# Polynomial spline spaces of non-uniform bi-degree on T-meshes: Combinatorial bounds on the dimension

## TL;DR

This paper investigates the dimension of polynomial spline spaces on T-meshes with non-uniform bi-degrees, providing combinatorial bounds and conditions for exact dimension calculation, enhancing understanding for geometric design and analysis.

## Contribution

It generalizes previous uniform bi-degree spline frameworks to non-uniform bi-degrees using homological algebra, deriving bounds and conditions for spline space dimension.

## Key findings

- Derived combinatorial bounds on spline space dimension
- Established conditions for bounds to be tight
- Extended framework to non-uniform bi-degree splines

## Abstract

Polynomial splines are ubiquitous in the fields of computer aided geometric design and computational analysis. Splines on T-meshes, especially, have the potential to be incredibly versatile since local mesh adaptivity enables efficient modeling and approximation of local features. Meaningful use of such splines for modeling and approximation requires the construction of a suitable spanning set of linearly independent splines, and a theoretical understanding of the spline space dimension can be a useful tool when assessing possible approaches for building such splines. Here, we provide such a tool. Focusing on T-meshes, we study the dimension of the space of bivariate polynomial splines, and we discuss the general setting where local mesh adaptivity is combined with local polynomial degree adaptivity. The latter allows for the flexibility of choosing non-uniform bi-degrees for the splines, i.e., different bi-degrees on different faces of the T-mesh. In particular, approaching the problem using tools from homo-logical algebra, we generalize the framework and the discourse presented by Mourrain (2014) for uniform bi-degree splines. We derive combinatorial lower and upper bounds on the spline space dimension and subsequently outline sufficient conditions for the bounds to coincide.