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Orthogonal Matrices and Rotations in R3

Worksheet by Frank Rooney

restart: with(LinearAlgebra): with(plots): with(plottools):

Warning, the name changecoords has been redefined

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The objectives are:

1) How to find the rotation in R3 represented by a given proper orthogonal matrix.

2) How to find the orthogonal matrix that represents a rotation in R3

<Text-field style="Heading 1" family="Arial" layout="Heading 1">Rotations in R3 corresponding to a given orthogonal matrix</Text-field>

An orthogonal matrix Q satisfies

Q.QT = QT.Q = I

The vector corresponding to the axis of rotation is left unchanged by the matrix in other words it is an eigenvector of Q with eigenvector 1.

Firstly let's generate a random orthogonal matrix

va := RandomVector(3): vb := RandomVector(3): vec1 := Normalize(va,Euclidean): vc := vb - (vec1.vb)*vec1: vec2 := Normalize(vc,Euclidean): vec3 := CrossProduct(vec1,vec2): Q := <vec1 | vec2 | vec3>;

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Just to check.

QT := Transpose(Q): QT.Q; Q.QT;

Now the axis of rotation is the eigenvector corresponding to the eigenvalue 1.

EigQ := Eigenvectors (Q): evalf(%);

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The axis is the eigenvector corresponding to 1.(Note the complex eigenvalues are of the form cos \316\270 \302\261 i sin \316\270, with \316\270 the rotation angle)

A := DeleteColumn(EigQ[2],1..2): B := <A[1,1],A[2,1],A[3,1]>: Normalize(B,Euclidean): axis1 := evalf(%);

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There is another way to find the axis of rotation. If w is the eigenvector of Q corresponding to 1 then Q.w = w = QT.w so (Q - QT).w = 0 or if q is the axial vector corresponding to Q - QT this means q x w = 0 or w is parallel to q

Qsk := Q - QT: q := < Qsk[2,3],Qsk[3,1],Qsk[1,2]>: Normalize(q,Euclidean): axis := evalf(%);

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

To find the angle of rotation we rely on the invariance of the trace. A rotation about the z axis has the form 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There is some orthogonal matrix P that rotates the z axis to the axis of rotation for Q. Then Q = P. Rot . PT Trace (Q) = Trace (P. Rot . PT) = Trace (PT.P. Rot) = Trace (Rot) = 1 + 2 cos(\316\270)

trQ := Trace(Q) : arccos((trQ-1)/2): theta = evalf(%);

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

<Text-field style="Heading 1" family="Arial" layout="Heading 1">Finding the matrix corresponding to a given rotation in R3</Text-field>

The rotation matrix will depend only on the axis n and the angle of rotation \316\261 so it has to have the form

Q = A I + B n \342\212\227n + C N where N is the skew symmetic matrix with n as its axial vector and A, B, and C are functions of \316\261. Using the facts that Q.n = n, Trace (Q) = 1 + 2 cos \316\261, and Q . QT= I, we calculate that A = cos \316\261, B = 1 - cos \316\261 and C = sin \316\261 First get a random axis and random angle of rotation.

vec := RandomVector(3): n := Normalize(vec,Euclidean): nbyn := << n[1]*n[1],n[2]*n[1],n[3]*n[1]>|<n[1]*n[2],n[2]*n[2],n[3]*n[2]>|<n[1]*n[3],n[2]*n[3],n[3]*n[3]>>: N := << 0,n[3],-n[2]>|<-n[3],0,n[1]>|<n[2],-n[1],0>>: alpha := Pi/6: Qrot := cos(alpha) * IdentityMatrix(3) + (1-cos(alpha))*nbyn+sin(alpha)*N;

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JiMiIyYpRl1hbEZmYGxGZ2BsRmZgbEZmYGwjIiNLRlxibEZmYGxJJ01hdHJpeEc2JCUqcHJvdGVjdGVkR0YqNyMtRl5kbDYjL0kkJWlkR0YoRltgbA==

Check this has the correct properties.

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