Transform Matrix into component pieces

Question

Given an affine 2D transform matrix such as:

[a  b  tx]
[c  d  ty]
[0  0  1 ]

• For a clockwise rotation about the origin, a is transformed by cos (θ) and b is transformed by sin (θ)

• For a scaleX of scaleFactor sx, a is transformed by sx

• For a shear parallel to the x axis, x' = x + ky b is transformed by k

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In my example, a was transformed twice, by the rotation and the scale-x, b was transformed twice, once by the rotation, once by the shear.

Rotation is no longer just arcsin(b)

ScaleX is no longer just 1 / a

ShearX is no longer just x - ky

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How can I get the values of rotation, shearX, and scaleX back from that matrix?

Answer 1

So rotation matrix (full) will be ( I leave out the boring part)

R=
a=cos(θ)  c=sin(θ)
b=-sin(θ) d=cos(θ)

while scale and shear matrix will be (again, leaving out the boring part)

S=
a=s  b=k 
c=0  d=1

Now applying FIRST rotation (R), THEN scale and shear (S) will just be multiplying the matrices, which gives resulting matrix

S times R
a=s cos(θ) - k sin(θ)   b=s sin(θ)+k cos(θ)
c=-sin(theta)    d=cos(theta)

If you would want to get back θ, s and k from that, you can determine θ =arcsin(-c). You know sin(θ) and cos(θ), so you can solve two linear equations (a=s cos(θ) - k sin(θ) b=s sin(θ)+k cos(θ)) with two unknowns to find s and k.