Determine the dimension of bivariate C1 spline spaces

Determine the dimension of the space of continuously differentiable bivariate splines of degree at most k over a polygonal domain without holes, thereby resolving the general dimension problem motivating Strang’s conjecture.

Background

The paper studies the dimension of the space of continuously differentiable bivariate splines over a triangulated polygonal domain. This dimension is important in applications including computer-aided geometric design, algebraic geometry, and finite element methods.

Strang’s conjecture provides a combinatorial formula for this dimension, with an additional correction term accounting for singular vertices. The paper establishes the conjecture for quadratic splines on a class of strongly collapsible triangulations and provides computational tools for testing it on individual meshes, but does not resolve the general dimension problem.