Z’’, the first rotation is perpendicular to "x" and the last one to "Z". Therefore the planes are the yz and the XY, and the line of nodes is the intersection of these two.
Proper Euler angles There are six possibilities of choosing the proper Euler angles. Using the static definition they correspond to the three possible homogeneous combinations of planes (XY, XZ and YZ) with the two possible options to measure the angles from (given the line of nodes by the XY planes for example, it can be taken X-N or Y-N as first angle). Hence the six possibilities. There is a intrinsic rotations equivalence which is normally used to name the possible conventions of Euler Angles. If we are told that some angles are given using the convention Z-X’-Z’’, this means that they are equivalent to three concatenated intrinsic rotations around some moving axes Z, X’ and Z’’ in that order. This composition is noncommutative. It has to be applied in such a way that in the beginning one of the intrinsic axis moves together with the line of nodes. The above diagram convention is usually named this way. Nevertheless, sometimes the extrinsic rotations equivalence could be used. If this is the case, the given angles are backwards, meaning that the first angle is the intrinsic rotation and the last one the precession. The name of the convention would be indistinguishable from the previous one, even if the angles' order is the opposite, being something like z-xz (here lowercase is used to remark extrinsic composition). To specify that the given order means intrinsic composition, sometimes a similar notation is used, but stating explicitly which rotation axis are different for each step, as in Z-X’Z’’. Using this notation, Z-X-Z would mean extrinsic composition.
Tait-Bryan angles
Tait-Bryan angles statically defined. Z-X’-Y’’ convention There are also six Tait-Bryan combinations. They come from the two possible nonhomogeneous planes that exist when one is given (given XY, there are two nonhomogeneous, XZ and YZ). The three possible planes at the reference frame multiplied by the two options for each one yield the six possible conventions. There are six possible combinations of this kind, and all of them behave in an identical way. Using the intrinsic rotations equivalence, Tait-Bryan angles correspond with the three rotations with a different axis. Z-X-Y for example. There are also six possibilities of this kind. The enclosed image shows the ZXY convention. The other five proper conventions are obtained by selecting different axes of rotation. These three angles are normally called Heading, Elevation and Bank, or Yaw, Pitch and Roll. The second terms have to be used carefully because they are also the names for the three aircraft principal axes. For Tait-Bryan angles, also intrinsic and extrinsic conventions can be used, giving therefore two meanings for every convention name. For example, X-Y-Z, using intrinsic convention, means that a X-rotation is performed, composing intrinsic rotations Y and Z later, but using extrinsic convention means that after the X rotation, extrinsic rotations Y and Z are performed. The meaning is different in both cases.
Geometric derivation
Projections of Z vector.
Projections of Y vector. The fastest way to get the Euler Angles of a given frame is to write the three given vectors as columns of a matrix and compare it with the expression of the theoretical matrix. Hence the three Euler Angles can be calculated. Nevertheless, the same result can be reached avoiding matrix calculus, which is more geometrical. Assuming a frame with unitary vectors (X, Y, Z) as in the main diagram, it can be seen that:
and since sin2x = 1 − cos2x
as Z2 is a double projection of an unitary vector:
There is a similar construction for Y3, projecting it first over the plane defined by the axis Z and the line of nodes. As the angle between the planes is 90 − β and cos(90 − β) = sin(β), this leads to:
and finally, using the cosine inverse function arc cos:
It is interesting to note that the cosine inverse function yields two possible values for the argument. In this geometrical description only one of the solutions is valid. When Euler angles are defined as a sequence of rotations all the solutions can be valid, but there will be only one inside the angles ranges. This is because the sequence of rotations to reach the target frame is not unique if the ranges are not previously defined.
Relationship with physical motions Euler angles can be considered as the result of three composed rotations and these conventions are named according to this composition. As intrinsic rotations produce the same result than extrinsic rotations inverted there are two possible names for any static convention. For example the intrinsic ZXY and the extrinsic yxz have the same static parameters.
Euler angles as composition of intrinsic rotations
Any target frame can be reached using a specific sequence of intrinsic rotations (mobile frame rotations), whose values are exactly the Euler Angles of the target frame. Using ZX'-Z" convention in this example. Starting with an initial set of mobile axes, say XYZ overlapping the reference axes xyz, a composition of three intrinsic rotations (rotations only about the mobile frame axes, assuming active composition) can be used to reach any target frame with an origin coincident with that of XYZ from the reference frame. The value of the rotations are the Euler Angles. The position of the mobile axes can be reached using three rotations with angles α, β, γ in three ways equivalent to the former definition, as follows:
The XYZ system rotates while the xyz is fixed. Starting with the XYZ system overlapping the reference frame xyz, the same rotations as before can be performed using only rotations around the mobile axes XYZ.
Rotate the XYZ-system about the Z-axis by α. The X-axis now lies on the line of nodes. Rotate the XYZ-system again about the now rotated X-axis by β. The Z-axis is now in its final orientation, and the x-axis remains on the line of nodes. Rotate the XYZ-system a third time about the new Z-axis by γ.
Any convention for proper Euler angles is equivalent to three such rotations that one axis is repeated (ZXZ for example). Tait-Bryan angles are also equivalent to three composed rotations, but in this case, all three rotations are around different axes (ZXY). Usually conventions are named according with this equivalence.
Euler angles as composition of extrinsic rotations
A rotation represented by Euler angles with (φ,θ,ψ)=(−60°, 30°, 45°) using the 3-1-3 (ZX-Z) co-moving axes rotations
The same rotation alternatively expressed by (φ,θ,ψ)=(45°, 30°, −60°) using the 3-1-3 (ZX-Z) fixed axes rotations Also composition of extrinsic rotations (rotations about the reference frame axes, assuming active composition) can be used to reach any target frame. Let xyz system be
fixed while the XYZ system rotates. Start with the rotating XYZ system coinciding with the fixed xyz system.
Rotate the XYZ-system about the z-axis by γ. The X-axis is now at angle γ with respect to the x-axis. Rotate the XYZ-system again about the x-axis by β. The Z-axis is now at angle β with respect to the z-axis. Rotate the XYZ-system a third time about the z-axis by α. The first and third axes are identical.
This can be shown to be equivalent to the previous statement: Let us call (e), (f), (g), (h), the successive frames deduced from the initial (e) reference frame by the successive intrinsic rotations described above. We call u, v, w, t, the successive vectors obtained with that rotation. We note (x)e the column matrix representing a vector x in the frame (e). If necessary we add also a lower index to any matrix we wish to operate in a specific frame. We call (Zα), (Xβ), (Zγ) the successive rotations of our example. Thus we can write when describing the intrinsic operations :
When describing the intrinsic rotations in the (e) reference frame we must of course transform the matrices used to represent the rotations. Then by the rules of matrix algebra we get :
The relation (5) can then of course be interpreted in extrinsic manner as a succession of rotations around the (e) axes. Again, proper Euler angles repeat an axis and Tait-Bryan angles do not. As before, this kind of composition is non-commutative.
Euler rotations
Euler rotations of the Earth. Intrinsic (green), Precession (blue) and Nutation (red) Euler rotations are defined as the movement obtained by changing one of the Euler angles while leaving the other two constant. Euler rotations are never expressed in terms of the external frame, or in terms of the co-moving rotated body frame, but in a mixture. They constitute a mixed axes of rotation system, where the first angle moves the line of nodes around the external axis z, the second rotates around the line of nodes and the third one is an intrinsic rotation around an axis fixed in the body that moves. These rotations are called Precession, Nutation, and intrinsic rotation. While they are rotations when they are applied over individual frames, only precession is valid as a
rotation operator, and only precession can be expressed in general as a matrix in the basis of the space.
Gimbal analogy
Three axes z-x-z-gimbal showing Euler angles. External frame and external axis 'x' are not shown. Axes 'Y' are perpendicular to each gimbal ring, together with a simple diagram showing how the axes 'Y' of intermediate frames are located in the main diagram. If we suppose a set of frames, able to move each with respect to the former according to just one angle, like a gimbal, there will be one initial, one final and two in the middle, which are called intermediate frames. The two in the middle work as two gimbal rings that allow the last frame to reach any orientation in space.
Intermediate frames The gimbal rings indicate some intermediate frames. They can be defined statically too. Taking some vectors i, j and k over the axes x, y and z, and vectors I, J, K over X, Y and
Z, and a vector N over the line of nodes, some intermediate frames can be defined using the vector cross product, as following:
origin: [i,j,k] (where k = i × j) first: [N,k × N,k] second: [N,K × N,K] final: [I,J,K]
These intermediate frames are equivalent to those of the gimbal. They are such that they differ from the previous one in just a single elemental rotation. This proves that:
Any target frame can be reached from the reference frame just composing three rotations. The values of these three rotations are exactly the Euler angles of the target frame.
Relationship to other representations Euler angles are one way to represent orientations. There are others, and it is possible to change to and from other conventions.
Matrix orientation Using the equivalence between Euler angles and rotation composition, it is possible to change to and from matrix convention. Fixed (world) axes and column vectors, with intrinsic composition (composition of rotations about body axes) of active rotations and the right-handed rule for the positive sign of the angles are assumed. This means for example that a convention named (YXZ) is the result of performing first an intrinsic Y rotation, followed by an X and a Z rotations, in the moving axes. Its matrix is the product of Rot(Y,θ1) Rot(X,θ2) Rot(Z,θ3) like this:
.
.
Subindexes refer to the order in which the angles are applied. Trigonometric notation has been simplified. For example, c1 means cos(θ1) and s2 means sin(θ2). As we assumed intrinsic and active compositions, θ1 is the external angle of the static definition (angle between fixed axis x and line of nodes) and θ3 the internal angle (from the line of nodes to rotated axis X). The following table can be used both ways, to obtain an orientation matrix from Euler angles and to obtain Euler angles from the matrix. The possible combinations of rotations equivalent to Euler angles are shown here.
