Outer polyhedral representations of a given polynomial curve are extensively exploited in computer graphics rendering, computer gaming, path planning for robots, and finite element simulations. B\'ezier curves (which use the Bernstein basis) or B-Splines are a very common choice for these polyhedral representations because their non-negativity and partition-of-unity properties guarantee that each interval of the curve is contained inside the convex hull of its control points. However, the convex hull provided by these bases is not the one with smallest volume, producing therefore undesirable levels of conservatism in all of the applications mentioned above. This paper presents the MINVO basis, a polynomial basis that generates the smallest $n$-simplex that encloses any given $n^\text{th}$-order polynomial curve. The results obtained for $n=3$ show that, for any given $3^{\text{rd}}$-order polynomial curve, 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.

中文翻译：

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MINVO 基础：寻找具有最小体积封闭多项式曲线的单纯形

给定多项式曲线的外多面体表示广泛用于计算机图形渲染、计算机游戏、机器人路径规划和有限元模拟。B\'ezier 曲线（使用 Bernstein 基）或 B-Splines 是这些多面体表示的非常常见的选择，因为它们的非负性和统一性属性保证曲线的每个区间都包含在凸包内其控制点。然而，由这些基提供的凸包并不是体积最小的那个，因此在上述所有应用中都产生了不希望有的保守性水平。本文介绍了 MINVO 基，这是一种多项式基，可生成包含任何给定 $n^\text{th}$ 阶多项式曲线的最小 $n$-单纯形。$n=3$ 获得的结果表明，对于任何给定的 $3^{\text{rd}}$ 阶多项式曲线，MINVO 基能够获得体积为 $2.36$ 和 $254.9$ 倍的封闭单纯形比分别通过 Bernstein 和 B-Spline 基获得的值。当 $n=7$ 时，这些比率分别增加到 $902.7$ 和 $2.997\cdot10^{21}$。