Here, rotating around a line uses the concept of a change of basis. See Change of basis for further information.
The steps to perform a rotation around a line through the origin are
-
Find any basis (x, y, z) such that the positive x-axis direction of the basis is codirectional with the direction of the line:
|x.x| |y.x| |z.x| x = |x.y| y = |y.y| z = |z.y| |x.z| |y.z| |z.z| -
Perform a change of basis from the implied reference basis to the new basis:
| x.x x.y x.z | B = | y.x y.y y.z | | z.x z.y z.z | -
Perform the rotation as a rotation around the x-axis:
| 1 0 0 | R = | 0 cos(θ) -sin(θ) | | 0 sin(θ) cos(θ) | -
Perform a change of basis back to the implied reference basis:
| x.x y.x z.x | B⁻¹ = | x.y y.y z.y | | x.z y.z z.z |
Combine these steps into one matrix M:
M = B⁻¹ R B
| m00 m01 m02 | | x.x y.x z.x | | 1 0 0 | | x.x x.y x.z |
| m10 m11 m12 | = | x.y y.y z.y | | 0 c -s | | y.x y.y y.z |
| m20 m21 m22 | | x.z y.z z.z | | 0 s c | | z.x z.y z.z |
where
c = cos(θ)
s = sin(θ)
k10 = c*y.x - s*z.x
k11 = c*y.y - s*z.y
k12 = c*y.z - s*z.z
k20 = s*y.x + c*z.x
k21 = s*y.y + c*z.y
k22 = s*y.z + c*z.z
m00 = x.x*x.x + y.x*k10 + z.x*k20
m01 = x.x*x.y + y.x*k11 + z.x*k21
m02 = x.x*x.z + y.x*k12 + z.x*k22
m10 = x.y*x.x + y.y*k10 + z.y*k20
m11 = x.y*x.y + y.y*k11 + z.y*k21
m12 = x.y*x.z + y.y*k12 + z.y*k22
m20 = x.z*x.x + y.z*k10 + z.z*k20
m21 = x.z*x.y + y.z*k11 + z.z*k21
m22 = x.z*x.z + y.z*k12 + z.z*k22
The steps to perform a rotation around a general line are
-
Find any frame (x, y, z, o) such that the origin of the frame coincides with the line and the frame's positive x-axis direction is codirectional with the direction of the line:
|x.x| |y.x| |z.x| |o.x| x = |x.y| y = |y.y| z = |z.y| o = |o.y| |x.z| |y.z| |z.z| |o.z| | 1 | | 1 | | 1 | | 1 | -
Perform a change of frame from the implied reference frame to the new frame:
(Here we translate the new frame's origin to the implied reference frame's origin and then change to the new frame's basis.)
| x.x x.y x.z 0 | | 1 0 0 -o.x | F = | y.x y.y y.z 0 | | 0 1 0 -o.y | | z.x z.y z.z 0 | | 0 0 1 -o.z | | 0 0 0 1 | | 0 0 0 1 | | x.x x.y x.z -(x.x*o.x + x.y*o.y + x.z*o.z) | = | y.x y.y y.z -(y.x*o.x + y.y*o.y + y.z*o.z) | | z.x z.y z.z -(z.x*o.x + z.y*o.y + z.z*o.z) | | 0 0 0 1 | -
Perform the rotation as a rotation around the x-axis:
| 1 0 0 0 | R = | 0 cos(θ) -sin(θ) 0 | | 0 sin(θ) cos(θ) 0 | | 0 0 0 1 | -
Perform a change of frame back to the implied reference frame:
| 1 0 0 o.x | | x.x y.x z.x 0 | F⁻¹ = | 0 1 0 o.y | | x.y y.y z.y 0 | | 0 0 1 o.z | | x.z y.z z.z 0 | | 0 0 0 1 | | 0 0 0 1 | | x.x y.x z.x o.x | = | x.y y.y z.y o.y | | x.z y.z z.z o.z | | 0 0 0 1 |
Combine these steps into one matrix M:
M = F⁻¹ R F
| x.x y.x z.x o.x | | 1 0 0 0 | | x.x x.y x.z t0 |
= | x.y y.y z.y o.y | | 0 c -s 0 | | y.x y.y y.z t1 |
| x.z y.z z.z o.z | | 0 s c 0 | | z.x z.y z.z t2 |
| 0 0 0 1 | | 0 0 0 1 | | 0 0 0 1 |
| m00 m01 m02 m03 |
= | m10 m11 m12 m13 |
| m20 m21 m22 m23 |
| 0 0 0 1 |
where
c = cos(θ)
s = sin(θ)
t0 = -(x.x*o.x + x.y*o.y + x.z*o.z)
t1 = -(y.x*o.x + y.y*o.y + y.z*o.z)
t2 = -(z.x*o.x + z.y*o.y + z.z*o.z)
k10 = c*y.x - s*z.x
k11 = c*y.y - s*z.y
k12 = c*y.z - s*z.z
k13 = c*t1 - s*t2
k20 = s*y.x + c*z.x
k21 = s*y.y + c*z.y
k22 = s*y.z + c*z.z
k23 = s*t1 + c*t2
m00 = x.x*x.x + y.x*k10 + z.x*k20
m01 = x.x*x.y + y.x*k11 + z.x*k21
m02 = x.x*x.z + y.x*k12 + z.x*k22
m03 = x.x*t0 + y.x*k13 + z.x*k23 + o.x
m10 = x.y*x.x + y.y*k10 + z.y*k20
m11 = x.y*x.y + y.y*k11 + z.y*k21
m12 = x.y*x.z + y.y*k12 + z.y*k22
m13 = x.y*t0 + y.y*k13 + z.y*k23 + o.y
m20 = x.z*x.x + y.z*k10 + z.z*k20
m21 = x.z*x.y + y.z*k11 + z.z*k21
m22 = x.z*x.z + y.z*k12 + z.z*k22
m23 = x.z*t0 + y.z*k13 + z.z*k23 + o.z