转换3D 4x4的旋转矩阵为二维 [英] Convert 3d 4x4 Rotation Matrix into 2d

问题描述

假设我们有像这样索引的4x4矩阵：

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

一个人如何转换的旋转数据（忽略z轴，是否有帮助）包含在此基质成一个单一的二维旋转角度（弧度）？

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)?

背景：我有一个3D .dae动画从搅拌机出口到Collada的格式。该动画在技术上是2D，所有的Z轴值都为0。我想4×4矩阵转换成2D平移，旋转和缩放数据。

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.

推荐答案

比例矩阵取值是这样的：

sx 0  0  0
0  sy 0  0
0  0  sz 0
0  0  0  1

翻译矩阵 T 是这样的：

1  0  0  0
0  1  0  0
0  0  1  0
tx ty tz 1

Z轴旋转矩阵研究是这样的：

 cos(a) sin(a)  0  0
-sin(a) cos(a)  0  0
   0      0     1  0
   0      0     0  1

如果你有一个变换矩阵 M ，它是众多研究的乘法结果， T 和取值矩阵。纵观 M ，这些乘法的顺序和数量不详。但是，如果我们假设 M = S * R *牛逼，我们可以分解为单独的矩阵。首先让我们来计算 S * R *牛逼：

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)

因为我们知道这是一个2D转换，让翻译很简单：

Since we know it's a 2D transformation, getting translation is straightforward:

translation = vector2D(tx, ty) = vector2D(m41, m42)

要计算旋转和缩放，我们可以使用罪（一）^ 2 + COS（一）^ 2 = 1 ：

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)

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