Geodesic
Multifocal Surfaces and Algorithms for Displaying Them
Journal for geometry and graphics, Tome 10 (2006) no. 1, pp. 37-62 Cet article a éte moissonné depuis la source Heldermann Verlag

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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.
Classification : 53A05, 68U05, 51M04
Mots-clés : String surfaces, product surfaces, octree algorithm, iso-surfaces, Cassini surfaces, lemniscate, implicit functions, pseudofocus, Fermat's point, radial projection 
@article{JGG_2006_10_1_JGG_2006_10_1_a3,
     author = {W. M. Pieper },
     title = {Multifocal {Surfaces} and {Algorithms} for {Displaying} {Them}},
     journal = {Journal for geometry and graphics},
     pages = {37--62},
     year = {2006},
     volume = {10},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/JGG_2006_10_1_JGG_2006_10_1_a3/}
}
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W. M. Pieper . Multifocal Surfaces and Algorithms for Displaying Them. Journal for geometry and graphics, Tome 10 (2006) no. 1, pp. 37-62. http://geodesic.mathdoc.fr/item/JGG_2006_10_1_JGG_2006_10_1_a3/