Rotation About an Arbitrary Axis in 3 Dimensions Glenn Murray June 6, 2013
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Introduction
The problem of rotation about an arbitrary axis in three dimensions arises in many fields including computer graphics and molecular simulation. In this article we give an algorithm and matrices for doing the movement. Many of the results were initially obtained with Mathematica. An algorithm (See Figure 1):
Figure 1: Moving the axis of rotation A to the z-axis.
(1) Translate space so that the rotation axis passes through the origin. 1
(2) Rotate space about the z axis so that the rotation axis lies in the xz plane. (3) Rotate space about the y axis so that the rotation axis lies along the z axis. (4) Perform the desired rotation by θ about the z axis. (5) Apply the inverse of step (3). (6) Apply the inverse of step (2). (7) Apply the inverse of step (1). We will write our three-dimensional points in four homogeneous coordinates; i.e., (x, y, z) will be written as (x, y, z, 1). This enables us to do coordinate transformations using 4x4 matrices. Note that these are really only necessary for translations, if we omitted translations from our movements we could do the motions with 3x3 rotation matrices obtained by deleting the last rows and last columns of the 4x4 matrices. In this article vectors are multiplied by matrices on the vector’s left.
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A translation matrix
The product TP1 ·v is equivalent to the vector sum h−a, −b, −c, 0i+v, i.e., this transformation moves the point P1 (a, b, c) to the origin. 1 0 0 −a 0 1 0 −b TP1 = 0 0 1 −c 0 0 0 1 1 0 0 −a 0 1 0 −b TP1 = 0 0 1 −c 0 0 0 1
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3D Coordinate axes rotation matrices
Here are the matrices for rotation by α around the x-axis, β around the y-axis, and γ around the z-axis. 1 0 0 0 0 cos α − sin α 0 Rx (α) = 0 sin α cos α 0 0 0 0 1 cos β 0 sin β 0 0 1 0 0 Ry (β) = − sin β 0 cos β 0 0 0 0 1
2
cos γ − sin γ sin γ cos γ Rz (γ) = 0 0 0 0
0 0 1 0
0 0 0 1
The general rotation matrix depends on the order of rotations. The first matrix rotates about x, then y, then z; the second rotates about z, then y, then x.
0 0 0 1
0 0 0 1
cos β cos γ cos γ sin α sin β − cos α sin γ cos α cos γ sin β + sin α sin γ cos β sin γ cos α cos γ + sin α sin β sin γ − cos γ sin α + cos α sin β sin γ Rz Ry Rx = − sin β cos β sin α cos α cos β 0 0 0 cos β cos γ − cos β sin γ sin β cos α sin γ + sin α sin β cos γ cos α cos γ − sin α sin β sin γ − sin α cos β Rx Ry Rz = sin α sin γ − cos α sin β cos γ sin α cos γ + cos α sin β sin γ cos α cos β 0 0 0
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Transformations for rotating a vector to the z -axis
In this section we introduce matrices to move a rotation vector hu, v, wi to the z-axis. Note that we use the components to form expressions for the cosines and sines to avoid using inverse trigonometric functions. We require that the rotation vector not be parallel to the z-axis, else u = v = 0 and the denominators vanish.
4.1
The matrix to rotate a vector about the z -axis to the xz -plane √ √ u √u2 + v 2 v √u2 + v 2 −v u2 + v 2 u u2 + v 2 = 0 0 0 0
Txz
4.2
0 0 0 1
The matrix to rotate the vector in the xz -plane to the z -axis
√
u2 + v 2 + w 2 √ √0 Tz = u2 + v 2 u2 + v 2 + w2 0
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0 0 1 0
w
√ √ 0 − u2 + v 2 u2 + v 2 + w2 1 √ 0 0 w u2 + v 2 + w2 0 0
0 0 0 1
Rotations about the origin
In this section we rotate the point (x, y, z) about the vector hu, v, wi by the angle θ. 3
5.1
The matrix for rotations about the origin
−1 −1 This is the product Txz Tz Rz (θ)Tz Txz .
