Three-Dimensional Orthogonal Transformations

A P P E N D I X II

Three-Dimensional Orthogonal Transformations

A nonsingular linear transformation R> defined with respect to an w-dimensional real vector space, is said to be orthogonal if i ?^{- 1} = R.

This condition implies that

RR = 7, (l.a)

RR = /. (l.b)

If R is represented by an « Χ η matrix, (l.a) and (l.b) may be expanded to yield the orthogonality relations

η

X RikRjk = &ij » (2-a)

k=l

η

X RkiRkj = &ij > (2.b) where /, j = 1 , 2 , . . . , « . Equation (2.a) states that if the rows of R are

interpreted as w-dimensional vectors, each row is of unit length and orthogonal to every other row. Equation (2.b) shows that a similar remark applies to the columns of R. It follows that an η Χ η orthogonal matrix is determined by \n(n — 1) independent real parameters.1

An important property of an orthogonal transformation is that it preserves the scalar product (ξ, η). For if ξ' = and η = P^ are the transforms of ξ and η, then

(f, η) = {Ri, R^V) = (ξ, R~RV) = (ξ, ^v). (3) In particular, if ξ and η are orthogonal, then ξ' and η are also orthogonal.

1 See t h e discussion of u n i t a r y t r a n s f o r m a t i o n s in A p p e n d i x I.

4 7 2

T H R E E - D I M E N S I O N A L O R T H O G O N A L TRANSFORMATIONS 473

If ξ = η⁹ (3) states that

(ί,

i

Thus orthogonal transformations preserve lengths and angles in the generalized sense.

A second property of orthogonal transformations follows upon taking the determinant of both sides of RR = 1:

det RR = (deti?)2 = 1,

where use has been made of the rule for the determinant of a matrix product, and the fact that det A = det Ä. It follows that

detÄ = ± l . (4) Orthogonal transformations are classified as proper or improper,

accordingly as the determinant is + 1 or — 1 . Proper orthogonal trans- formations correspond to τζ-dimensional rotations; improper orthogonal transformations correspond to a rotation followed by a reflection.

Let the ^-dimensional space be subjected to a succession of r orthogonal transformations R^x, R², R^r, and let the product of these transformations be denoted

R = RfR^i ··· R2R\ · (5) Since each Ri is orthogonal, R^Ri = 1> and

RR = R^R2 m" RrRr ··* R2RX = 1 = RR.

Hence the product of any number of orthogonal transformations is an orthogonal transformation. Furthermore, the determinant of each Ri

is ± 1 , so that a product of orthogonal transformations is proper or improper, accordingly as the product contains an even or odd number of improper transformations.

A proper orthogonal transformation in three dimensions may be interpreted as a rotation of a given cartesian-coordinate system (xyz) into a second (x'y'z') cartesian system. T h e condition det R = + 1 requires the (x'y'z') system to be right- or left-handed, accordingly as the (xyz) system is right- or left-handed. T h e following discussion will be restricted to proper orthogonal transformations and right-handed coordinate systems.

The matrix elements of a three-dimensional orthogonal transformation represent the direction cosines of the new axes relative to the original axes:

cos(*/, xj) = R^ijt (6)

474 APPENDIX II

where (x1x²x^) = {^xy^z) ^ana* (x^x^Xz) = (x'y'z). Only three parameters are required to specify an orthogonal transformation in three dimensions, so that there must be six equations relating the nine direction cosines.

These equations are provided by the orthogonality relations. In the special case where the rotation is about the direction of one of the original coordinate axes, the rotation matrices are easily written down:

,10 Ox

Rx(x) =

X

XI (7)

\0 —sin χ cos χ '

(

0 1 0 , (8)

sin

0

0 /

/ cos φ sin φ 0\

Rz(<p) — —sin φ cos φ 01. (9)

\ 0

01/

All three rotations are taken in the positive or right-hand screw sense;

the matrices for the corresponding negative rotations are obtained by the transcription: χ, 0, φ -> — χ, —0, —φ.

The matrices for Rx , Ry , and Rz , or any other orthogonal matrix, are applied to column vectors using the rule for matrix multiplication. Thus

#11

R21 #22 tfJUJ. (10)

R³¹ R%2 R33' \xz'

T h e matrices for Rx , R^y , and Rz reveal an important property of three-dimensional rotation matrices—the trace of each matrix is twice the cosine of the angle of rotation plus one. N o w Rx , Ry , and Rz are special cases of the general three-dimensional rotation matrix and, since the trace is an invariant,

tr{R(n0)} = 1 + 2cos<2>, (11) where R(n<P) is the matrix for a rotation through an angle Φ about a

direction defined by the unit vector n.

