For example, here’s an attempt on a more complex shape.
On the left is the relaxed result, and on the right is the curve used to identify which points to pull to the boundary (essentially the cutting pattern of the mesh).
The more closely this grid aligned cutting pattern matches the target boundary curve, the less variation in spacing there will be in the final result.
Here I’ve used some turns of 30°, so some half triangles. Using only 60° turns can give a calmer look towards the boundary, but since it doesn’t let you approximate the curved shape so closely, there will be greater variation in distance between points.