XZX
XZY
XYX
XYZ
YXY
YXZ
YZY
YZX
ZYZ
ZYX
ZXZ
ZXY
Quaternions Unit quaternions, also known as Euler-Rodrigues parameters, provide another mechanism for representing 3D rotations. This is equivalent to the special unitary group description. Expressing rotations in 3D as unit quaternions instead of matrices has some advantages:
Concatenating rotations is computationally faster and numerically more stable. Extracting the angle and axis of rotation is simpler. Interpolation is more straightforward.
Geometric algebra Other representation comes from the Geometric algebra(GA). GA is a higher level abstraction, in which the quaternions are an even subalgebra. The principal tool in GA is the rotor rotation axis (unitary vector) and
where
angle of rotation,
pseudoscalar (trivector in
Properties The Euler angles form a chart on all of SO(3), the special orthogonal group of rotations in 3D space. The chart is smooth except for a polar coordinate style singularity along β=0. The space of rotations is called in general "The Hypersphere of rotations", though this is a misnomer: the group Spin(3) is isometric to the hypersphere S3, but the rotation space
SO(3) is instead isometric to the real projective space RP3 which is a 2-fold quotient space of the hypersphere. This 2-to-1 ambiguity is the mathematical origin of spin in physics. A similar three angle decomposition applies to SU(2), the special unitary group of rotations in complex 2D space, with the difference that β ranges from 0 to 2π. These are also called Euler angles. The Haar measure for Euler angles has the simple form sin(β)dαdβdγ, usually normalized by a factor of 1/8π². For example, to generate uniformly randomized orientations, let α and γ be uniform from 0 to 2π, let z be uniform from −1 to 1, and let β = arccos(z).
Higher dimensions It is possible to define parameters analogous to the Euler angles in dimensions higher than three. The number of degrees of freedom of a rotation matrix is always less than the dimension of the matrix squared. That is, the elements of a rotation matrix are not all completely independent. For example, the rotation matrix in dimension 2 has only one degree of freedom, since all four of its elements depend on a single angle of rotation. A rotation matrix in dimension 3 (which has nine elements) has three degrees of freedom, corresponding to each independent rotation, for example by its three Euler angles or a magnitude one (unit) quaternion. In SO(4) the rotation matrix is defined by two quaternions, and is therefore 6-parametric (three degrees of freedom for every quaternion). The 4x4 rotation matrices have therefore 6 out of 16 independent components. Any set of 6 parameters that define the rotation matrix could be considered an extension of Euler angles to dimension 4. In general, the number of euler angles in dimension D is quadratic in D; since any one rotation consists of choosing two dimensions to rotate between, the total number of rotations available in dimension D is D=2,3,4 yields Nrot = 1,3,6.
, which for
Applications
A gyroscope keeps its rotation axis constant. Therefore, angles measured in this frame are equivalent to angles measured in the lab frame
Vehicles and moving frames Their main advantage over other orientation descriptions is that they are directly measurable from a gimbal mounted in a vehicle. As gyroscopes keep their rotation axis constant, angles measured in a gyro frame are equivalent to angles measured in the lab frame. Therefore gyros are used to know the actual orientation of moving spacecrafts, and Euler angles are directly measurable. Intrinsic rotation angle cannot be read from a single gimbal, so there has to be more than one gimbal in a spacecraft. Normally there are at least three for redundancy. There is also a relation to the well-known gimbal lock problem of Mechanical Engineering.
Heading, elevation and bank for an aircraft with axes DIN 9300 The most popular application is to describe aircraft attitudes, normally using a Tait-Bryan convention so that zero degrees elevation represents the horizontal attitude. Tait-Bryan angles represent the orientation of the aircraft respect a reference axis system (world frame) with three angles which in the context of an aircraft are normally called Heading, Elevation and Bank. When dealing with vehicles, different axes conventions are possible. When studying rigid bodies in general, one calls the xyz system space coordinates, and the XYZ system body coordinates. The space coordinates are treated as unmoving, while the body coordinates are considered embedded in the moving body. Calculations involving acceleration, angular acceleration, angular velocity, angular momentum, and kinetic energy are often easiest in body coordinates, because then the moment of inertia
tensor does not change in time. If one also diagonalizes the rigid body's moment of inertia tensor (with nine components, six of which are independent), then one has a set of coordinates (called the principal axes) in which the moment of inertia tensor has only three components. The angular velocity of a rigid body takes a simple form using Euler angles in the moving frame. Also the Euler's rigid body equations are simpler because the inertia tensor is constant in that frame.
Industrial robot operating in a foundry.
Others Euler angles, normally in the Tait-Bryan convention, are also used in robotics for speaking about the degrees of freedom of a wrist. They are also used in Electronic stability control in a similar way. Gun fire control systems require corrections to gun-order angles (bearing and elevation) to compensate for deck tilt (pitch and roll). In traditional systems, a stabilizing gyroscope with a vertical spin axis corrects for deck tilt, and stabilizes the optical sights and radar antenna. However, gun barrels point in a direction different from the line of sight to the target, to anticipate target movement and fall of the projectile due to gravity, among other factors. Gun mounts roll and pitch with the deck plane, but also require stabilization. Gun
orders include angles computed from the vertical gyro data, and those computations involve Euler angles. Euler angles are also used extensively in the quantum mechanics of angular momentum. In quantum mechanics, explicit descriptions of the representations of SO(3) are very important for calculations, and almost all the work has been done using Euler angles. In the early history of quantum mechanics, when physicists and chemists had a sharply negative reaction towards abstract group theoretic methods (called the Gruppenpest), reliance on Euler angles was also essential for basic theoretical work. In materials science, crystallographic texture (or preferred orientation) can be described using Euler angles. In texture analysis, the Euler angles provide the necessary mathematical depiction of the orientation of individual crystallites within a polycrystalline material, allowing for the quantitative description of the macroscopic material.
Chapter 5
Euler's Rotation Theorem
Euler pole. In geometry, Euler's rotation theorem states that, in three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about a fixed axis that runs through the fixed point. It also means that the composition of two rotations is also a rotation. Therefore the set of rotations has a structure known as a rotation group. The theorem is named after Leonhard Euler, who proved it in 1775 by an elementary geometric argument. The axis of rotation is known as an Euler pole. The extension of the theorem to kinematics yields the concept of Instant axis of rotation. In linear algebra terms, the theorem states that, in 3D space, any two Cartesian coordinate systems with a common origin are related by a rotation about some fixed axis. This also
means that the product of two rotation matrices is again a rotation matrix and that a nonidentity rotation matrix has only one real eigenvalue which is equal to unity. The eigenvector corresponding to this eigenvalue is the axis of rotation connecting the two systems.
Euler's theorem (1776)
Construction showing the theorem for a rotated sphere whose Euler angles are [ψ,θ,φ]. Frame in blue is attached to the fixed sphere. Frame in red is fixed to the rotated sphere. The line of nodes N determines the point A of the theorem. The arcs Aa and Aα must be equal Euler states the theorem as follows:
Theorema. Quomodocunque sphaera circa centrum suum conuertatur, semper assignari potest diameter, cuius directio in situ translato conueniat cum situ initiali. or (in free translation): To prove this, Euler considers a great circle on the sphere and the great circle to which it is transported by the movement. These two circles intersect in two (opposite) points of which one, say A, is chosen. This point lies on the initial circle and thus is transported to a point a on the second circle. On the other hand, A lies also on the translated circle, and thus corresponds to a point α on the initial circle. Notice that the arc aA must be equal to the arc Aα. Now Euler needs to construct point O in the surface of the sphere that is in the same position in reference to the arcs aA and αA. If such a point exists then:
it is necessary, that the distances OA and Oa are equal to each other; the arcs Oa and OA must be equal, it is necessary that the angles OaA and OAα are equal.
Now Euler points out that the angles OAa and OaA must also be equal, since Oa and OA have the same length. Thus OAa and OAα are equal, proving O lies on the angle bisecting αAa. To provide a complete construction for O, we need only note that the arc aO may also be constructed such that AaO is the same as αAO. This completes the proof. Euler provides a further construction that might be easier in practice. He proposes two planes:
the symmetry plane of the angle αAa (which passes through the centre C of the sphere), and the symmetry plane of the arc Aa (which also passes through C).
Proposition. These two planes intersect in a diameter. This diameter is the one we are looking for. Proof. Let's call O to any of the endpoints (there are two) of this diameter over the sphere surface. Since αA is mapped on Aa and the triangles have the same angles, it follows that the triangle OαA is transported onto the triangle OAa. Therefore the point O has to remain fixed under the movement. Corollaries This also shows that the rotation of the sphere can be seen as two consecutive reflections about the two planes described above. Points in a mirror plane are invariant under reflection, and hence the points on their intersection (a line: the axis of rotation) are invariant under both the reflections, and hence under the rotation.
Given that for a rigid body any movement that leaves an axis invariant is a rotation, this also proves that any arbitrary composition of rotations is equivalent to a single rotation around a new axis.
Matrix proof An algebraic proof starts from the fact that a rotation is a linear map in one-to-one correspondence with a 3×3 rotation matrix R, i.e., a matrix for which
where E is the 3×3 identity matrix and superscript T indicates the transposed matrix. Clearly a rotation matrix has determinant ±1, for invoking some properties of determinants, one can prove
The matrix with positive determinant is a proper rotation and with a negative determinant an improper rotation (is equal to a reflection times a proper rotation). It will now be shown that a rotation matrix R has at least one invariant vector n, i.e., R n = n. Note that this is equivalent to stating that the vector n is an eigenvector of the matrix R with eigenvalue λ = 1. A proper rotation matrix R has at least one unit eigenvalue. Using the two relations:
we find
From this follows that λ = 1 is a root (solution) of the secular equation, that is,
In other words, the matrix R − E is singular and has a non-zero kernel, that is, there is at least one non-zero vector, say n, for which
The line μn for real μ is invariant under R, i.e., μn is a rotation axis. This proves Euler's theorem.
Equivalence of an orthogonal matrix to a rotation matrix Two matrices (representing linear maps) are said to be equivalent if there is change of basis that makes one equal to the other. A proper orthogonal matrix is always equivalent (in this sense) to either the following matrix or to its vertical reflection:
Then, any orthogonal matrix is either a rotation or a reflection. It can be seen that any orthogonal matrix has only one real eigenvalue, which is +1 or -1. When it is +1 the matrix is a rotation. When -1, the matrix is a reflection. If R has more than one invariant vector then φ = 0 and R = E. Any vector is an invariant vector of E.
