<p>Toric patch is a kind of rational multisided patch, which is associated with a finite integer lattice points set <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\cal{A}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="script">A</mi> </mrow> </math></EquationSource> </InlineEquation>. A set of weights is defined which depend on a parameter according to regular decomposition of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\cal{A}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="script">A</mi> </mrow> </math></EquationSource> </InlineEquation>. When all weights of the patch tend to infinity, we obtain the limiting form of toric patch which is called its regular control surface. The different weights may induce the different regular control surfaces of the same toric patch. It prompts us to consider that how many regular control surfaces of a toric patch. In this paper, we study the regular decompositions of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\cal{A}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="script">A</mi> </mrow> </math></EquationSource> </InlineEquation> by using integer programming method firstly, and then provide the relationship between all regular decompositions of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\cal{A}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="script">A</mi> </mrow> </math></EquationSource> </InlineEquation> and corresponding state polytope. Moreover, we present that the number of regular control surfaces of a toric patch associated with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\cal{A}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="script">A</mi> </mrow> </math></EquationSource> </InlineEquation> is equal to the number of regular decompositions of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\cal{A}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="script">A</mi> </mrow> </math></EquationSource> </InlineEquation>. An algorithm to calculate the number of regular control surfaces of toric patch is provided. The algorithm also presents a method to construct all of the regular control surfaces of a toric patch. At last, the application of proposed result in shape deformation is demonstrated by several examples.</p>

Regular control surfaces of a toric patch and integer programming

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发表日期：

2025年12月1日

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Han Wang,
Chun-gang Zhu

摘要

Toric patch is a kind of rational multisided patch, which is associated with a finite integer lattice points set $\cal{A}$ $A$ . A set of weights is defined which depend on a parameter according to regular decomposition of $\cal{A}$ $A$ . When all weights of the patch tend to infinity, we obtain the limiting form of toric patch which is called its regular control surface. The different weights may induce the different regular control surfaces of the same toric patch. It prompts us to consider that how many regular control surfaces of a toric patch. In this paper, we study the regular decompositions of $\cal{A}$ $A$ by using integer programming method firstly, and then provide the relationship between all regular decompositions of $\cal{A}$ $A$ and corresponding state polytope. Moreover, we present that the number of regular control surfaces of a toric patch associated with $\cal{A}$ $A$ is equal to the number of regular decompositions of $\cal{A}$ $A$ . An algorithm to calculate the number of regular control surfaces of toric patch is provided. The algorithm also presents a method to construct all of the regular control surfaces of a toric patch. At last, the application of proposed result in shape deformation is demonstrated by several examples.