Advances in Geometric Modeling and Processing: 6th by Bernard Mourrain, Scott Schaefer, Guoliang Xu

By Bernard Mourrain, Scott Schaefer, Guoliang Xu

This ebook constitutes the refereed court cases of the sixth overseas convention on Geometric Modeling and Processing, GMP 2010, held in Castro Urdiales, Spain, in June 2010. The 20 revised complete papers provided have been rigorously reviewed and chosen from a complete of 30 submissions. The papers hide a large spectrum within the zone of geometric modeling and processing and deal with subject matters equivalent to strategies of transcendental equations; quantity parameterization; delicate curves and surfaces; isogeometric research; implicit surfaces; and computational geometry.

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Mathematical Methods for Curves and Surfaces, pp. 275–286. Nashboro Press (2005) 12. : Complex rational B´ezier curves. Computer Aided Geometric Design 26, 865–876 (2009) 13. : Curves with rational chord-length parametrization. cz Abstract. The Tschirnhausen cubic represents all non-degenerate Pythagorean Hododgraph cubics. We determine its support function and represent it as a convolution of a centrally symmetrical curve and a curve with linear normals. We use the support function to parametrize the Tschirnhausen cubic by normals.

We determine its B. Mourrain, S. Schaefer, and G. ): GMP 2010, LNCS 6130, pp. 29–42, 2010. c Springer-Verlag Berlin Heidelberg 2010 30 ˇ ˇ ır E. Cernohorsk´ a and Z. S´ support function and then fully describe all possible data which can be interpolated and give the number of solutions. We thus solve the interpolation problem in a top-down way and extend results of [14]. The remainder of this paper is organized as follows. In Section 2 we recall some basic facts about PH curves and support function.

Lemma 2. The set of all points Y satisfying ∀(i, j) ∈ {(1, 2), (2, 3), (3, 1)} : i (Y)Rj (X) = j (Y)Ri (X) (7) is a circle which passes through X and is perpendicular to any sphere containing the vertices of the patch. If X lies on the circumcircle of the vertex triangle, then the circle Y shrinks to the single point X. Proof. Recall that for any two points M, N in the plane, the set of all points Z satisfying ||Z − M||2 = c ||Z − N||2 (8) 22 B. Bastl et al. for some positive constant c is a circle (Apollonius’ definition) which intersects any circle through M and N orthogonally.

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