CAGD Handbook

References (p. 23) 22. Index (p. 25) This chapter promotes, details and exploits the fact that (univariate) splines, i.e., smooth piecewise polynomial functions, are weighted sums of B-splines. 1. Piecewise polynomials A piecewise polynomial of order k with break sequence # # # # # (necessarily strictly increasing) is, by definition, any function f that, on each of the half-open intervals [# j . . # j+1 ), agrees with some polynomial of degree < k. The term `order' used here is not standard but handy. Note that this definition makes a piecewise polynomial function right-continuous, meaning that, for any x, f(x) = f(x+) := lim h#0 f(x+h). This choice is arbitrary, but has become standard. Keep in mind that, at its break # j , the piecewise polynomial function f has, in e#ect, two values, namely its limit from the left, f(# j -), and its limit from the right, f(# j +) = f(# j ). The set of all piecewise polynomial functions of order k with break sequence # # # # # is denote

Computing the Radius of Pointedness of a

Abstract. Let Ξ(H) denote the set of all nonzero closed convex cones in a finite dimensional Hilbert space H. Consider this set equipped with the bounded Pompeiu-Hausdorff metric δ. The collection of all pointed cones forms an open set in the metric space (Ξ(H), δ). One possible way of measuring the degree of pointedness of a cone K is by evaluating the distance from K to the set of all nonpointed cones. The number ρ(K) obtained in this way is called the radius of pointedness of the cone K. The evaluation of this number is, in general, a very cumbersome task. In this note, we derive a simple formula for computing ρ(K), and we propose also a method for constructing a nonpointed cone at minimal distance from K. Our results apply to any cone K whose maximal angle does not exceed 120 degrees. 1

Calculating Voronoi Diagrams using

Introduction The Voronoi diagram is an important data structure in computational geometry. Given n sites in the plane, the Voronoi diagram partitions the plane into n regions. The region of a site p consists of all those points that lie closer to p than to any of the other sites. For a survey on Voronoi diagrams and their applications we refer to Aurenhammer [1]. A generalization of the sweep line method of Fortune [2] was developed in [4] which will be presented here. To achieve this generalization, the sweep line was replaced by a general sweep curve. In order to look for sweep curves that are feasable for such an algorithm, equidistant curves were introduced. It turned out that there are two useful forms of sweep curves for the Euclidian plane, lines and circles. As a further result, a sweep algorithm based on sweep circles was implemented; this algorithms runs with O(n log n) time and O(n) space requirements, which is optimal. 2 Sweep Curves When sweeping with a curve acros

Blind Equalization Of Constant Modulus Signals Via Restricted

In this paper, we formulate the blind equalization of Constant Modulus (CM) signals as a convex optimization problem. This is done by performing an algebraic transformation on the direct formulation of the equalization problem and then restricting the set of design variables to a subset of the original feasible set. In particular, we express the blind equalization problem as a linear objective function subject to some linear and semidefiniteness constraints. Such Semidefinite Programs (SDPs) can be efficiently solved using interior point methods. Simulations indicate that our method performs better than the standard methods, whilst requiring significantly fewer data samples