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Journal for Geometry and Graphics 10 (2006), No. 1, 037--062
Copyright Heldermann Verlag 2006

Multifocal Surfaces and Algorithms for Displaying Them

Wilhelm M. Pieper
Fachhochschule Giessen-Friedberg, Wiesenstr. 14, 35390 Giessen, Germany

This article investigates mathematical properties of three-dimensional generalizations of an ellipse (called string surfaces) and a lemniscate (called product surfaces) with more than two foci. They are defined by implicit functions. The singular points of both kinds of surfaces with three foci are calculated analytically. It is explained how the string surfaces can easily be constructed by use of a string. In addition some special cases of surfaces with more than three foci are studied and the transition is made to a continuous distribution of foci. 3D-views of the surfaces with equilateral, linear or isosceles arrangement of the three foci are presented. In Section 4 "axial surfaces" of the form f(x, y) = c(z) are discussed, where the constant c of the 2D-string surfaces f(x, y) is varying with z. Finally a simplification of the polygonization used in the known algorithms for displaying implicit surfaces is described and a new method of radial projection is presented.

Keywords: String surfaces, product surfaces, octree algorithm, iso-surfaces, Cassini surfaces, lemniscate, implicit functions, pseudofocus, Fermat's point, radial projection.

MSC: 53A05; 68U05, 51M04

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