Reflections and Rotations in R 3

Transcription

1 Reflections and Rotations in R 3 P. J. Ryan May 29, 21 Rotations as Compositions of Reflections Recall that the reflection in the hyperplane H through the origin in R n is given by f(x) = x 2 ξ, x ξ (1) where ξ is a unit normal for H. Let u be a unit vector in R 3, v a unit vector orthogonal to u, and let w = u v. Then (u, v, w) is an orthonormal triple. Let T be the rotation which is the composition of refections T 1 and T 2 in planes through the origin whose unit normals are ξ and η respectively, both orthogonal to w. We may choose numbers θ and ϕ so that η = sin θ u + cos θ v ξ = sin ϕ u + cos ϕ v. If the planes are distinct, then T is not the identity rotation and the axis of T is the line through the origin with direction vector w. Using the formula (1), we have f(u) = (1 2 sin 2 ϕ) u + 2 sin ϕ cos ϕ v f(v) = 2 sin ϕ cos ϕ u + (1 2 cos 2 ϕ) v f(w) = w (2) That is, f(u) = cos 2ϕ u + sin 2ϕ v f(v) = sin 2ϕ u cos 2ϕ v f(w) = w To get the formula for the reflection involving η, just replace ϕ by θ. Then T = T 1 T 2, and we compute T (u) = cos 2(ϕ θ) u + sin 2(ϕ θ) v T (v) = sin 2(ϕ θ) u + cos 2(ϕ θ) v T (w) = w. (3) 1

2 This is a tedious verification if carried out directly. For the moment, consider the special case where u = ɛ 1, v = ɛ 2, w = ɛ 3. Then, the calculation reduces to the matrix multiplication [ ] [ ] [ ] cos 2ϕ sin 2ϕ cos 2θ sin 2θ cos 2(ϕ θ) sin 2(ϕ θ) =. sin 2ϕ cos 2ϕ sin 2θ cos 2θ sin 2(ϕ θ) cos 2(ϕ θ) From (3) we conclude that T (u), u = T (v), v = cos 2(ϕ θ). (4) and T (u), v = T (v), u. (5) Proposition 1. For all unit vectors x orthogonal to w in R 3, T (x), x = cos 2(ϕ θ). To see this, first recall that (u, v, w) is an orthonormal triple. Thus, x = x, u u + x, v v + x, w w = x, u u + x, v v (6) since x is orthogonal to w. By linearity of T, we have T (x) = x, u T (u) + x, v T (v) from which we get, using (5) to eliminate the cross terms, T (x), x = x, u 2 T (u), u + x, v 2 T (v), v. Taking (4) into account and the fact that x, u 2 + x, v 2 = x 2 = 1, we have verified the proposition. This justifies the idea that T is a rotation by 2(ϕ θ) about its axis. In general, for any unit orthogonal vectors u and v, and any number α, the transformation T (u) = cos α u + sin α v T (v) = sin α u + cos α v T (w) = w (7) is called a rotation by α. As long as T is not the identity, the axis of the rotation will be the line through the origin with direction vector u v. 2

3 Counter-Clockwise and Clockwise Rotations Although the notions of clockwise and counterclockwise rotations are not defined in any absolute sense in R 3, we give the following definition: Definition 2. Let T be the rotation described in (7). If ω is any positive multiple of u v, then T is said to be counter-clockwise with respect to ω and clockwise with respect to ω. This means that a counter-clockwise rotation by π is also a clockwise rotation by π. It also means that a counter-clockwise rotation by α is also a clockwise rotation by 2π α. In order to test if a given rotation is counter-clockwise, we now derive a more general criterion. Proposition 3. Let T be a rotation by α, where < α π, counter-clockwise with respect to ω. Then x T (x), ω for all x orthogonal to ω. Proof. From (7) we compute u T (u) = sin α w u T (v) = cos α w v T (u) = cos α w v T (v) = sin α w Note that u T (u), w = v T (v), w = sin α. Now let x be a unit vector orthogonal to w. We can write (6) in the form (8) x = cos β u + sin β v for some suitable number β. Then, x T (x) = cos 2 β u T (u) + cos β sin β v T (u) + sin β cos β u T (v) + sin 2 β v T (v) = (cos 2 β + sin 2 β) sin α w = sin α w where we have used (8) to simplify the expression. This gives x T (x), ω = ω sin α as required. Thus, to determine if a given rotation is clockwise or counter-clockwise, first express it in such a way that the angle of rotation satisfies < α π. Then pick a convenient x orthogonal to the axis of rotation, and choose a particular direction vector ω for the axis. Now check the sign of x T (x), ω. Unless α = π, there are two rotations (one clockwise, one counter-clockwise) by α about a given axis. These rotations are inverses of each other. For example, in (7), we could construct the inverse by replacing α by α, then rewriting as T (u) = cos α u + sin α ( v) T ( v) = sin α u + cos α ( v) T (w) = w (9) thus maintaining < α π. This will be the clockwise rotation by α with respect to w. 3

4 Standard Matrices of Reflections and Rotations We now show how to compute the standard matrix of a reflection or rotation in R 3. In particular, we derive the matrices of counter-clockwise rotations by α about the coordinate axes. Let A be the standard matrix of the reflection defined in formula (1). Then the equations (2) give cos 2ϕ sin 2ϕ AP = P sin 2ϕ cos 2ϕ. (1) Note that P is an orthogonal matrix, since its columns are an orthonormal set. In other words, P T P = I, i.e. the inverse of P is its transpose. Thus, we may express the standard matrix of the reflection as cos 2ϕ sin 2ϕ A = P sin 2ϕ cos 2ϕ P T and hence, the standard matrix of the rotation in (3) is cos 2(ϕ θ) sin 2(ϕ θ) A = P sin 2(ϕ θ) cos 2(ϕ θ) P T Three special cases are worth mentioning. First, look at the case where w = ɛ 3, so that we can take u = ɛ 1 and v = ɛ 2. Then P is the identity and the standard matrix of the counter-clockwise rotation by α with respect to ɛ 3 is cos α sin α sin α cos α To find the standard matrix of the counter-clockwise rotation by α with respect to ɛ 1, we choose w = ɛ 1, u = ɛ 2, and v = ɛ 3. Then 1 P = 1 1 so that the standard matrix is cos α sin α 1 1 sin α cos α = cos α sin α 1 1 sin α cos α. 4

5 To find the standard matrix of the counter-clockwise rotation by α with respect to ɛ 2, we choose w = ɛ 2, u = ɛ 3, and v = ɛ 1. Then 1 P = 1 so that the standard matrix is 1 cos α sin α cos α sin α sin α cos α 1 = sin α cos α 5