IMProVe 2011 - Proceedings

Geometric Modelling and Analysis
Shape optimization of smooth surfaces with arbitrary topology
P. Kiciak (a)
(a) Institut Matematyki Stosowanej i Mechaniki, Uniwersytet Warszawski
Abstract:
A popular method of constructing a smooth surface with arbitrary topology is choosing an
irregular mesh and applying iteratively the Catmull-Clark algorithm. The sequence of
meshes obtained in this way converges to a surface, whose curvature is continuous,
except in a vicinity of special mesh elements; apart from the curvature discontinuity there,
the limiting surface exhibits undesirable undulations, visible on curvature images.
In this paper a shape optimization method is described, modifying vertices of the mesh to
produce a surface with the curvature continuous and the undulations significantly
reduced.
Keywords: B-spline surfaces, Curvature continuity, Shape optimization
Corresponding Author: Przemysław Kiciak
Tel.: +48 22 55 44 501
e-mail: przemek@mimuw.edu.pl
Address: Institut Matematyki Stosowanej i Mechaniki, Uniwersytet Warszawski, ul. Banacha 2, 02-
097 Warszawa, Poland.
Characteristics of conic segments in Bézier form
J. Sánchez Reyes (a)
(a) IMACI, ETS Ingenieros Industriales de Ciudad Real. Universidad de Castilla-La Mancha (Spain)
Abstract:
The rational Bézier form has become a standard in CAD-CAM packages and data exchange
formats, because it encompasses both conic segments (in the quadratic case) and general
free-form geometry. We present several results on the relationship between the quadratic
rational Bézier form and the classical definition of conics in terms of their characteristics,
such as foci, centre, axis and eccentricity. First, we recall a simple geometric procedure to
compute arbitrary conic segments of given focus in Bézier form. Second, from this
procedure we derive the geometric characteristics of a given Bézier conic in a
straightforward manner, by employing complex arithmetic. For a central conic, a simple
quadratic equation defines the foci location, and its solution furnishes not only an explicit
formula for the foci, but also for the centre, axis direction and linear eccentricity.
Keywords: rational Bézier, conic, focus, axis, eccentricity
Corresponding Author: Javier Sánchez Reyes
Tel.: +34 926-295463
June 15 th – 17 th , 2011, Venice, Italy
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IMProVe 2011 - Proceedings