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.
翻译:计算机图形、计算机赌博、机器人路径规划和有限元素模拟中广泛利用了特定多面曲线的外表。B\'ezier曲线(使用伯恩斯坦基)或B-Splines是这些多面图的一个非常常见的选择,因为其非常态和单线性能保证曲线的每一个间隔都包含在其控制点的圆柱体内。然而,这些基地提供的圆柱体不是体积最小的船体,因此在上述所有应用中产生不可取的保守程度。本文展示了MINO基础,一个多面基,它产生最小的美元块状,其中附有任何给定的美元(text{th}美元-单线性多面性能曲线)。 美元=3美元的计算结果显示,对于任何给定的 $(text{rd_10__rd___rd__zyrmaxyl曲线), MINOB基底体体体体体体积不能包含一个简单x,其体积分别为2.36美元, 和254.9美元/Sdea 美元,这些基数比获得的基数小2美元。