剪切矩阵作为基本变换的组合? [英] Shear Matrix as a combination of basic transformation?

问题描述

我知道旋转、缩放、平移等的变换矩阵.我也知道剪切变换的矩阵.现在，我需要剪切矩阵--

[1 Sx 0][0 1 0][0 0 1]

以其他上述转换的组合形式.尝试过搜索，尝试过头脑风暴，但无法罢工！谢谢！

解决方案

剪切角 theta 的 x 剪切操作减少为旋转和缩放为如下:

(a) 逆时针旋转 theta/2.

(b) 使用 x-scaling factor = sin(theta/2) 和 y-scaling factor = cos(theta/2) 进行缩放.

(c) 顺时针旋转 45 度.

(d) 使用 x-scaling factor = sqrt(2)/sin(theta) 和 y-scaling factor= sqrt(2) 进行缩放.>

I know the transformation matrices for rotation, scaling, translation etc. I also know the matrix for shear transformation. Now, I need to have the shear matrix--

[1 Sx 0]
[0 1  0]
[0 0  1]

in the form of a combination of other aforesaid transformations. Tried searching, tried brainstorming, but unable to strike! Thanks!

解决方案

The x-shear operation for a shearing angle thetareduces to rotations and scaling as follows:

(a) Rotate by theta/2 counter-clockwise.

(b) Scale with x-scaling factor = sin(theta/2) and y-scaling factor = cos(theta/2).

(c) Rotate by 45 degree clockwise.

(d) Scale with x-scaling factor = sqrt(2)/sin(theta) , and y-scaling factor= sqrt(2).

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