T h e nine matrix elements of R(n<&) can be expressed in terms of the angle of rotation Φ and the direction cosines of n:

η = (cos a, cos ß, cos y),

cos2 oc -h cos² β + cos2 γ = 1.

(12)

T H R E E - D I M E N S I O N A L O R T H O G O N A L T R A N S F O R M A T I O N S 475

T h u s there are only three independent parameters in the set (α, ßf y, Φ).

If η is defined in terms of its polar angles θ, φ, then

cos α = sin θ cos φ, cos β = sin θ sin φ, cos γ = cos Θ, (13) and the matrix elements of R(n<P) can be expressed in terms of φ, 0, and Φ.

T h e rotation ϋ ( η Φ ) may be described as a rigid rotation of the (xyz) system about the direction η in such a way that the positive coordinate axes trace out conical sectors on the infinite cones defined by η and the positive axes; the semiangles of these cones are the direction angles (x^y β, y, and the angle of rotation Φ is the angle between the initial and final planes of the conical sections thus generated (Fig. AII.l). T h e deduction of the matrix for R(n<P) by geometric considerations is straight- forward but extremely tedious. There is, however, a procedure that involves much simpler computations and is more instructive.

Consider a coordinate system (x'y'z') whose z' axis is parallel to n. In this coordinate system, the matrix for R(n<P) has the simple form

X

F I G. A I I . l . L o c u s of t h e positive ζ axis u n d e r a rotation Φ a b o u t a n axis defined by t h e u n i t vector n . T h e positive χ a n d y axes trace o u t analogous conical sections w i t h η as t h e i r axis of s y m m e t r y , b u t w i t h γ replaced b y <x a n d ]8, respectively.

476 A P P E N D I X II

where Τ is the orthogonal transformation which sends the (xyz) system into the (x'y'z) system. If Τ can be determined, Ρ(ηΦ) is given by

Κ{μΦ) = TR\n0)T-\ (16) T h e orthogonal transformation Τ is immediately apparent from

Fig. AII.2: Τ consists of a rotation through π about an axis defined by

ζ

y χ

F I G . A I I. 2 . T h e t r a n s f o r m a t i o n Τ rotates t h e (xyz) s y s t e m a b o u t m b y an angle π, s e n d i n g t h e ζ axis i n t o t h e direction n.

the unit vector m with direction angles (α', /?', y') given by

, . 0 . 0 . , θ cosa = sin - cos 99, cos ρ = sin - sin φ, cosy = cos ^ . (17) If R(n0) is denoted i?(<p, 0, Φ), then

Τ = Α (^{φ >} θ/2, ΤΓ). (18)

Furthermore, Τ2 is a rotation by 2τγ, SO that Τ2 = 1. Since T i s orthogonal and its own inverse, Τ must be symmetrical:

Τ = Ä (9 > 0/2, π) = &(φ, 0/2,

ττ) =

T h u s

Α(ηΦ) = R(<p, 0/2, n)Rf(ri0)R(<p, 0/2, ττ). (20) T h e matrix elements of 7?(φ, 0/2, ττ) are easily determined. Since z'

is in the direction n, the last row of P(<p, 0/2, ττ) consists of cos a, cos ßy

cos y, and since the matrix is symmetrical, the third column is also

T H R E E - D I M E N S I O N A L O R T H O G O N A L TRANSFORMATIONS 477

known. T h e matrix element Ru = c o s ^ / , xx) is determined by noting that both the initial and final χ axes make an angle <x with the vector m (by the definition of a rotation), and that the angle of rotation is π; hence

/ / \ ο ' c o s 2a 1 cosi*! , χΛ) = cos 2oc = -—• 1,

v 1 u 1 + cos γ

where the last form is obtained by using (13) and (17). Similarly, cos² ß

cos(#o', Xo) = cos IB' = -—• 1.

