matrixgraphicslinear-algebratransformationrotational-matrices

Shear Matrix as a combination of basic transformation?

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!

Solution

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

    1. Rotate by theta/2 counter-clockwise.

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

    3. Rotate by 45 degrees clockwise.

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