Fast and Simple Methods For Computing Control Points

The purpose of this paper is to present simple and fast methods for computing control points for polynomial curves and polynomial surfaces given explicitly in terms of polynomials (written as sums of monomials). We give recurrence formulae w.r.t. arbitrary affine frames. As a corollary, it is amusing that we can also give closed-form expressions in the case of the frame (r, s) for curves, and the frame ((1, 0, 0), (0, 1, 0), (0, 0, 1) for surfaces. Our methods have the same low polynomial (time and space) complexity as the other best known algorithms, and are very easy to implement.Comment: 15 page

The apolar bilinear form in geometric modeling

Some recent methods of Computer Aided Geometric Design are related to the apolar bilinear form, an inner product on the space of homogeneous multivariate polynomials of a fixed degree, already known in 19th century invariant theory. Using a generalized version of this inner product, we derive in a straightforward way some of the recent results in CAGD, like Marsden's identity, the expression for the de Boor-Fix functionals, and recursion schemes for the computation of B-patches and their derivatives

Approximate Reachability Computation for Polynomial Systems

Abstract. In this paper we propose an algorithm for approximating the reachable sets of systems defined by polynomial differential equations. Such systems can be used to model a variety of physical phenomena. We first derive an integration scheme that approximates the state reachable in one time step by applying some polynomial map to the current state. In order to use this scheme to compute all the states reachable by the system starting from some initial set, we then consider the problem of computing the image of a set by a multivariate polynomial. We propose a method to do so using the Bézier control net of the polynomial map and the blossoming technique to compute this control net. We also prove that our overall method is of order 2. In addition, we have successfully applied our reachability algorithm to two models of a biological system.

Change of Basis Algorithms for Surfaces in CAGD

The computational complexity of general change of basis algorithms from one bivariate polynomial basis of degree n to another bivariate polynomial basis of degree n using matrix multiplication is O(n 4 ). Applying blossoming and duality, we derive change of basis algorithms with computational complexity O(n 3 ) between two important classes of polynomial bases used for representing surfaces in CAGD: Bbases and L-bases. Change of basis algorithms for B-bases follow from their blossoming property; change of basis algorithms for L-bases follow from the duality between L-bases and B-bases. The B'ezier and multinomial bases are special cases of both B-bases and L-bases, so these algorithms can be used to convert between the B'ezier and multinomial forms. We also show that the bivariate Horner evaluation algorithm for the multinomial basis is dual to the bivariate de Boor evaluation algorithm for B-patches. Keywords: algorithms, blossoming, CAGD, change of basis, de Boor evaluation, de..