By Maxim Aizenshtein, Michael Bartoň (auth.), Bernard Mourrain, Scott Schaefer, Guoliang Xu (eds.)
This quantity includes the papers offered at sixth convention on Geometric Modeling and Processing (GMP 2010) held in Castro Urdiales, Spain in the course of June16–18,2010. GeometricModelingandProcessingisabiannualinternational convention sequence on geometric modeling, simulation and computing. formerly, GMPhasbeenheldinHongKong(2000),Saitama,Japan(2002),Beijing,China (2004), Pittsburgh, united states (2006) and Hangzhou, China (2008). GMP 2010 acquired a complete of 30 submissions that have been reviewed through 3 to 4 application Committee individuals on regular. whereas the variety of subm- sions dropped signi?cantly from prior years, the standard didn't and used to be nonetheless relatively excessive total. in line with the studies bought, the committee determined to - cept 20 papers for inclusion within the court cases. also, prolonged models of chosen papers have been thought of for a distinct factor of Computer-Aided - signal (CAD) and Computer-Aided Geometric layout (CAGD). The paper subject matters spanned a wide selection and contain: – options of transcendental equations – quantity parameterization – delicate curves and surfaces – Isogeometric research – Implicit surfaces – Computational geometry many of us helped make this convention occur and we're thankful for his or her support. we'd specifically wish to thank the convention Chair, the entire authors who submitted papers, the ProgramCommittee participants who reviewed the papers and all the contributors on the conference.
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Extra info for Advances in Geometric Modeling and Processing: 6th International Conference, GMP 2010, Castro Urdiales, Spain, June 16-18, 2010. Proceedings
Given G1 data and the offset distance d, we can easily obtain the corresponding data for the PH cubic by shifting the end points perpendicularly to the end point vectors (angle φ will not change). We will not analyze in details this problem and limit ourselves to one example of data and four Hermite interpolants (two for the left and two for the right offset), see Figure 8, left. 6 Conclusion We have determined the support function representation of the Tschirnhausen cubic and applied it to the computation of PH cubic Hermite interpolants.
2 Preliminaries In this section we will recall some basic facts about Pythagorean Hodograph curves and the support function representation. Recall, that a polynomial planar curve c = (x(t), y(t)) is called Pythagorean-Hodograph curve (PH), if there exists a polynomial σ so that 2 2 x (t) + y (t) = σ 2 (t) (1) see  and citations therein. All polynomial PH curves can be constructed using the following lemma proved for general unique factorization domains by Kubota in . Lemma 1. The condition (1) is satisfied if and only if there exist polynomials u(t), v(t), w(t) such that x (t) = 2u(t)v(t)w(t) and y (t) = [u2 (t) − v 2 (t)]w(t).
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 . The remainder of this paper is organized as follows. In Section 2 we recall some basic facts about PH curves and support function. In Section 3 we determine the support function and suitable parameterization of the Tschirnhausen cubic.
Advances in Geometric Modeling and Processing: 6th International Conference, GMP 2010, Castro Urdiales, Spain, June 16-18, 2010. Proceedings by Maxim Aizenshtein, Michael Bartoň (auth.), Bernard Mourrain, Scott Schaefer, Guoliang Xu (eds.)