How to calculate a grid based on 4 curved edges?

Answer 1

You can use a parametric representation to map points from a unit square with corners [1, 1], [-1, 1], [-1, -1], [1, -1] onto the Bezier quad by transforming from one coordinate system to the other by calculating the Jacobean from one to the other.

This article shows you how. It's a common thing to do in finite element analysis, because most modern elements are expressed in terms of shape functions in a parametric space. You have to transform from the parametric to element coordinates using the techniques shown in that article.