This demonstrates the polar decomposition of 2D matrices into their angle, scale, and shear components. The polarDecomp algorithm is taken from Shoemake & Duff “Matrix Animation and Polar Decomposition“, 1992.

I found that the formula for the polar decomposition, on page 3, produces rotation matrices that don’t account for diagonal axis flipping (swapping x with y). The article seems to factor out the flipping, but I didn’t want to have a fourth component. Removing the sign(det(M)) factor eliminated this problem by incorporating the flipping into the rotation matrix.

It’s very desirable to decompose a matrix generated by {angle, scale, shear} back into the same part values, but this happens only when scale is positive and shear is zero. If not, the rotation is affected. At least, the parts are stable once derived from the first matrix. This may help in UI, and certainly in interpolation.

This page is a Mathematica-generated HTML file, with CSS that I tweaked. Unfortunately, it shows Mathematica code as images, to preserve the original appearance. The original Mathematica notebook is here. If you don’t have Mathematica, you can view it using the free Mathematica Player.