Excursion into matrix theory In order to prove the previous equation some facts from matrix theory must be recalled. An m×m matrix A has m orthogonal eigenvectors if and only if A is normal, that is, if A†A = AA†. This result is equivalent to stating that normal matrices can be brought to diagonal form by a unitary similarity transformation:
and U is unitary, that is,
The eigenvalues α1, ..., αm are roots of the secular equation. If the matrix A happens to be unitary (and note that unitary matrices are normal), then
and it follows that the eigenvalues of a unitary matrix are on the unit circle in the complex plane:
Also an orthogonal (real unitary) matrix has eigenvalues on the unit circle in the complex plane. Moreover, since its secular equation (an mth order polynomial in λ) has real
coefficients, it follows that its roots appear in complex conjugate pairs, that is, if α is a root then so is α∗. After recollection of these general facts from matrix theory, we return to the rotation matrix R. It follows from its realness and orthogonality that
with the third column of the 3×3 matrix U equal to the invariant vector n. Writing u1 and u2 for the first two columns of U, this equation gives
If u1 has eigenvalue 1, then φ= 0 and u2 has also eigenvalue 1, which implies that in that case R = E. Finally, the matrix equation is transformed by means of a unitary matrix,
which gives
The columns of U′ are orthonormal. The third column is still n, the other two columns are perpendicular to n. This result implies that any orthogonal matrix R is equivalent to a rotation over an angle φ around an axis n.
Equivalence classes It is of interest to remark that the trace (sum of diagonal elements) of the real rotation matrix given above is 1 + 2cosφ. Since a trace is invariant under an orthogonal matrix transformation:
it follows that all matrices that are equivalent to R by an orthogonal matrix transformation have the same trace. The matrix transformation is clearly an equivalence relation, that is, all equivalent matrices form an equivalence class. In fact, all proper rotation 3×3 rotation matrices form a group, usually denoted by SO(3) (the special orthogonal group in 3 dimensions) and all matrices with the same trace form an equivalence class in this group. Elements of such an equivalence class share their rotation angle, but all rotations are around different axes. If n is a eigenvector of R with eigenvalue 1, then An is an eigenvector of ARAT, also with eigenvalue 1. Unless A = E, n and An are different.
Applications Generators of rotations Suppose we specify an axis of rotation by a unit vector [x, y, z] , and suppose we have an infinitely small rotation of angle Δθ about that vector. Expanding the rotation matrix as an infinite addition, and taking the first order approach, the rotation matrix ΔR is represented as:
A finite rotation through angle θ about this axis may be seen as a succession of small rotations about the same axis. Approximating Δθ as θ/N where N is a large number, a rotation of θ about the axis may be represented as:
It can be seen that Euler's theorem essentially states that all rotations may be represented in this form. The product is the "generator" of the particular rotation, being the vector (x,y,z) associated with the matrix A. This shows that the rotation matrix and the axisangle format are related by the exponential function. Analysis is often easier in terms of these generators, rather than the full rotation matrix. Analysis in terms of the generators is known as the Lie algebra of the rotation group.
Quaternions It follows from Euler's theorem that the relative orientation of any pair of coordinate systems may be specified by a set of three independent numbers. Sometimes a redundant fourth number is added to simplify operations with quaternion algebra. Three of these numbers are the direction cosines that orient the eigenvector. The fourth is the angle
about the eigenvector that separates the two sets of coordinates. Such a set of four numbers is called a quaternion. While the quaternion as described above, does not involve complex numbers, if quaternions are used to describe two successive rotations, they must be combined using the non-commutative quaternion algebra derived by William Rowan Hamilton through the use of imaginary numbers. Rotation calculation via quaternions has come to replace the use of direction cosines in aerospace applications through their reduction of the required calculations, and their ability to minimize round-off errors. Also, in computer graphics the ability to perform spherical interpolation between quaternions with relative ease is of value.
Generalizations In higher dimensions, any rigid motion that preserve a point in dimension 2n or 2n+1 is a composition of at most n rotations in orthogonal planes of rotation, though these planes need not be uniquely determined, and a rigid motion may fix multiple axes.
A screw motion. A rigid motion in 3 dimensions that does not necessarily fix a point is a "screw motion". This is because a composition of a rotation with a translation perpendicular to the axis is a rotation about a parallel axis, while composition with a translation parallel to the axis yields a screw motion. This gives rise to screw theory.
Chapter 6
Rotation Representation (Mathematics)
In geometry a rotation representation expresses a rotation as a mathematical transformation. In physics, this concept extends to classical mechanics where rotational (or angular) kinematics is the science of describing with numbers the purely rotational motion of an object. According to Euler's rotation theorem the general displacement of a rigid body (or threedimensional coordinate system) with one point fixed is described by a rotation about some axis. This allows the use of rotations to express orientations as a single rotation from a reference placement in space of the rigid body (or coordinate system). Furthermore, such a rotation may be uniquely described by a minimum of three parameters. However, for various reasons, there are several ways to represent it. Many of these representations use more than the necessary minimum of three parameters, although each of them still has only three degrees of freedom. An example where rotation representation is used is in computer vision, where an automated observer needs to track a target. Let's consider a rigid body, with an orthogonal right-handed triad , , and of unit vectors fixed to its body (representing the three axes of the object's coordinate system). The basic problem is to specify the orientation of this triad, and hence the rigid body, in terms of the reference coordinate system (in our case the observer's coordinate system).
Rotation matrix The above mentioned triad of unit vectors is also called a basis. Specifying the coordinates (scalar components) of this basis in its current (rotated) position, in terms of the reference (non-rotated) coordinate axes, will completely describe the rotation. The three unit vectors , and which form the rotated basis each consist of 3 coordinates, yielding a total of 9 parameters. These parameters can be written as the elements of a matrix , called a rotation matrix. Typically, the coordinates of each of these vectors are arranged along a column of the matrix.
The elements of the rotation matrix are not all independent - as Euler's rotation theorem dictates, the rotation matrix has only three degrees of freedom. The rotation matrix has the following properties:
A is a real, orthogonal matrix, hence each of its rows or colums represents a unit vector. The eigenvalues of A are where i is the standard imaginary unit with the property i 2 = −1
The determinant of A is +1, equivalent to the product of its eigenvalues. The trace of A is 1 + 2cos(θ), equivalent to the sum of its eigenvalues.
The angle θ which appears in the eigenvalue expression corresponds to the angle of the Euler axis and angle representation. The eigenvector corresponding with the eigenvalue of 1 is the accompanying Euler axis, since the axis is the only (nonzero) vector which remains unchanged by left-multiplying (rotating) it with the rotation matrix. The above properties are equivalent to:
which is another way of stating that form a 3D orthonormal basis. Note that the statements above constitute a total of 6 conditions (the cross product contains 3), leaving the rotation matrix with just 3 degrees of freedom as required. and are easily combined as Two successive rotations represented by matrices follows: (Note the order, since the vector being rotated is multiplied from the right). The ease by which vectors can be rotated using a rotation matrix, as well as the ease of combining successive rotations, make the rotation matrix a very useful and popular way to represent rotations, even though it is less concise than other representations.
Euler axis and angle (rotation vector)
A visualization of a rotation represented by an Euler axis and angle. From Euler's rotation theorem we know that any rotation can be expressed as a single rotation about some axis. The axis is the unit vector (unique except for sign) which remains unchanged by the rotation. The magnitude of the angle is also unique, with its sign being determined by the sign of the rotation axis. The axis can be represented as a three-dimensional unit vector the angle by a scalar .
, and
Since the axis is normalized, it has only two degrees of freedom. The angle adds the third degree of freedom to this rotation representation. One may wish to express rotation as a rotation vector, a non-normalized threedimensional vector the direction of which specifies the axis, and the length of which is :
The rotation vector is in some contexts useful, as it represents a three-dimensional rotation with only three scalar values (its scalar components), representing the three degrees of freedom. This is also true for representations based on sequences of three Euler angles. If the rotation angle θ is zero, the axis is not uniquely defined. Combining two successive rotations, each represented by an Euler axis and angle, is not straightforward, and in fact does not satisfy the law of vector addition, which shows that finite rotations are not really vectors at all. It is best to employ the rotation matrix or quaternion notation, calculate the product, and then convert back to Euler axis and angle.
Euler rotations
Euler rotations of the Earth. Intrinsic (green), Precession (blue) and Nutation (red) The idea behind Euler rotations is to split the complete rotation of the coordinate system into three simpler constitutive rotations, called Precession, Nutation, and intrinsic rotation, being each one of them an increment on one of the Euler angles. Notice that the outer matrix will represent a rotation around one of the axes of the reference frame, and the inner matrix represents a rotation around one of the moving frame axis. The middle matrix represent a rotation around an intermediate axis called line of nodes.
Unfortunately, the definition of Euler angles is not unique and in the literature many different conventions are used. These conventions depend on the axes about which the rotations are carried out, and their sequence (since rotations are not commutative). The convention being used is usually indicated by specifying the axes about which the consecutive rotations (before being composed) take place, referring to them by index (1,2,3) or letter (X,Y,Z). The engineering and robotics communities typically use 3-1-3 Euler angles. Notice that after composing the independent rotations, they do not rotate about their axis anymore. The most external matrix rotates the other two, leaving the second rotation matrix over the line of nodes, and the third one in a frame comoving with the body. There are 3x3x3 = 27 possible combinations of three basic rotations but only 3x2x2 = 12 of them can be used for representing arbitrary 3D rotations as Euler angles. These 12 combinations avoid consecutive rotations around the same axis (such as XXY) which would reduce the degrees of freedom that can be represented. Therefore Euler angles are never expressed in terms of the external frame, or in terms of the co-moving rotated body frame, but in a mixture. Other conventions (e.g., rotation matrix or quaternions) are used to avoid this problem.
