How to measure distortion of a parametrization?

Question

If you have a map from a mesh into the 2D plane. How do you measure its distortion? I know that theoretically what you do is express the differential map of the transformation then use SVD to figure out your components and then find the magnitudes of the principal axes.

But how do I even define the differential map on a mesh?

When talking about distortion I mean the metrics of the differential map. i.e. things like the eigen values of the matrix obtained from the matrix representation of the differential map.

https://www.researchgate.net/post/How-can-I-measure-distortion-of-mesh-parameterization

Answer 1

Two non-parallel edges originating from a common vertex are mapped to two corresponding edges. From the two pairs of vectors, you derive a linear transformation, which gives you a local deformation matrix.

[Depending on your mesh topology, you can find more pairs from a given vertex. That will give you slightly different deformation matrices.]

You can get better accuracy by fitting an affine transform on more local vectors, or even with a nonlinear model.