Papers › MINVO Basis: Finding Simplexes with Minimum Volume Enclosing Polynomial Curves
MINVO Basis: Finding Simplexes with Minimum Volume Enclosing Polynomial Curves
Jesus Tordesillas, Jonathan P. How
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This paper studies the polynomial basis that generates the smallest n-simplex enclosing a given nᵗʰ-degree polynomial curve in ℝⁿ. 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 nᵗʰ-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 nᵗʰ-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∈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 3ʳᵈ-degree polynomial curve in ℝ³, 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 2.997·10²¹, respectively.
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