u2 +(v 2 +w2 ) cos θ u2 +v 2 +w2 √ uv(1−cos θ)+w u2 +v 2 +w2 sin θ u2 +v 2 +w2 √ uw(1−cos θ)−v u2 +v 2 +w2 sin θ u2 +v 2 +w2
0
√ uv(1−cos θ)−w u2 +v 2 +w2 sin θ u2 +v 2 +w2 2
2
2
v +(u +w ) cos θ u2 +v 2 +w2 √ vw(1−cos θ)+u u2 +v 2 +w2 sin θ u2 +v 2 +w2
0
√ uw(1−cos θ)+v u2 +v 2 +w2 sin θ u2 +v 2 +w2 √ vw(1−cos θ)−u u2 +v 2 +w2 sin θ u2 +v 2 +w2 w2 +(u2 +v 2 ) cos θ u2 +v 2 +w2
0
0
0 0 1
If we multiply this times hx, y, zi we can obtain a function of of seven variables that yields the result of rotating the point (x, y, z) about the axis hu, v, wi by the angle θ. f (x, y, z, u, v, w, θ) =
5.2
√ u(ux+vy+wz)(1−cos θ)+(u2 +v 2 +w2 )x cos θ+ u2 +v 2 +w2 (−wy+vz) sin θ u2 +v 2 +w2 √ v(ux+vy+wz)(1−cos θ)+(u2 +v 2 +w2 )y cos θ+ u2 +v 2 +w2 (wx−uz) sin θ u2 +v 2 +w2 √ w(ux+vy+wz)(1−cos θ)+(u2 +v 2 +w2 )z cos θ+ u2 +v 2 +w2 (−vx+uy) sin θ u2 +v 2 +w2
The normalized matrix for rotations about the origin
At this point we would like to simplify the expressions by making the assumption that hu, v, wi is a unit vector; i.e., that u2 + v 2 + w2 = 1. With this simplification, we obtain the −1 −1 following expression for Txz Tz Rz (θ)Tz Txz . 2 u + (1 − u2 ) cos θ uv(1 − cos θ) − w sin θ uw(1 − cos θ) + v sin θ 0 uv(1 − cos θ) + w sin θ 2 2 v + (1 − v ) cos θ vw(1 − cos θ) − u sin θ 0 2 2 uw(1 − cos θ) − v sin θ vw(1 − cos θ) + u sin θ w + (1 − w ) cos θ 0 0 0 0 1 If we multiply this times hx, y, zi we can obtain a function of of seven variables that yields the result of rotating the point (x, y, z) about the axis hu, v, wi (where u2 + v 2 + w2 = 1) by the angle θ. 4
f (x, y, z, u, v, w, θ) = u(ux + vy + wz)(1 − cos θ) + x cos θ + (−wy + vz) sin θ v(ux + vy + wz)(1 − cos θ) + y cos θ + (wx − uz) sin θ w(ux + vy + wz)(1 − cos θ) + z cos θ + (−vx + uy) sin θ
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Rotation about an arbitrary line
We will define an arbitrary line by a point the line goes through and a direction vector. If the axis of rotation is given by two points P1 = (a, b, c) and P2 = (d, e, f ), then a direction vector can be obtained by hu, v, wi = hd − a, e − b, f − ci. We can now write a transformation for the rotation of a point about this line.
6.1
The matrix for rotation about an arbitrary line
−1 −1 This is given by the product TP−1 Tz Rz (θ)Tz Txz TP1 . In hopes of fitting the matrix onto Txz 1 the page we make the substitution L = u2 + v 2 + w2 .
u2 +(v 2 +w2 ) cos θ L
√ uv(1−cos θ)+w L sin θ L uw(1−cos θ)−v√L sin θ L 0
√ uv(1−cos θ)−w L sin θ L
√ uw(1−cos θ)+v L sin θ L
√ a(v 2 +w2 )−u(bv+cw) (1−cos θ)+(bw−cv) L sin θ
v 2 +(u2 +w2 ) cos θ L
√ vw(1−cos θ)−u L sin θ L
√ b(u2 +w2 )−v(au+cw) (1−cos θ)+(cu−aw) L sin θ
√ vw(1−cos θ)+u L sin θ L
w2 +(u2 +v 2 ) cos θ L
0
0
L
L
√ (1−cos θ)+(av−bu) L sin θ L
c(u2 +v 2 )−w(au+bv)
1
If we multiply this times hx, y, zi we can obtain a function of of ten variables that yields the result of rotating the point (x, y, z) about the line through (a, b, c) with direction vector hu, v, wi by the angle θ. f (x, y, z, a, b, c, u, v, w, θ) =
√ a(v 2 +w2 )−u(bv+cw−ux−vy−wz) (1−cos θ)+Lx cos θ+ L(−cv+bw−wy+vz) sin θ L √ 2 2 b(u +w )−v(au+cw−ux−vy−wz) (1−cos θ)+Ly cos θ+ L(cu−aw+wx−uz) sin θ L √ 2 2 c(u +v )−w(au+bv−ux−vy−wz) (1−cos θ)+Lz cos θ+ L(−bu+av−vx+uy) sin θ L 5
6.2
The normalized matrix for rotation about an arbitrary line
Assuming that hu, v, wi is a unit vector so that L = 1, we obtain a more practical result for −1 −1 Tz Rz (θ)Tz Txz TP1 . TP−1 Txz 1 u2 + (v 2 + w2 ) cos θ
uv(1 − cos θ) − w sin θ
uw(1 − cos θ) + v sin θ
uv(1 − cos θ) + w sin θ uw(1 − cos θ) − v sin θ 0
v 2 + (u2 + w2 ) cos θ
vw(1 − cos θ) − u sin θ
vw(1 − cos θ) + u sin θ
w2 + (u2 + v 2 ) cos θ
0
0
a(v 2 + w2 ) − u(bv + cw) (1 − cos θ) + (bw − cv) sin θ b(u2 + w2 ) − v(au + cw) (1 − cos θ) + (cu − aw) sin θ c(u2 + v 2 ) − w(au + bv) (1 − cos θ) + (av − bu) sin θ 1
If we multiply this times hx, y, zi we can obtain a function of of ten variables that yields the result of rotating the point (x, y, z) about the line through (a, b, c) with direction vector hu, v, wi (where u2 + v 2 + w2 = 1) by the angle θ. f (x, y, z, a, b, c, u, v, w, θ) = a(v 2 + w2 ) − u(bv + cw − ux − vy − wz) (1 − cos θ) + x cos θ + (−cv + bw − wy + vz) sin θ 2 2 b(u + w ) − v(au + cw − ux − vy − wz) (1 − cos θ) + y cos θ + (cu − aw + wx − uz) sin θ 2 2 c(u + v ) − w(au + bv − ux − vy − wz) (1 − cos θ) + z cos θ + (−bu + av − vx + uy) sin θ
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Code and visualization
A graphic visualization of rotating a point about a line can be found at http://twist-and-shout. appspot.com/. Tested Java code for the matrices and formulas, released under the Apache license, is at https://sites.google.com/site/glennmurray/Home/rotation-matrices-and-formulas.
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