Quaternions Quaternions (Euler symmetric parameters) have proven very useful in representing rotations due to several advantages above the other representations mentioned here. A quaternion representation of rotation is written as a normalized four-dimensional vector . In terms of the Euler axis
and angle θ this vector's elements are expressed as follows:
The above definition follows the convention as used in (Wertz 1980) and (Markley 2003). An alternative definition used in some publications defines the "scalar" term as the first quaternion element, with the other elements shifted down one position. (Coutsias 1999), (Schmidt 2001) Inspection shows that the quaternion parametrization obeys the following constraint:
The last term (in our definition) is often called the scalar term, which has its origin in quaternions when understood as the mathematical extension of the complex numbers, written as a + bi + cj + dk. with
,
and where {i,j,k} are the hypercomplex numbers satisfying
Quaternion multiplication is performed in the same manner as multiplication of complex numbers, except that the order of elements must be taken into account, since multiplication is not commutative. In matrix notation we can write quaternion multiplication as
Combining two consecutive quaternion rotations is therefore just as simple as using the followed by , rotation matrix. Remember that two successive rotation matrices, are combined as follows:
We can represent this quaternion parameters in a similarly concise way. Please note the inverse ordering of quaternion multiplication when compared to matrix multiplication.
Quaternions are a very popular parametrization due to the following properties:
More compact than the matrix representation and less susceptible to round-off errors The quaternion elements vary continuously over the unit sphere in , (denoted by S3) as the orientation changes, avoiding discontinuous jumps (inherent to threedimensional parameterizations) Expression of the rotation matrix in terms of quaternion parameters involves no trigonometric functions It is simple to combine two individual rotations represented as quaternions using a quaternion product
Like rotation matrices, quaternions must sometimes be re-normalized due to rounding errors, to make sure that they correspond to valid rotations. The computational cost of renormalizing a quaternion, however, is much less than for normalizing a matrix.
Rodrigues parameters Rodrigues parameters (also called Gibbs vector) can be expressed in terms of Euler axis and angle as follows:
The Gibbs vector is undefined for attitude representation.
rotations, which is undesirable for global
Modified Rodrigues parameters (MRPs) can be expressed in terms of Euler axis and angle by:
The modified Rodrigues parametrization shares many characteristics with the rotation vector parametrization, including the occurrence of discontinuous jumps in the parameter space when incrementing the rotation.
Cayley-Klein parameters Rotors in a geometric algebra The formalism of geometric algebra (GA) provides an extension and interpretation of the quaternion method. Central to GA is the geometric product of vectors, an extension of the traditional inner and cross products, given by
where the symbol denotes the outer product. This product of vectors a,b produces two terms: a scalar part from the inner product and a bivector part from the outer product. This bivector describes the plane perpendicular to what the cross product of the vectors would return. Bivectors in GA have some unusual properties compared to vectors. Under the geometric describes the xy-plane. Its square product, bivectors have negative square: the bivector . Because the unit basis vectors are orthogonal to each other, the is geometric product reduces to the antisymmetric outer product-- and can be swapped since the basis freely at the cost of a factor of − 1. The square reduces to vectors themselves square to + 1. This result holds generally for all bivectors, and as a result the bivector plays a role similar to the imaginary unit. Geometric algebra uses bivectors in its analogue to the quaternion, the rotor, given by , where is a unit bivector that describes the plane of rotation. Because u squares to − 1, the power series expansion of R generates the trigonometric functions. The rotation formula that maps a vector a to a rotated vector a' is then
is the reverse of R (reversing the where order of the vectors in u is equivalent to changing its sign). Example. A rotation about the axis can be accomplished by , where is the unit volume converting to its dual bivector, element, the only trivector in three-dimensional space. The result is . In three-dimensional space, however, it is often simpler to leave the expression for , using the fact that i commutes with all objects in 3D and also squares to − 1. A rotation of the vector in this plane by an angle θ is then
and that is the reflection of about Recognizing that the plane perpendicular to gives a geometric interpretation to the rotation operation: the rotation preserves the components that are parallel to and changes only those that are perpendicular. The terms are then computed:
The result of the rotation is then
A simple check on this result is the angle θ = 2π / 3. Such a rotation should map the to . Indeed, the rotation reduces to
exactly as expected. This rotation formula is valid not only for vectors but for any multivector. In addition, when Euler angles are used, the complexity of the operation is much reduced. Compounded rotations come from multiplying the rotors, so the total rotor from Euler angles is
but like so:
and
. These rotors come back out of the exponentials
where Rβ refers to rotation in the original coordinates. Similarly for the γ rotation, . Noting that Rγ and Rα commute (rotations in the same plane must commute), and the total rotor becomes R = RαRβRγ Thus, the compounded rotations of Euler angles become a series of equivalent rotations in the original fixed frame. While rotors in geometric algebra work almost identically to quaternions in three dimensions, the power of this formalism is its generality: this method is appropriate and valid in spaces with any number of dimensions. In 3D, rotations have three degrees of freedom, a degree for each linearly independent plane (bivector) the rotation can take place in. It has been known that pairs of quaternions can be used to generate rotations in 4D, yielding six degrees of freedom, and the geometric algebra approach verifies this result: in 4D, there are six linearly independent bivectors that can be used as the generators of rotations.
Conversion formulae between representations Rotation matrix ↔ Euler angles The Euler angles (φ,θ,ψ) can be extracted from the rotation matrix rotation matrix in analytical form.
by inspecting the
Using the x-convention, the 3-1-3 Euler angles φ, θ and ψ (around the Z, X and again the Z-axis) can be obtained as follows:
is equivalent to arctan(a / b) where it also takes into account Note that the quadrant in which the point (a,b) is in. When implementing the conversion, one has to take into account several situations:
There are generally two solutions in ( − π,π > 3 interval. The above formula works only when θ is from the interval < 0,π)3. For special case A33 = 0, φ,ψ shall be derived from A11,A12. There is infinitely many but countably many solutions outside of interval ( − π,π > 3 . Whether all mathematical solutions apply for given application depends on the situation.
The rotation matrix is generated from the Euler angles by multiplying the three matrices generated by rotations about the axes.
The axes of the rotation depend on the specific convention being used. For the xconvention the rotations are about the X, Y and Z axes with angles φ, θ and ψ, the individual matrices are as follows:
This yields
Note: This is valid for the left-hand system.
Rotation matrix ↔ Euler axis/angle If the Euler angle θ is not a multiple of π, the Euler axis can be computed from the elements of the rotation matrix
and angle θ as follows:
Alternatively, the following method can be used: Eigen-decomposition of the rotation matrix yields the eigenvalues 1, and . The Euler axis is the eigenvector corresponding to the eigenvalue of 1, and the θ can be computed from the remaining eigenvalues. The Euler axis can be also found using Singular Value Decomposition since it is the normalized vector spanning the null-space of the matrix I − A. To convert the other way the rotation matrix corresponding to an Euler axis and angle θ can be computed according to the Rodrigues' rotation formula as follows:
with
the
identity matrix, and
is the cross-product matrix.
Rotation matrix ↔ quaternion When computing a quaternion from the rotation matrix there is a sign ambiguity, since and represent the same rotation. One way of computing the quaternion is as follows:
from the rotation matrix
There are three other mathematically equivalent ways to compute . Numerical inaccuracy can be reduced by avoiding situations in which the denominator is close to zero. One of the other three methods looks as follows:
The rotation matrix corresponding to the quaternion computed as follows:
with
the
identity matrix, and
can be
which gives
or equivalently
.
Euler angles ↔ quaternion We will consider the x-convention 3-1-3 Euler Angles for the following algorithm. The terms of the algorithm depend on the convention used. We can compute the quaternion follows:
Given the rotation quaternion angles (φ,θ,ψ) can be computed by
Euler axis/angle ↔ quaternion Given the Euler axis and angle θ, the quaternion
from the Euler angles (φ,θ,ψ) as
, the x-convention 3-1-3 Euler
can be computed by
Given the rotation quaternion the Euler axis and angle θ can be computed by
, define
. Then
Conversion formulae between derivatives Rotation matrix ↔ Angular velocities The angular velocity vector the rotation matrix
can be extracted from the derivative of
by the following relation:
The derivation is adapted from as follows: For any vector r0 consider
and differentiate it:
The derivative of a vector is the linear velocity of its tip. Since A is a rotation matrix, by definition the length of r(t) is always equal to the length of r0, and hence it does not change with time. Thus, when r(t) rotates, its tip moves along a circle, and the linear velocity of its tip is tangential to the circle, i.e. always perpendicular to r(t). In this specific case, the relationship between the linear velocity vector and the angular velocity vector is
By the transitivity of the above mentioned equations,
which implies (QED),
Chapter 7
Orthogonal Group
In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. This is a subgroup of the general linear group GL(n,F) given by
where QT is the transpose of Q. The classical orthogonal group over the real numbers is usually just written O(n). More generally the orthogonal group of a non-singular quadratic form over F is the group of linear operators preserving the form – the above group O(n, F) is then the orthogonal group of the sum-of-n-squares quadratic form. The Cartan–Dieudonné theorem describes the structure of the orthogonal group for non-singular form. Here we, only discusses definite forms – the orthogonal group of the positive definite form (equivalent to sum of n squares) and negative definite forms (equivalent to the negative sum of n squares) are identical – O(n,0) = O(0,n) – though the associated Pin groups differ; for other nonsingular forms O(p,q). Every orthogonal matrix has determinant either 1 or −1. The orthogonal n-by-n matrices with determinant 1 form a normal subgroup of O(n,F) known as the special orthogonal group, SO(n,F). (More precisely, SO(n,F) is the kernel of the Dickson invariant, discussed below.) By analogy with GL/SL (general linear group, special linear group), the orthogonal group is sometimes called the general orthogonal group and denoted GO, though this term is also sometimes used for indefinite orthogonal groups O(p,q). The derived subgroup Ω(n,F) of O(n,F) is an often studied object because when F is a finite field Ω(n,F) is often a central extension of a finite simple group. Both O(n,F) and SO(n,F) are algebraic groups, because the condition that a matrix be orthogonal, i.e. have its own transpose as inverse, can be expressed as a set of polynomial equations in the entries of the matrix.
