Shear Matrix as a combination of basic transformation?

matrix graphics linear-algebra transformation rotational-matrices
Kunal S. Kushwah picture Kunal S. Kushwah · Aug 21, 2013 · Viewed 13.3k times · Source

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!

Answer

Kunal S. Kushwah picture Kunal S. Kushwah · Dec 7, 2013

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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