MINVO Basis: Finding Simplexes with Minimum Volume Enclosing Polynomial Curves

TL;DR

This paper introduces the MINVO basis, a polynomial basis that generates the smallest enclosing simplexes for polynomial curves in  space, significantly reducing conservativeness in CAD applications.

Contribution

The paper formulates and solves the MINVO basis problem, proving its optimality for small dimensions and providing high-quality solutions for higher dimensions using advanced optimization techniques.

Findings

MINVO basis produces much smaller enclosing simplexes than Bernstein and B-Spline bases.

For 3rd-degree curves in , MINVO reduces volume by factors of 2.36 and 254.9.

For 7th-degree curves, volume ratios increase to 902.7 and 2.99710^{21}.

Abstract

This paper studies the polynomial basis that generates the smallest $n$-simplex enclosing a given $n^{\text{th}}$-degree polynomial curve in $\mathbb{R}^n$. Although the Bernstein and B-Spline polynomial bases provide feasible solutions to this problem, the simplexes obtained by these bases are not the smallest possible, which leads to overly conservative results in many CAD (computer-aided design) applications. We first prove that the polynomial basis that solves this problem (MINVO basis) also solves for the $n^\text{th}$-degree polynomial curve with largest convex hull enclosed in a given $n$-simplex. Then, we present a formulation that is independent of the $n$-simplex or $n^{\text{th}}$-degree polynomial curve given. By using Sum-Of-Squares (SOS) programming, branch and bound, and moment relaxations, we obtain high-quality feasible solutions for any $n\in\mathbb{N}$, and prove (numerical) global optimality for $n=1,2,3$ and (numerical) local optimality for $n=4$. The results obtained for $n=3$ show that, for any given $3^{\text{rd}}$-degree polynomial curve in $\mathbb{R}^3$, the MINVO basis is able to obtain an enclosing simplex whose volume is $2.36$ and $254.9$ times smaller than the ones obtained by the Bernstein and B-Spline bases, respectively. When $n=7$, these ratios increase to $902.7$ and $2.997\cdot10^{21}$, respectively.