Over the real number field Over the field R of real numbers, the orthogonal group O(n,R) and the special orthogonal group SO(n,R) are often simply denoted by O(n) and SO(n) if no confusion is possible. They form real compact Lie groups of dimension n(n − 1)/2. O(n,R) has two connected components, with SO(n,R) being the identity component, i.e., the connected component containing the identity matrix. The real orthogonal and real special orthogonal groups have the following geometric interpretations O(n,R) is a subgroup of the Euclidean group E(n), the group of isometries of Rn; it contains those that leave the origin fixed – O(n,R) = E(n) ∩ GL(n,R). It is the symmetry group of the sphere (n = 3) or hypersphere and all objects with spherical symmetry, if the origin is chosen at the center. SO(n,R) is a subgroup of E+(n), which consists of direct isometries, i.e., isometries preserving orientation; it contains those that leave the origin fixed – It is the rotation group of the sphere and all objects with spherical symmetry, if the origin is chosen at the center. { I, −I } is a normal subgroup and even a characteristic subgroup of O(n,R), and, if n is even, also of SO(n,R). If n is odd, O(n,R) is the direct product of SO(n,R) and { I, −I }. The cyclic group of k-fold rotations Ck is for every positive integer k a normal subgroup of O(2,R) and SO(2,R). Relative to suitable orthogonal bases, the isometries are of the form:
where the matrices R1,...,Rk are 2-by-2 rotation matrices in orthogonal planes of rotation. As a special case, known as Euler's rotation theorem, any (non-identity) element of SO(3,R) is rotation about a uniquely defined axis. The orthogonal group is generated by reflections (two reflections give a rotation), as in a Coxeter group, and elements have length at most n (require at most n reflections to generate; this follows from the above classification, noting that a rotation is generated by
2 reflections, and is true more generally for indefinite orthogonal groups, by the Cartan– Dieudonné theorem). A longest element (element needing the most reflections) is reflection through the origin (the map ), though so are other maximal combinations of rotations (and a reflection, in odd dimension). The symmetry group of a circle is O(2,R), also called Dih (S1), where S1 denotes the multiplicative group of complex numbers of absolute value 1. SO(2,R) is isomorphic (as a Lie group) to the circle S1 (circle group). This isomorphism sends the complex number exp(φi) = cos(φ) + i sin(φ) to the orthogonal matrix
The group SO(3,R), understood as the set of rotations of 3-dimensional space, is of major importance in the sciences and engineering. In terms of algebraic topology, for n > 2 the fundamental group of SO(n,R) is cyclic of order 2, and the spinor group Spin(n) is its universal cover. For n = 2 the fundamental group is infinite cyclic and the universal cover corresponds to the real line (the spinor group Spin(2) is the unique 2-fold cover).
Even and odd dimension The structure of the orthogonal group differs in certain respects between even and odd dimensions – for example, − I (reflection through the origin) is orientation-preserving in even dimension, but orientation-reversing in odd dimension. When this distinction wishes to be emphasized, the groups are generally denoted O(2k) and O(2k+1), reserving n for the dimension of the space (n = 2k or n = 2k + 1). The letters p or r are also used, indicating the rank of the corresponding Lie algebra; in odd dimension the corresponding Lie algebra is Br = so2r + 1, while in even dimension the Lie algebra is Dr = so2r.
Lie algebra The Lie algebra associated to the Lie groups O(n,R) and SO(n,R) consists of the skewsymmetric real n-by-n matrices, with the Lie bracket given by the commutator. This Lie algebra is often denoted by o(n,R) or by so(n,R), and called the orthogonal Lie algebra or special orthogonal Lie algebra. These Lie algebras are the compact real forms of two of the four families of semisimple Lie algebras: in odd dimension Br = so2r + 1, while in even dimension Dr = so2r. More intrinsically, given a vector space with an inner product, the special orthogonal Lie algebra is given by the bivectors on the space, which are sums of simple bivectors (2. The correspondence is given by the map blades)
where v * is the covector dual to the vector v; in coordinates these are exactly the elementary skew-symmetric matrices. This characterization is used in interpreting the curl of a vector field (naturally a 2-vector) as an infinitesimal rotation or "curl", hence the name. Generalizing the inner product with a nondegenerate form yields the indefinite orthogonal Lie algebras sop,q. The representation theory of the orthogonal Lie algebras includes both representations corresponding to linear representations of the orthogonal groups, and representations corresponding to projective representations of the orthogonal groups (linear representations of spin groups), the so-called spin representation, which are important in physics.
3D isometries that leave the origin fixed Isometries of R3 that leave the origin fixed, forming the group O(3,R), can be categorized as:
SO(3,R): o identity o rotation about an axis through the origin by an angle not equal to 180° o rotation about an axis through the origin by an angle of 180° the same with inversion in the origin (x is mapped to −x), i.e. respectively: o inversion in the origin o rotation about an axis by an angle not equal to 180°, combined with reflection in the plane through the origin perpendicular to the axis o reflection in a plane through the origin.
The 4th and 5th in particular, and in a wider sense the 6th also, are called improper rotations.
Conformal group Being isometries (preserving distances), orthogonal transforms also preserve angles, and are thus conformal maps, though not all conformal linear transforms are orthogonal. In classical terms this is the difference between congruence and similarity, as exemplified by SSS (Side-Side-Side) congruence of triangles and AAA (Angle-Angle-Angle) similarity of triangles. The group of conformal linear maps of Rn is denoted CO(n) for the conformal orthogonal group, and consists of the product of the orthogonal group with the group of dilations. If n is odd, these two subgroups do not intersect, and they are a direct product: subgroups intersect in
, while if n is even, these , so this is not a direct product, but it is a direct product with
the subgroup of dilation by a positive scalar:
.
Similarly one can define CSO(n); note that this is always :
.
Over the complex number field Over the field C of complex numbers, O(n,C) and SO(n,C) are complex Lie groups of dimension n(n − 1)/2 over C (which means the dimension over R is twice that). O(n,C) has two connected components, and SO(n,C) is the connected component containing the identity matrix. For n ≥ 2 these groups are noncompact. Just as in the real case SO(n,C) is not simply connected. For n > 2 the fundamental group of SO(n,C) is cyclic of order 2 whereas the fundamental group of SO(2,C) is infinite cyclic. The complex Lie algebra associated to O(n,C) and SO(n,C) consists of the skewsymmetric complex n-by-n matrices, with the Lie bracket given by the commutator.
Topology Low dimensional The low dimensional (real) orthogonal groups are familiar spaces:
The group SO(4) is double covered by
.
There are numerous charts on SO(3), due to the importance of 3-dimensional rotations in engineering applications. Here Sn denotes the n-dimensional sphere, RPn the n-dimensional real projective space, and SU(n) the special unitary group of degree n.
Homotopy groups The homotopy groups of the orthogonal group are related to homotopy groups of spheres, and thus are in general hard to compute. However, one can compute the homotopy groups of the stable orthogonal group (aka the infinite orthogonal group), defined as the direct limit of the sequence of inclusions
(as the inclusions are all closed inclusions, hence cofibrations, this can also be interpreted as a union). Sn is a homogeneous space for O(n + 1), and one has the following fiber bundle:
which can be understood as "The orthogonal group O(n + 1) acts transitively on the unit sphere Sn, and the stabilizer of a point (thought of as a unit vector) is the orthogonal group of the perpendicular complement, which is an orthogonal group one dimension lower". The map
is the natural inclusion.
is (n − 1)-connected, so the homotopy groups Thus the inclusion stabilize, and πk(O) = πk(O(n)) for n > k + 1: thus the homotopy groups of the stable space equal the lower homotopy groups of the unstable spaces. Via Bott periodicity, , thus the homotopy groups of O are 8-fold periodic, meaning πk + 8O = πkO, and one needs only to compute the lower 8 homotopy groups to compute them all.
Relation to KO-theory Via the clutching construction, homotopy groups of the stable space O are identified with stable vector bundles on spheres (up to isomorphism), with a dimension shift of 1: πkO = πk + 1BO. Setting obtains:
(to make π0 fit into the periodicity), one
Computation and Interpretation of homotopy groups Low-dimensional groups
The first few homotopy groups can be calculated by using the concrete descriptions of low-dimensional groups.
from orientation-preserving/reversing (this class survives to O(2) and hence stably) yields
, which is spin π2(O) = π2(SO(3)) = 0, which surjects onto π2(SO(4)); this latter thus vanishes.
Lie groups
From general facts about Lie groups, π2G always vanishes, and π3G is free (free abelian). Vector bundles
From the vector bundle point of view, π0(KO) is vector bundles over S0, which is two points. Thus over each point, the bundle is trivial, and the non-triviality of the bundle is the difference between the dimensions of the vector spaces over the two points, so is dimension. Loop spaces
Using concrete descriptions of the loop spaces in Bott periodicity, one can interpret higher homotopy of O as lower homotopy of simple to analyze spaces. Using π0, O and
O/U have two components, components, and the rest are connected.
and
have
Interpretation of homotopy groups In a nutshell: is dimension
is orientation
is spin is topological quantum field theory.
Let line
, and let LF be the tautological line bundle over the projective , and [LF] its class in K-theory. Noting that , these yield vector bundles
over the corresponding spheres, and
π1(KO) is generated by π2(KO) is generated by π4(KO) is generated by π8(KO) is generated by
can be From the point of view of symplectic geometry, interpreted as the Maslov index, thinking of it as the fundamental group of the stable Lagrangian Grassmannian π1(U / O), as
so π1(U / O) = π1 + 7(KO).
Over finite fields Orthogonal groups can also be defined over finite fields , where q is a power of a prime p. When defined over such fields, they come in two types in even dimension: O + (2n,q) and O − (2n,q); and one type in odd dimension: O(2n + 1,q). If V is the vector space on which the orthogonal group G acts, it can be written as a direct orthogonal sum as follows:
where Li are hyperbolic lines and W contains no singular vectors. If W = 0, then G is of plus type. If W = < w > then G has odd dimension. If W has dimension 2, G is of minus type. In the special case where n = 1, Oε(2,q) is a dihedral group of order 2(q − ε).
