问题
Say we have a 4x4 matrix with indices like so:
00 01 02 03
10 11 12 13
20 21 22 23
30 31 32 33
How does one convert the rotation data (ignoring the z axis, if that helps) contained in this matrix into a single 2d rotational angle (in radians)?
Background: I have a 3D .dae animation exported from Blender into the Collada format. The animation is technically 2d, all of the z axis values are 0. I'm trying to convert the 4x4 matrices into 2d translation, rotation and scale data.
回答1:
Scale matrix S looks like this:
sx 0 0 0
0 sy 0 0
0 0 sz 0
0 0 0 1
Translation matrix T looks like this:
1 0 0 0
0 1 0 0
0 0 1 0
tx ty tz 1
Z-axis rotation matrix Rlooks like this:
cos(a) sin(a) 0 0
-sin(a) cos(a) 0 0
0 0 1 0
0 0 0 1
If you have a transformation matrix M, it is a result of a number of multiplications of R, T and S matrices. Looking at M, the order and number of those multiplications is unknown. However, if we assume that M=S*R*T we can decompose it into separate matrices. Firstly let's calculate S*R*T:
( sx*cos(a) sx*sin(a) 0 0) (m11 m12 m13 m14)
S*R*T = (-sy*sin(a) sy*cos(a) 0 0) = M = (m21 m22 m23 m24)
( 0 0 sz 0) (m31 m32 m33 m34)
( tx ty tz 1) (m41 m42 m43 m44)
Since we know it's a 2D transformation, getting translation is straightforward:
translation = vector2D(tx, ty) = vector2D(m41, m42)
To calculate rotation and scale, we can use sin(a)^2+cos(a)^2=1:
(m11 / sx)^2 + (m12 / sx)^2 = 1
(m21 / sy)^2 + (m22 / sy)^2 = 1
m11^2 + m12^2 = sx^2
m21^2 + m22^2 = sy^2
sx = sqrt(m11^2 + m12^2)
sy = sqrt(m21^2 + m22^2)
scale = vector2D(sx, sy)
rotation_angle = atan2(sx*m22, sy*m12)
回答2:
this library has routines for converting a 4x4 matrix into its 5 components - rotation, translation, scale, shear, and perspective. You should be able to take the formulas and just drop the 3rd component of the 3d vectors.
来源：https://stackoverflow.com/questions/10629737/convert-3d-4x4-rotation-matrix-into-2d