v 2 ' 2} r 1 + cos γ

T h e remaining matrix element, R12 = R²i , can be obtained by using the orthogonality of the third row of i?(ç?, 0/2, π) with the first or second rows. T h u s

cos² α. ^Λ cos α cos β

1 cos a 1

1 + cos γ 1 + cos γ

#(<p, 0/2, tt) = I cos a cos β cos2 β I. (21)

1 + cos γ 1 + cos γ

cos öl cos β cos y y Substituting (14) and (21) into (20), one obtains

(

— cos γ sin Φ + cos α cos 0(1 —cos Φ) s i n2 j8 cos Φ -f- c o s2 β

cos j3 sin Φ -h cos α cos y(l —cos Φ) —cos α sin Φ + cos β cos y(l —cos Φ)

— cos β sin Φ + cos α cos y(l — cos Φ)\

cos α sin Φ + cos β cos y(l — cos Φ) I . (22)

s i n2 y cos Φ + c o s2 y /

The matrix for 2?(ηΦ) is fundamental in the theory of proper orthogonal matrices in three dimensions. For if any proper orthogonal matrix A is given, the direction cosines of the axis of rotation and the angle of rotation may be obtained by equating the matrix elements of A to the corresponding matrix elements in Ζ?(ηΦ). T h e angle of rotation is determined from the trace relation:

1 + 2 COS Φ = a^u + Λ2 2 + a33 ·

T h e proper rotation of an (xyz) system to a new {x'y'z') system can also be expressed as a product of three rotations about axes defined by unit vectors a , b , and c :

R(n0) = # ( c < / O # ( b 0 ) i ? ( a ? ) . (23)

478 A P P E N D I X II

If the factorization on the right side of (23) is to be expressible in terms of three independent parameters, it is necessary that two successive rotations take place about different axes. This restriction also holds for the associative groupings into a product of two factors. For example, a cannot be parallel to b, and the axis of rotation of the product R(b6)R(a(p) cannot be parallel to c. On the other hand, it is possible for a to be parallel to c, since the independence of φ and φ is ensured by the noncommutivity of finite rotations. Even with these restrictions, there are many ways in which R(nO) can be expressed as a product of three factors. Whichever factorization is selected, Φ and η may be determined by the method indicated above.

One of the most useful factorizations is the Euler decomposition. Let (x'y'z) be the axes of the (xyz) system after the rotation #(ηΦ), and let the line determined by the intersection of the xy' plane with the xy plane be denoted N> as shown in Fig. AII.3. The line N> called the line of nodes,

ζ

X

Ν

F I G . Α Π .3 . T h e E u l e r angles φ, 0, a n d φ.

is perpendicular to ζ and ζ'. T h e Euler angles ψ, θ, φ are defined as follows:

(1) A rotation about the ζ axis through an angle φ, sending the (xyz) system into the (χ&χζ) system.

(2) A rotation about the yx axis through an angle 0, sending the (x^y^z) system into the (x^y^z) system.

(3) A rotation about the z' axis through an angle φ, sending the (χ<$\ζ') system into the (x'y'z') system.

T H R E E - D I M E N S I O N A L O R T H O G O N A L TRANSFORMATIONS 479

T h e angles θ and φ are the polar angles of z\ so that 0 < θ < π, 0 < φ < 2τ7, 0 < ⁰ < 2ττ.

T h e matrices for 7?Α(φ), Ryfê), and Ρζ>(ψ) may be obtained from (8) and (9); their product

gives the matrix for a three-dimensional rotation in terms of Euler's angles. Upon carrying out the matrix multiplications, one finds

(

— sin 0 cos 0 cos φ — cos </f sin φ —sin ^ cos θ sin φ + cos ψ cos 93 sin 0 sin θ I.

sin 0 cos φ sin 0 sin φ cos 0 /

(24) T h e angle of rotation in terms of the Euler angles may be obtained by equating 1 + 2 cos Φ to the trace of (24). One finds, after some trigonometric reductions,

Φ θ 1

cos — = ± c o s 2 cos 2 + Φ)-

In the special case θ = φ = 0, the angle of rotation is φ, so that the positive sign must be chosen. T h e direction cosines of the axis of rotation may be determined by equating corresponding matrix elements of (22) and (24). T h e final results are

cos α sin ^Φ = sin \Q sin \(φ — φ), cos β sin \Φ = sin \θ cos ^(φ — φ), cos y sin \Φ = cos \θ sin ^ ( 0 + φ), cos | Φ = cos ^ 0 cos \(φ + <p).