We have the following formulas for the order of these groups, O(n,q) = { A in GL(n,q) : A·At=I }, when the characteristic is greater than two
If −1 is a square in
If −1 is a nonsquare in
The Dickson invariant For orthogonal groups, the Dickson invariant is a homomorphism from the orthogonal group to Z/2Z, and is 0 or 1 depending on whether an element is the product of an even or odd number of reflections. More concretely, the Dickson invariant can be defined as where I is the identity (Taylor 1992, Theorem 11.43). Over fields that are not of characteristic 2 it is equivalent to the determinant: the determinant is −1 to the power of the Dickson invariant. Over fields of characteristic 2, the determinant is always 1, so the Dickson invariant gives extra information. The special orthogonal group is the kernel of the Dickson invariant and usually has index 2 in O(n,F). When the characteristic of F is not 2, the Dickson Invariant is 0 whenever the determinant is 1. Thus when the characteristic is not 2, SO(n,F) is commonly defined to be the elements of O(n,F) with determinant 1. Each element in O(n,F) has determinant −1 or 1. Thus in characteristic 2, the determinant is always 1. The Dickson invariant can also be defined for Clifford groups and Pin groups in a similar way (in all dimensions).
Orthogonal groups of characteristic 2 Over fields of characteristic 2 orthogonal groups often behave differently. Here we, lists some of the differences. Traditionally these groups are known as the hypoabelian groups but this term is no longer used for these groups.
Any orthogonal group over any field is generated by reflections, except for a unique example where the vector space is 4 dimensional over the field with 2
elements and the Witt index is 2. Note that a reflection in characteristic two has a slightly different definition. In characteristic two, the reflection orthogonal to a vector u takes a vector v to v+B(v,u)/Q(u)·u where B is the bilinear form and Q is the quadratic form associated to the orthogonal geometry. Compare this to the Householder reflection of odd characteristic or characteristic zero, which takes v to v − 2·B(v,u)/Q(u)·u.
The center of the orthogonal group usually has order 1 in characteristic 2, rather than 2, since I = − I.
In odd dimensions 2n+1 in characteristic 2, orthogonal groups over perfect fields are the same as symplectic groups in dimension 2n. In fact the symmetric form is alternating in characteristic 2, and as the dimension is odd it must have a kernel of dimension 1, and the quotient by this kernel is a symplectic space of dimension 2n, acted upon by the orthogonal group.
In even dimensions in characteristic 2 the orthogonal group is a subgroup of the symplectic group, because the symmetric bilinear form of the quadratic form is also an alternating form.
The spinor norm The spinor norm is a homomorphism from an orthogonal group over a field F to F*/F*2, the multiplicative group of the field F up to square elements, that takes reflection in a vector of norm n to the image of n in F*/F*2. For the usual orthogonal group over the reals it is trivial, but it is often non-trivial over other fields, or for the orthogonal group of a quadratic form over the reals that is not positive definite.
Galois cohomology and orthogonal groups In the theory of Galois cohomology of algebraic groups, some further points of view are introduced. They have explanatory value, in particular in relation with the theory of quadratic forms; but were for the most part post hoc, as far as the discovery of the phenomena is concerned. The first point is that quadratic forms over a field can be identified as a Galois H1, or twisted forms (torsors) of an orthogonal group. As an algebraic group, an orthogonal group is in general neither connected nor simplyconnected; the latter point brings in the spin phenomena, while the former is related to the discriminant. The 'spin' name of the spinor norm can be explained by a connection to the spin group (more accurately a pin group). This may now be explained quickly by Galois
cohomology (which however postdates the introduction of the term by more direct use of Clifford algebras). The spin covering of the orthogonal group provides a short exact sequence of algebraic groups.
Here μ2 is the algebraic group of square roots of 1; over a field of characteristic not 2 it is roughly the same as a two-element group with trivial Galois action. The connecting homomorphism from H0(OV), which is simply the group OV(F) of F-valued points, to H1(μ2) is essentially the spinor norm, because H1(μ2) is isomorphic to the multiplicative group of the field modulo squares. There is also the connecting homomorphism from H1 of the orthogonal group, to the H2 of the kernel of the spin covering. The cohomology is non-abelian, so that this is as far as we can go, at least with the conventional definitions.
Related groups The orthogonal groups and special orthogonal groups have a number of important subgroups, supergroups, quotient groups, and covering groups. These are listed below. The inclusions
and
are part of a sequence of 8 inclusions used in a geometric proof of the Bott periodicity theorem, and the corresponding quotient spaces are symmetric spaces of independent interest – for example, U(n) / O(n) is the Lagrangian Grassmannian.
Lie subgroups In physics, particularly in the areas of Kaluza–Klein compactification, it is important to find out the subgroups of the orthogonal group. The main ones are: – preserves an axis – U(n) are those that preserve a compatible complex structure or a compatible symplectic structure – see 2-out-of-3 property; SU(n) also preserves a complex orientation.
Lie supergroups The orthogonal group O(n) is also an important subgroup of various Lie groups:
Discrete subgroups As the orthogonal group is compact, discrete subgroups are equivalent to finite subgroups. These subgroups are known as point group and can be realized as the symmetry groups of polytopes. A very important class of examples are the finite Coxeter groups, which include the symmetry groups of regular polytopes. Dimension 3 is particularly studied –point groups in three dimensions, polyhedral groups, and list of spherical symmetry groups. In 2 dimensions, the finite groups are either cyclic or dihedral –point groups in two dimensions. Other finite subgroups include:
Permutation matrices (the Coxeter group An) Signed permutation matrices (the Coxeter group Bn); also equals the intersection of the orthogonal group with the integer matrices.
Covering and quotient groups The orthogonal group is neither simply connected nor centerless, and thus has both a covering group and a quotient group, respectively:
Two covering Pin groups, Pin+(n) → O(n) and Pin−(n) → O(n), The quotient projective orthogonal group, O(n) → PO(n).
These are all 2-to-1 covers. For the special orthogonal group, the corresponding groups are:
Spin group, Spin(n) → SO(n), Projective special orthogonal group, SO(n) → PSO(n).
Spin is a 2-to-1 cover, while in even dimension, PSO(2k) is a 2-to-1 cover, and in odd dimension PSO(2k+1) is a 1-to-1 cover, i.e., isomorphic to SO(2k+1). These groups, Spin(n), SO(n), and PSO(n) are Lie group forms of the compact special orthogonal Lie – Spin is the simply connected form, while PSO is the centerless form, algebra, and SO is in general neither.
Applications to string theory The group O(10) is of special importance in superstring theory because it is the symmetry group of 10 dimensional space-time.
Principal homogeneous space: Stiefel manifold The principal homogeneous space for the orthogonal group O(n) is the Stiefel manifold of orthonormal bases (orthonormal n-frames). In other words, the space of orthonormal bases is like the orthogonal group, but without a choice of base point: given an orthogonal space, there is no natural choice of orthonormal basis, but once one is given one, there is a one-to-one correspondence between bases and the orthogonal group. Concretely, a linear map is determined by where it sends a basis: just as an invertible map can take any basis to any other basis, an orthogonal map can take any orthogonal basis to any other orthogonal basis. for k < n of incomplete orthonormal bases The other Stiefel manifolds (orthonormal k-frames) are still homogeneous spaces for the orthogonal group, but not principal homogeneous spaces: any k-frame can be taken to any other k-frame by an orthogonal map, but this map is not uniquely determined.
Chapter 8
Point Groups in Three Dimensions
In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group of all isometries that leave the origin fixed, or correspondingly, the group of orthogonal matrices. O(3) itself is a subgroup of the Euclidean group E(3) of all isometries. Symmetry groups of objects are isometry groups. Accordingly, analysis of isometry groups is analysis of possible symmetries. All isometries of a bounded 3D object have one or more common fixed points. We choose the origin as one of them. The symmetry group of an object is sometimes also called full symmetry group, as opposed to its rotation group or proper symmetry group, the intersection of its full symmetry group and the rotation group SO(3) of the 3D space itself. The rotation group of an object is equal to its full symmetry group if and only if the object is chiral. The point groups in three dimensions are heavily used in chemistry, especially to describe the symmetries of a molecule and of molecular orbitals forming covalent bonds, and in this context they are also called molecular point groups.
Group structure SO(3) is a subgroup of E+(3), which consists of direct isometries, i.e., isometries preserving orientation; it contains those that leave the origin fixed. O(3) is the direct product of SO(3) and the group generated by inversion (denoted by its matrix −I): O(3) = SO(3) × { I , −I } Thus there is a 1-to-1 correspondence between all direct isometries and all indirect isometries, through inversion. Also there is a 1-to-1 correspondence between all groups of direct isometries H and all groups K of isometries that contain inversion:
K = H × { I , −I } H = K ∩ SO(3) If a group of direct isometries H has a subgroup L of index 2, then, apart from the corresponding group containing inversion there is also a corresponding group that contains indirect isometries but no inversion: M = L ∪ ( (H \ L) × { − I } ) where isometry ( A , I ) is identified with A. Thus M is obtained from H by inverting the isometries in H \ L. This group M is as abstract group isomorphic with H. Conversely, for all isometry groups that contain indirect isometries but no inversion we can obtain a rotation group by inverting the indirect isometries. This is clarifying when categorizing isometry groups. In 2D the cyclic group of k-fold rotations Ck is for every positive integer k a normal subgroup of O(2,R) and SO(2,R). Accordingly, in 3D, for every axis the cyclic group of k-fold rotations about that axis is a normal subgroup of the group of all rotations about that axis, and also of the group obtained by adding reflections in planes through the axis.
3D isometries that leave origin fixed The isometries of R3 that leave the origin fixed, forming the group O(3,R), can be categorized as follows:
SO(3,R): o identity o rotation about an axis through the origin by an angle not equal to 180° o rotation about an axis through the origin by an angle of 180° the same with inversion (x is mapped to −x), i.e. respectively: o inversion o rotation about an axis by an angle not equal to 180°, combined with reflection in the plane through the origin perpendicular to the axis o reflection in a plane through the origin
The 4th and 5th in particular, and in a wider sense the 6th also, are called improper rotations.
