What is the significance of 4/3rds and 4/5ths here? Are they just arbitrary parameters greater than and less than the mean edge length, chosen by 'prettiness,' as we've seen before, or are they in some way special values?
I remember the discussion about the need to resample. Even though the mesh on the left better approximates the geometry, the one on the right has a smooth flow which is what is needed for finite element simulation.
For example when performing automotive crash simulation using finite element, a mesh on the left would typically take much more time for the solution to converge as compared to mesh on the right. Sometimes, the solution might not even converge with the mesh on the left.
@zbp Yes, they are heuristics; they come from the paper "A Remeshing Approach to Multiresolution Modeling" by Botsch and Kobbelt. (If you implement this scheme for A2, you will see that it works quite well!)
@sjip Right. You can find a more detailed discussion of this trade off in Shewchuk, "What is a Good Linear Element?" which does a careful analysis of the effect of triangle shape on both geometric approximation error as well as conditioning of the Laplace matrix. Mesh quality is also linked to important guarantees about properties of the solution such as the maximum principle, which is discussed in Wardetzky et al, "Discrete Laplace Operators: No Free Lunch"