B\'ezier representation of the constrained dual Bernstein polynomials

Explicit formulae for the B\'ezier coefficients of the constrained dual Bernstein basis polynomials are derived in terms of the Hahn orthogonal polynomials. Using difference properties of the latter polynomials, efficient recursive scheme is obtained to compute these coefficients. Applications of this result to some problems of CAGD is discussed.Comment: 10 page

The computation of multiple roots of a Bernstein basis polynomial

This paper describes the algorithms of Musser and Gauss for the computation of multiple roots of a theoretically exact Bernstein basis polynomial ˆ 5 f(y) when the coefficients of its given form f(y) are corrupted by noise. The exact roots of f(y) can therefore be assumed to be simple, and thus the problem reduces to the calculation of multiple roots of a polynomial f˜(y) that is near f(y), such that the backward error is small. The algorithms require many greatest common divisor (GCD) computations and polynomial deconvolutions, both of which are implemented by a structure-preserving matrix method. The motivation of these algorithms arises from the unstructured and structured condition numbers of a multiple root of a polynomial. These condition numbers have an elegant interpretation in terms of the pejorative manifold of ˆ 12 f(y), which allows the geometric significance of the GCD computations and polynomial deconvolutions to be considered. A variant of the Sylvester resultant matrix is used for the GCD computations because it yields better results than the standard form of this matrix, and the polynomial deconvolutions can be computed in several different ways, sequentially or simultaneously, and with the inclusion or omission of the preservation of the structure of the coefficient matrix. It is shown that Gauss’ algorithm yields better results than Musser’s algorithm, and the reason for these superior results is explained

Compact Parameterized Black-Box Modeling via Fourier-Rational Approximations

We present a novel black-box modeling approach for frequency responses that depend on additional parameters with periodic behavior. The methodology is appropriate for representing with compact low-order equivalent models the behavior of electromagnetic systems observed at well-defined ports and/or locations, including dependence on geometrical parameters with rotational symmetry. Examples can be polarization or incidence angles of a plane wave, or stirrer rotation in reverberation chambers. The proposed approach is based on fitting a Fourier-rational model to sampled frequency responses, where frequency dependence is represented through rational functions and parameter dependence through a Fourier series. Several examples from different applications are used to validate and demonstrate the approach

Adaptive isocurves based rendering for freeform surfaces

technical reportFreeform surface rendering is traditionally performed by approximating the surface with polygons and then rendering the polygons This approach is extremely common because of the complexity in accurately rendering the surfaces directly Recently?? several papers presented methods to render surfaces as sequences of isocurves Unfortunately?? these methods start by assuming that an appropriate collection of isocurves has already been derived The algorithms themselves neither automatically create an optimal or almost optimal set of isocurves so the whole surface would be correctly rendered without having pixels redundantly visited nor automatically compute the parameter spacing required between isocurves to guarantee such coverage In this paper?? a new algorithm is developed to ll these needs An algorithm is introduced that automat ically computes a set of almost optimal isocurves covering the entire surface area This algorithm can be combined with a fast curve rendering method?? to make surface rendering without polygonal approximation practica

Continuous collision detection for ellipsoids

We present an accurate and efficient algorithm for continuous collision detection between two moving ellipsoids. We start with a highly optimized implementation of interference testing between two stationary ellipsoids based on an algebraic condition described in terms of the signs of roots of the characteristic equation of two ellipsoids. Then we derive a time-dependent characteristic equation for two moving ellipsoids, which enables us to develop a real-time algorithm for computing the time intervals in which two moving ellipsoids collide. The effectiveness of our approach is demonstrated with several practical examples. © 2006 IEEE.published_or_final_versio

Adaptive isocurves based rendering for freeform surfaces

Journal ArticleFreeform surface rendering is traditionally performed by approximating the surface with polygons and then rendering the polygons. This approach is extremely common because of the complexity in accurately rendering the surfaces directly. Recently, several papers presented methods to render surfaces as sequences of isocurves. Unfortunately, these methods start by assuming that an appropriate collection of isocurves has already been derived. The algorithms themselves neither automatically create an optimal or almost optimal set of isocurves so t h e whole surface would be correctly rendered without having pixels redundantly visited nor automatically compute the parameter spacing required between isocurves to guarantee such coverage. In this paper, a new algorithm is developed to fill these needs. An algorithm is introduced that automatically computes a set of almost optimal isocurves covering the entire surface area. This algorithm can be combined with a fast curve rendering method, to make surface rendering without polygonal approximation practical