Conjugacy When comparing the symmetry type of two objects, the origin is chosen for each separately, i.e. they need not have the same center. Moreover, two objects are considered to be of the same symmetry type if their symmetry groups are conjugate subgroups of O(3) (two subgroups H1, H2 of a group G are conjugate, if there exists g ∈ G such that H1 = g−1H2g ).
Thus two 3D objects have the same symmetry type:
if both have mirror symmetry, but with respect to a different mirror plane if both have 3-fold rotational symmetry, but with respect to a different axis.
In the case of multiple mirror planes and/or axes of rotation, two symmetry groups are of the same symmetry type if and only if there is a single rotation mapping this whole structure of the first symmetry group to that of the second. The conjugacy definition would also allow a mirror image of the structure, but this is not needed, the structure itself is achiral. For example, if a symmetry group contains a 3-fold axis of rotation, it contains rotations in two opposite directions. (The structure is chiral for 11 pairs of space groups with a screw axis.)
Infinite isometry groups We restrict ourselves to isometry groups that are closed as topological subgroups of O(3). This excludes for example the group of rotations by an irrational number of turns about an axis. The whole O(3) is the symmetry group of spherical symmetry; SO(3) is the corresponding rotation group. The other infinite isometry groups consist of all rotations about an axis through the origin, and those with additionally reflection in the planes through the axis, and/or reflection in the plane through the origin, perpendicular to the axis. Those with reflection in the planes through the axis, with or without reflection in the plane through the origin, perpendicular to the axis, are the symmetry groups for the two types of cylindrical symmetry.
Finite isometry groups For point groups, being finite corresponds to being discrete; infinite discrete groups as in the case of translational symmetry and glide reflectional symmetry do not apply. Symmetries in 3D that leave the origin fixed are fully characterized by symmetries on a sphere centered at the origin. For finite 3D point groups. Up to conjugacy the set of finite 3D point groups consists of:
7 infinite series with at most one more-than-2-fold rotation axis; they are the finite symmetry groups on an infinite cylinder, or equivalently, those on a finite cylinder. 7 point groups with multiple 3-or-more-fold rotation axes; they can also be characterized as point groups with multiple 3-fold rotation axes, because all 7 include these axes; with regard to 3-or-more-fold rotation axes the possible combinations are: o 4×3 o 4×3 and 3×4
o
10×3 and 6×5
A selection of point groups is compatible with discrete translational symmetry: 27 from the 7 infinite series, and 5 of the 7 others, the 32 so-called crystallographic point groups.
The seven infinite series The infinite series have an index n, which can be any integer; in each series, the nth symmetry group contains n-fold rotational symmetry about an axis, i.e. symmetry with respect to a rotation by an angle 360°/n. n=1 covers the cases of no rotational symmetry at all. For n = ∞ they correspond to the frieze groups. Schönflies notation is used, and, in parentheses, orbifold notation; the latter is not only conveniently related to its properties, but also to the order of the group it is a unified notation, also applicable for wallpaper groups and frieze groups. The 7 infinite series are:
Cn (nn ) of order n - n-fold rotational symmetry (abstract group Zn ); for n = 1: no symmetry (trivial group) Cnh (n* ) of order 2n (for odd n abstract group Z2n = Zn × Z2 , for even n abstract group Zn × Z2 ) Cnv (*nn ) of order 2n - pyramidal symmetry (abstract group Dihn ); in biology C2v is called biradial symmetry. Dn (22n ) of order 2n - dihedral symmetry (abstract group Dihn ) S2n (nx ) of order 2n (not to be confused with symmetric groups, for which the same notation is used; abstract group Z2n ) Dnh (*22n ) of order 4n - prismatic symmetry (for odd n abstract group Dih2n = Dihn × Z2 ; for even n abstract group Dihn × Z2 ) Dnd (or Dnv ) (2*n ) - antiprismatic symmetry of order 4n (abstract group Dih2n )
The terms horizontal (h) and vertical (v), and the corresponding subscripts, refer to the additional mirror plane, that can be parallel to the rotation axis (vertical) or perpendicular to the rotation axis (horizontal). Involutional symmetry (abstract group Z2 ):
Ci - inversion symmetry C2 - 2-fold rotational symmetry Cs - reflection symmetry, also called bilateral symmetry.
Patterns on a cylindrical band illustrating the case n = 6 for each of the 7 infinite families of point groups. The symmetry group of each pattern is the indicated group. The second of these is the first of the uniaxial groups (cyclic groups) Cn of order n (also applicable in 2D), which are generated by a single rotation of angle 360°/n. In addition to this, one may add a mirror plane perpendicular to the axis, giving the group Cnh of order 2n, or a set of n mirror planes containing the axis, giving the group Cnv, also of order 2n. The latter is the symmetry group for a regular n-sided pyramid. A typical object with symmetry group Cn or Dn is a propellor. If both horizontal and vertical reflection planes are added, their intersections give n axes of rotation through 180°, so the group is no longer uniaxial. This new group of order 4n is called Dnh. Its subgroup of rotations is the dihedral group Dn of order 2n, which still has
the 2-fold rotation axes perpendicular to the primary rotation axis, but no mirror planes. Note that in 2D Dn includes reflections, which can also be viewed as flipping over flat objects without distinction of front- and backside, but in 3D the two operations are distinguished: the group contains "flipping over", not reflections. There is one more group in this family, called Dnd (or Dnv), which has vertical mirror planes containing the main rotation axis, but instead of having a horizontal mirror plane, it has an isometry that combines a reflection in the horizontal plane and a rotation by an angle 180°/n. Dnh is the symmetry group for a regular n-sided prisms and also for a regular n-sided bipyramid. Dnd is the symmetry group for a regular n-sided antiprism, and also for a regular n-sided trapezohedron. Dn is the symmetry group of a partially rotated prism. Sn is generated by the combination of a reflection in the horizontal plane and a rotation by an angle 360°/n. For n odd this is equal to the group generated by the two separately, Cnh of order 2n, and therefore the notation Sn is not needed; however, for n even it is distinct, and of order n. Like Dnd it contains a number of improper rotations without containing the corresponding rotations. All symmetry groups in the 7 infinite series are different, except for the following four pairs of mutually equal ones:
C1h and C1v: group of order 2 with a single reflection (Cs ) D1 and C2: group of order 2 with a single 180° rotation D1h and C2v: group of order 4 with a reflection in a plane and a 180° rotation through a line in that plane D1d and C2h: group of order 4 with a reflection in a plane and a 180° rotation through a line perpendicular to that plane
S2 is the group of order 2 with a single inversion (Ci ) "Equal" is meant here as the same up to conjugacy in space. This is stronger than "up to algebraic isomorphism". For example, there are three different groups of order two in the first sense, but there is only one in the second sense. Similarly, e.g. S2n is algebraically isomorphic with Z2n.
The seven remaining point groups The remaining point groups are said to be of very high or polyhedral symmetry because they have more than one rotation axis of order greater than 2. Using Cn to denote an axis of rotation through 360°/n and Sn to denote an axis of improper rotation through the same, the groups are:
T (332) of order 12 - chiral tetrahedral symmetry. There are four C3 axes, each through two vertices of a cube (body diagonals) or one of a regular tetrahedron, and three C2 axes, through the centers of the cube's faces, or the midpoints of the
tetrahedron's edges. This group is isomorphic to A4, the alternating group on 4 elements, and is the rotation group for a regular tetrahedron.
Td (*332) of order 24 - full tetrahedral symmetry. This group has the same rotation axes as T, but with six mirror planes, each containing two edges of the cube or one edge of the tetrahedron, a single C2 axis and two C3 axes. The C2 axes are now actually S4 axes. This group is the symmetry group for a regular tetrahedron. Td is isomorphic to S4, the symmetric group on 4 letters.
Th (3*2) of order 24 - pyritohedral symmetry.
The structure of a volleyball has Th symmetry. This group has the same rotation axes as T, with mirror planes parallel to the cube faces. The C3 axes become S6 axes, and there is inversion symmetry. Th is isomorphic to A4 × C2. It is the symmetry of a cube with on each face a line segment dividing the face into two equal rectangles, such that the line segments of adjacent faces do not meet at the edge. The symmetries correspond to the even permutations of the body diagonals and the same combined with inversion. It is also the symmetry of a pyritohedron, which is similar to the cube described, with each rectangle replaced by a pentagon with one symmetry axis and 4 equal sides and 1 different side (the one corresponding to the line segment dividing the cube's face); i.e., the cube's faces bulge out at the dividing line and become narrower there. It is a subgroup of the full icosahedral symmetry group (as isometry group, not just as abstract group), with 4 of the 10 3-fold axes.
O (432) of order 24 - chiral octahedral symmetry. This group is like T, but the C2 axes are now C4 axes, and additionally there are 6 C2 axes, through the
midpoints of the edges of the cube. This group is also isomorphic to S4, and is the rotation group of the cube and octahedron.
Oh (*432) of order 48 - full octahedral symmetry. This group has the same rotation axes as O, but with mirror planes, comprising both the mirror planes of Td and Th. This group is isomorphic to S4 × C2, and is the symmetry group of the cube and octahedron.
I (532) of order 60 - chiral icosahedral symmetry; the rotation group of the icosahedron and the dodecahedron. It is a normal subgroup of index 2 in the full group of symmetries Ih. The group I is isomorphic to A5, the alternating group on 5 letters. The group contains 10 versions of D3 and 6 versions of D5 (rotational symmetries like prisms and antiprisms).
Ih (*532) of order 120 - full icosahedral symmetry; the symmetry group of the icosahedron and the dodecahedron. The group Ih is isomorphic to A5 × C2. The group contains 10 versions of D3d and 6 versions of D5d (symmetries like antiprisms).
Relation between orbifold notation and order The order of each group is 2 divided by the orbifold Euler characteristic; the latter is 2 minus the sum of the feature values, assigned as follows:
n without or before * counts as (n−1)/n n after * counts as (n−1)/(2n) * and x count as 1
Rotation groups The rotation groups, i.e. the finite subgroups of SO(3), are: the cyclic groups Cn (the rotation group of a regular pyramid), the dihedral groups Dn (the rotation group of a regular prism, or regular bipyramid), and the rotation groups T, O and I of a regular tetrahedron, octahedron/cube and icosahedron/dodecahedron. In particular, the dihedral groups D3, D4 etc. are the rotation groups of plane regular polygons embedded in three-dimensional space, and such a figure may be considered as a degenerate regular prism. Therefore it is also called a dihedron (Greek: solid with two faces), which explains the name dihedral group.
