An algebraic framework for geometrically continuous splines

Geometrically continuous splines are piecewise polynomial functions defined on a collection of patches which are stitched together through transition maps. They are called <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Superscript r"> <mml:semantics> <mml:msup> <mml:mi>G</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>r</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">G^{r}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -splines if, after composition with the transition maps, they are continuously differentiable functions to order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r"> <mml:semantics> <mml:mi>r</mml:mi> <mml:annotation encoding="application/x-tex">r</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on each pair of patches with stitched boundaries. This type of spline has been used to represent smooth shapes with complex topology for which (parametric) spline functions on fixed partitions are not sufficient. In this article, we develop new algebraic tools to analyze <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Superscript r"> <mml:semantics> <mml:msup> <mml:mi>G</mml:mi> <mml:mi>r</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">G^r</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -spline spaces. We define <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Superscript r"> <mml:semantics> <mml:msup> <mml:mi>G</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>r</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">G^{r}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -domains and transition maps using an algebraic approach, and establish an algebraic criterion to determine whether a piecewise function is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Superscript r"> <mml:semantics> <mml:msup> <mml:mi>G</mml:mi> <mml:mi>r</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">G^r</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -continuous on the given domain. In the proposed framework, we construct a chain complex whose top homology is isomorphic to the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Superscript r"> <mml:semantics> <mml:msup> <mml:mi>G</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>r</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">G^{r}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -spline space. This complex generalizes Billera-Schenck-Stillman homological complex used to study parametric splines. Additionally, we show how previous constructions of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Superscript r"> <mml:semantics> <mml:msup> <mml:mi>G</mml:mi> <mml:mi>r</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">G^r</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -splines fit into this new algebraic framework, and present an algorithm to construct a basis for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Superscript r"> <mml:semantics> <mml:msup> <mml:mi>G</mml:mi> <mml:mi>r</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">G^r</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -spline spaces. We illustrate how our algebraic approach works with concrete examples and prove a dimension formula for the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Superscript r"> <mml:semantics> <mml:msup> <mml:mi>G</mml:mi> <mml:mi>r</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">G^r</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -spline space in terms of invariants to the chain complex. In some special cases, explicit dimension formulas in terms of the degree of splines are also given.