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Bisectors and α-Sectors of Rational Varieties [chapter]

G. Elber, G. Barequet, M. S. Kim
2001 Geometric Modelling  
We also describe the a-sector, a generalization of the bisector, and consider a few interesting cases where a-sectors become quadratic curves or surfaces.  ...  This suggests the pseudo a-sector which approximates the a-sector with a rational variety. Both the exact and the pseudo a-sectors identify with the bisector when a 1a2.  ...  Acknowledgements The authors are grateful for the anonymous reviewer who pointed us at the classi®cation of line-line a-sectors and bisectors; Chasles, Journal de Math 1, 1836; Schoen¯ies, Zeitschrift  ... 
doi:10.1007/978-3-7091-6270-5_5 dblp:conf/gm/ElberBK99 fatcat:j63qxzsqkvctdnie3zbtovlnz4

Computing rational bisectors

G. Elber, Myung-Soo Kim
1999 IEEE Computer Graphics and Applications  
Bisector construction plays an important role in many geometric computations. This article explains how to compute rational bisectors of point-surface and spheresurface pairs.  ...  Acknowledgments We are thankful to Martin Peternell and Helmut Pottmann for providing us with their Maple program for constructing rational canal surfaces (and rational offsets also).  ...  This research was supported in part by Korea Science and Engineering Foundation (Kosef) under Grant 96-0100-01-01-2 and by the Fund for Promotion of Research at the Technion, Haifa, Israel.  ... 
doi:10.1109/38.799747 fatcat:5retp4ufyjeybmuyjtdex3lazi

Piecewise rational approximation of square-root parameterizable curves using the Weierstrass form

Michal Bizzarri, Miroslav Lávička, Jan Vršek
2017 Computer Aided Geometric Design  
We design a simple algorithm for computing an approximate (piecewise) rational parametrization using topological graphs of the Weierstrass curves.  ...  Our goal is not to give a comprehensive list of all such operations but to present at least selected interesting cases originated in geometric modelling and to show a unifying feature of the formulated  ...  We thank to all referees for their valuable comments, which helped us to improve the paper.  ... 
doi:10.1016/j.cagd.2017.08.001 fatcat:ppcxisy5jjchrozoxx46t345zi

The Intersection of Two Ringed Surfaces and Some Related Problems

Hee-Seok Heo, Sung Je Hong, Joon-Kyung Seong, Myung-Soo Kim, Gershon Elber
2001 Graphical Models  
We present an efficient and robust algorithm to compute the intersection curve of two ringed surfaces, each being the sweep ∪ u C u generated by a moving circle.  ...  A similar approach produces an efficient algorithm for the intersection of a ringed surface and a ruled surface, which can play an important role in accelerating the ray-tracing of ringed surfaces.  ...  All the algorithms and figures presented in this paper were implemented and generated using the IRIT solid modeling system [1] ,  ... 
doi:10.1006/gmod.2001.0553 fatcat:vnoieavetbe6jhx5qg5i7av5ja

Symmetries of Canal Surfaces and Dupin Cyclides [article]

Juan Gerardo Alcázar, Heidi E. I. Dahl, Georg Muntingh
2017 arXiv   pre-print
We develop a characterization for the existence of symmetries of canal surfaces defined by a rational spine curve and rational radius function.  ...  In turn, this characterization inspires an algorithm for computing the symmetries of such canal surfaces.  ...  Canal surfaces, and Dupin cyclides as a distinguished subfamily, form a class of surfaces also used in Geometric Modeling.  ... 
arXiv:1611.06768v2 fatcat:ncktkazjhngd3lfaqtpn4hwagy

Foundations of Dynamic Geometry

Ulrich Kortenkamp
2000 Journal für Mathematik-Didaktik  
A second important element of the new geometry was the theory of poles and polars for quadric curves and surfaces, leading to a general theory of duality.  ...  The locus that is generated by animating the linkage can be created by building a Lego model that draws the curve.  ...  Java An interpreted, platform independent computer language created by James Gosling (Sun Microsystems) [20] . Can be used inside web pages, but also for standalone applications.  ... 
doi:10.1007/bf03338916 fatcat:pqnhh26dyrfkvpyuq646jepeji

Structural optimization of grid shells based on genetic algorithms [article]

Milos Dimcic, Universität Stuttgart, Universität Stuttgart
Within this research, design and static analysis software are combined in order to perform a statical optimization of grid shells, generated over a given free form surface.  ...  A plug-in for Rhinoceros 3D (software based on NURBS geometry representation) is developed, that uses Genetic Algorithms as an optimization method and implements automated iterative calls to Oasys GSA  ...  We will start with the parametric definition of a curve, and see how that parameter (u for curve and u, v for surface) remains as a constant factor in all functions, while they are expanded with the addition  ... 
doi:10.18419/opus-81 fatcat:s2usdrsxcza3rkxaj3yj23t6x4