An object with symmetry group Cn, Cnh, Cnv or S2n has rotation group Cn. An object with symmetry group Dn, Dnh, or Dnd has rotation group Dn. An object with one of the other seven symmetry groups has as rotation group the corresponding one without subscript: T, O or I.
The rotation group of an object is equal to its full symmetry group if and only if the object is chiral. In other words, the chiral objects are those with their symmetry group in the list of rotation groups.
Correspondence between rotation groups and other groups The following groups contain inversion:
Cnh and Dnh for even n S2n and Dnd for odd n (S2 = Ci is the group generated by inversion; D1d = C2h) Th, Oh, and Ih
As explained above, there is a 1-to-1 correspondence between these groups and all rotation groups:
Cnh for even n and S2n for odd n correspond to Cn Dnh for even n and Dnd for odd n correspond to Dn Th, Oh, and Ih correspond to T, O, and I, respectively.
The other groups contain indirect isometries, but not inversion:
Cnv Cnh and Dnh for odd n S2n and Dnd for even n Td
They all correspond to a rotation group H and a subgroup L of index 2 in the sense that they are obtained from H by inverting the isometries in H \ L, as explained above:
Cn is subgroup of Dn of index 2, giving Cnv Cn is subgroup of C2n of index 2, giving Cnh for odd n and S2n for even n Dn is subgroup of D2n of index 2, giving Dnh for odd n and Dnd for even n T is subgroup of O of index 2, giving Td
Maximal symmetries There are two discrete point groups with the property that no discrete point group has it as proper subgroup: Oh and Ih. Their largest common subgroup is Th. The two groups are obtained from it by changing 2-fold rotational symmetry to 4-fold, and adding 5-fold symmetry, respectively. Alternatively the two groups are generated by adding for each a reflection plane to Th. There are two crystallographic point groups with the property that no crystallographic point group has it as proper subgroup: Oh and D6h. Their maximal common subgroups, depending on orientation, are D3d and D2h.
The groups arranged by abstract group type Below the groups explained above are arranged by abstract group type. The smallest abstract groups that are not any symmetry group in 3D, are the quaternion group (of order 8), the dicyclic group Dic3 (of order 12), and 10 of the 14 groups of order 16. The column "# of order 2 elements" in the following tables shows the total number of isometry subgroups of types C2 , Ci , Cs. This total number is one of the characteristics helping to distinguish the various abstract group types, while their isometry type helps to distinguish the various isometry groups of the same abstract group. Within the possibilities of isometry groups in 3D, there are infinitely many abstract group types with 0, 1 and 3 elements of order 2, there are two with 2n + 1 elements of order 2, and there are three with 2n + 3 elements of order 2 (for each n ≥ 2 ). There is never a positive even number of elements of order 2.
Symmetry groups in 3D that are cyclic as abstract group The symmetry group for n-fold rotational symmetry is Cn; its abstract group type is cyclic group Zn , which is also denoted by Cn. However, there are two more infinite series of symmetry groups with this abstract group type:
For even order 2n there is the group S2n (Schoenflies notation) generated by a rotation by an angle 180°/n about an axis, combined with a reflection in the plane perpendicular to the axis. For S2 the notation Ci is used; it is generated by inversion. For any order 2n where n is odd, we have Cnh; it has an n-fold rotation axis, and a perpendicular plane of reflection. It is generated by a rotation by an angle 360°/n about the axis, combined with the reflection. For C1h the notation Cs is used; it is generated by reflection in a plane.
Thus we have, with bolding of the 10 cyclic crystallographic point groups, for which the crystallographic restriction applies: Order Isometry groups Abstract group # of order 2 elements 1
C1
Z1
0
2
C2 , Ci , Cs
Z2
1
3
C3
Z3
0
4
C4 , S4
Z4
1
5
C5
Z5
0
6
C6 , S6 , C3h
Z6 = Z3 × Z2
1
7
C7
Z7
0
8
C8 , S8
Z8
1
9
C9
Z9
0
10
C10 , S10 , C5h
Z10 = Z5 × Z2
1
etc.
Symmetry groups in 3D that are dihedral as abstract group In 2D dihedral group Dn includes reflections, which can also be viewed as flipping over flat objects without distinction of front- and backside. However, in 3D the two operations are distinguished: the symmetry group denoted by Dn contains n 2-fold axes perpendicular to the n-fold axis, not reflections. Dn is the rotation group of the n-sided prism with regular base, and n-sided bipyramid with regular base, and also of a regular, n-sided antiprism and of a regular, n-sided trapezohedron. The group is also the full symmetry group of such objects after making them chiral by e.g. an identical chiral marking on every face, or some modification in the shape. The abstract group type is dihedral group Dihn, which is also denoted by Dn. However, there are three more infinite series of symmetry groups with this abstract group type:
Cnv of order 2n, the symmetry group of a regular n-sided pyramid Dnd of order 4n, the symmetry group of a regular n-sided antiprism Dnh of order 4n for odd n. For n = 1 we get D2, already covered above, so n ≥ 3.
Note the following property: Dih4n+2
Dih2n+1 × Z2
Thus we have, with bolding of the 12 crystallographic point groups, and writing D1d as the equivalent C2h: Order Isometry groups Abstract group # of order 2 elements 4
D2 , C2v , C2h
Dih2 = Z2 × Z2
3
6
D3 , C3v
Dih3
3
8
D4 , C4v , D2d
Dih4
5
10
D5 , C5v
Dih5
5
12
D6 , C6v , D3d , D3h Dih6 = Dih3 × Z2 7
14
D7 , C7v
Dih7
7
16
D8 , C8v , D4d
Dih8
9
18
D9 , C9v
Dih9
9
etc.
Other C2n,h of order 4n is of abstract group type Z2n × Z2. For n = 1 we get Dih2 , already covered above, so n ≥ 2. Thus we have, with bolding of the 2 cyclic crystallographic point groups: Order
Isometry group
# of order 2 elements
Abstract group
8
C4h
Z4 × Z2
3
12
C6h
Z6 × Z2 = Z3 × Z2 × Z2 = Z3 × Dih2
3
16
C8h
Z8 × Z2
3
20
C10h
Z10 × Z2 = Z5 × Z2 × Z2 3
Cycle diagram
etc. Dnh of order 4n is of abstract group type Dihn × Z2. For odd n this is already covered above, so we have here D2nh of order 8n, which is of abstract group type Dih2n × Z2 (n≥1). Thus we have, with bolding of the 3 dihedral crystallographic point groups: Order
Isometry group
Abstract group
# of order 2 elements
8
D2h
Dih2 × Z2
7
16
D4h
Dih4 × Z2
11
24
D6h
Dih6 × Z2
15
32
D8h
Dih8 × Z2
19
Cycle diagram
etc. The remaining seven are, with bolding of the 5 crystallographic point groups:
order 12: of type A4 (alternating group): T order 24: o of type S4 (symmetric group, not to be confused with the symmetry group with this notation): Td, O o of type A4 × Z2: Th . order 48, of type S4 × Z2: Oh order 60, of type A5: I order 120, of type A5 × Z2: Ih
Impossible discrete symmetries Since the overview is exhaustive, it also shows implicitly what is not possible as discrete symmetry group. For example:
a C6 axis in one direction and a C3 in another a C5 axis in one direction and a C4 in another a C3 axis in one direction and another C3 axis in a perpendicular direction
etc.
Fundamental domain The fundamental domain of a point group is a conic solid. An object with a given symmetry in a given orientation is characterized by the fundamental domain. If the object is a surface it is characterized by a surface in the fundamental domain continuing to its radial bordal faces or surface. If the copies of the surface do not fit, radial faces or surfaces can be added. They fit anyway if the fundamental domain is bounded by reflection planes. For a polyhedron this surface in the fundamental domain can be part of an arbitrary plane. For example, in the disdyakis triacontahedron one full face is a fundamental domain. Adjusting the orientation of the plane gives various possibilities of combining two or more adjacent faces to one, giving various other polyhedra with the same symmetry. The polyhedron is convex if the surface fits to its copies and the radial line perpendicular to the plane is in the fundamental domain. Also the surface in the fundamental domain may be composed of multiple faces.
Binary polyhedral groups The map Spin(3) → SO(3) is the double cover of the rotation group by the spin group in 3 dimensions. (This is the only connected cover of SO(3), since Spin(3) is simply connected.) By the lattice theorem, there is a Galois connection between subgroups of Spin(3) and subgroups of SO(3) (rotational point groups): the image of a subgroup of Spin(3) is a rotational point group, and the preimage of a point group is a subgroup of Spin(3).
The preimage of a finite point group is called a binary polyhedral group, and is called by the same name as its point group, with the prefix binary. For instance, the preimage of the icosahedral group is the binary icosahedral group. The binary polyhedral groups are:
An: binary cyclic group of an (n + 1)-gon Dn: binary dihedral group of an n-gon E6: binary tetrahedral group E7: binary octahedral group E8: binary icosahedral group
These are classified by the ADE classification, and the quotient of C2 by the action of a binary polyhedral group is a Du Val singularity. For point groups that reverse orientation, the situation is more complicated, as there are two pin groups, so there are two possible binary groups corresponding to a given point group. Note that this is a covering of groups, not a covering of spaces – the sphere is simply connected, and thus has no covering spaces. There is thus no notion of a "binary polyhedron" that covers a 3-dimensional polyhedron. Binary polyhedral groups are discrete subgroups of a Spin group, and under a representation of the spin group act on a vector space, and may stabilize a polyhedron in this representation – under the map Spin(3) → SO(3) they act on the same polyhedron that the underlying (non-binary) group acts on, while under spin representations or other representations they may stabilize other polyhedra. This is in contrast to projective polyhedra – the sphere does cover projective space (and also lens spaces), and thus a tessellation of projective space or lens space yields a distinct notion of polyhedron.
