Future Vision and Trends on Shapes, Geometry and Algebra by Raffaele Amicis & Giuseppe Conti

Author:Raffaele Amicis & Giuseppe Conti
Language: eng
Format: epub
Publisher: Springer London, London

Keywords

Bisector surfaceParametrizationQuadricCramer’s rule

Mathematics Subject Classification (2010) 65D17 68U07

1 Introduction

The (untrimmed) bisector of two smooth surfaces is the set of centers of spheres which are tangent to both surfaces (see Fig. 1). The purpose of this work is to present a new approach to compute the parametrization (rational or non-rational) for the bisector of two low degree rational surfaces, given by their parameterizations. We will give special attention to the case where one of the surfaces is a plane or a cylinder. The bisector surfaces are used in several areas of applications, such as tool path generation, motion planning, NC-milling, medial axis transform, Voronoi diagram computation, etc.

Most of the known methods for computing the exact description of the bisectors are devised only for those bisectors possessing rational parameterizations. Various approaches are appropriately used for very special cases to determine a rational representation for the bisector (see for example [3]). In other cases, symmetry considerations reduce the bisector computation to the following cases: point-line, point-surface, or curve-surface [4, 5], where the bisector is a rational surface.

A PN-surface is a surface admitting a parametrization such that the norm of the normal vector is rational. Using Laguerre geometry, Martin Peternell [10, 11] has shown the rationality of the bisector between: plane and PN-surfaces, two PN developable surfaces, two canal surfaces and some other cases.

Fig. 1Schematic illustration of the bisector definition

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.