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 applicationsmentioned above. This paper presents the MINVO basis, a polynomial basis that generates the smallest n-simplex that encloses any given 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\inN and prove numerical global optimality for n=1,2,3. } The results obtained for n=3 show that, for any given 3^{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.
This paper studies the problem of finding the smallest n-simplex enclosing a given n^{\text{th by these bases are not the smallest possible, which leads to undesirably conservative results in many applications . We first prove that the polynomial basis that solves this problem (MINVO basis) also solves for the n^\text{th 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\inN and prove numerical global optimality for n=1,2,3. } The results obtained for n=3 show that, for any given 3^{rd} -degree 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.
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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
This paper studies the problem of finding the smallest n-simplex enclosing a given n^{\text{th
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is not the one with smallest volume, producing therefore undesirable levels of conservatism in all of the applicationsmentioned above. This paper presents the MINVO basis, a polynomial basis that generates the
are not the
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n-simplex that encloses any given n^\text{th
possible, which leads to undesirably conservative results in many applications
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We first prove that the polynomial basis that solves this problem (MINVO basis) also solves for the n^\text{th
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independent
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of the n-simplex or n^{\text{th
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-order
-degree