{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/minvo-basis-finding-simplexes-with-minimum","title":"MINVO Basis: Finding Simplexes with Minimum Volume Enclosing Polynomial Curves","arxiv_id":"2010.10726","date":"2020-10-21","proceeding":null,"authors":["Jesus Tordesillas","Jonathan P. How"],"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.","url_abs":"https://arxiv.org/abs/2010.10726v5","url_pdf":"https://arxiv.org/pdf/2010.10726v5.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"minvo-basis-finding-simplexes-with-minimum","repo_url":"https://github.com/mit-acl/minvo","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}