A class of cubic Bézier-type curves based on the blending of algebraic and trigonometric polynomials, briefly CAT Bézier curves, is presented in this paper. The CAT Bézier curves retain the main superiority of cubic Bézier curves. The CAT Bézier curves can approximate the Bézier curves from the both sides, and the shapes of the curves can be adjusted totally or locally. With the shape parameter...

The paper describes a method to compute a basis of mutually orthogonal polynomials with respect to an arbitrary Jacobi weight on the simplex. This construction takes place entirely in terms of the coefficients with respect to the so–called Bernstein–Bézier form of a polynomial.

This paper uses the symmetry properties of circles and Bernstein polynomials to establish a series of interesting barycentric properties of rational biquadratic Bézier patches. A robust algorithm is presented, based on these properties, for the conversion of Dupin cyclide patches into Bézier form. A set of conversion examples illustrates the use of this algorithm.

We prove that the (logarithmic) Mahler measure m(P ) of P (x, y) = x + 1/x + y + 1/y + 3 is equal to the L-value 2L′(E, 0) attached to the elliptic curve E : P (x, y) = 0 of conductor 21. In order to do this we investigate the measure of a more general Laurent polynomial Pa,b,c(x, y) = a ( x + 1 x ) + b ( y + 1 y ) + c and show that the wanted quantity m(P ) is related to a “half-Mahler” measur...

This paper investigates the convergence condition for the polynomial approximation of rational functions and rational curves. The main result, based on a hybrid expression of rational functions (or curves), is that two-point Hermite interpolation converges if all eigenvalue moduli of a certain r_r matrix are less than 2, where r is the degree of the rational function (or curve), and where the e...

We consider Markov decision processes (MDPs) with multiple long-run average objectives. Such MDPs occur in design problems where one wishes to simultaneously optimize several criteria, for example, latency and power. The possible trade-offs between the different objectives are characterized by the Pareto curve. We show that every Pareto optimal point can be ε-approximated by a memoryless strate...

A new method is presented for identifying rigid motion of free-form curves based on "related-points" extracted from the decomposition of implicit polynomials of these curves. Polynomial decomposition expresses the curve as a unique sum of products of (possibly) complex lines. We show that each real intersection point of these lines, i.e. relatedpoints, undergoes the same motion with the curve, ...

For a given high-speed machinery, a significant source of the internally induced vibrational excitation is the presence of high frequency harmonics in the trajectories that the system is forced to follow. In this paper a special class of rational Brzier curves is presented that correspond to low-harmonic trajectory patterns. Harmonic Bernstein polynomials and harmonic deCasteljau algorithm are ...

In this paper we show that the best constrained degree reduction of a given Bézier curve f of degree from n to m with Cα−1-continuity at the boundary in L2-norm is equivalent to the best weighted Euclidean approximation of the vector of Bernstein–Bézier (BB) coefficients of f from the vector of degree raised BB coefficients of polynomials of degree m with Cα−1-continuity at the boundary.  2003...

Abstract. The author illustrates several results from the theory of elliptic curves, as well as the theory of chip-firing games on graphs. More specifically, in both of these cases, we obtain analogues of cyclotomic polynomials with several combinatorial and number theoretic properties. We also provide an analysis of zeta functions which highlights the connections between these two disparate fi...