Bisectors and Voronoï Diagram of a Family of Parallel Half-Lines

Abstract

AbstractThis work has three main parts: the bisectors of curves in the plane, the bisectors of surfaces in \(\mathbb{R}^{3}\), and the Voronoï diagram of a finite family of parallel half-lines in \(\mathbb{R}^{3}\). These subjects are closely related, and have applications in CAD/CAGD and Computational Geometry.We present a new approach to determine, using Cramer’s rule and suitable elimination steps, an exact algebraic parameterization (rational or non rational) of the bisector curve of two given planar rational curves. The approach is, then, generalized by using the generalized Cramer’s rule to determine an exact algebraic parameterization of the bisector surface of two low degree rational surfaces. We apply the method to obtain parametrizations of the bisector of two rational plane curves, when one of them is a circle or a line. On the other hand, we show how to easily obtain parametrizations of the bisector of the following pairs of surfaces: plane-quadric, plane-torus, circular cylinder-non developable quadric, circular cylinder-torus, cylinder-cylinder, cylinder-cone and cone-cone. These parametrizations are rational in most cases. In the remaining cases the parametrization involves one square root which is well-suited for a good approximation of the bisector.In addition, a different approach for the bisector curve problem will be presented. This new method uses dynamic color in GeoGebra for the computation of a geometric and numerical characterization of the bisector of two planar geometric objects (two curves, or a curve and a point). Even if it does not provide an algebraic representation, the method could lead to the computation of an approximate representation of the bisector curve.The Voronoï diagram (VD) is a fundamental data structure in computational geometry with various applications in theoretical and practical areas. We consider the VD of a set of parallel half-lines constrained to a compact domain \(\mathcal{D}_{0} \subset \mathbb{R}^{3}\), with respect to the Euclidean distance. This new kind of VD can be applied to the solution of some problems in the drilling industry, such as hydraulic or mining. We present an efficient approximate algorithm for computing such VD, using a box subdivision process, which produces a mesh representing the topology of the VD.KeywordsComputational GeometryInterval ArithmeticRational CurfRational SurfaceCompact DomainThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.