New PDF release: Advances in Geometric Modeling and Processing: 6th
By Bernard Mourrain, Scott Schaefer, Guoliang Xu
This booklet 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 offered have been rigorously reviewed and chosen from a complete of 30 submissions. The papers disguise a large spectrum within the zone of geometric modeling and processing and deal with themes similar to options of transcendental equations; quantity parameterization; tender curves and surfaces; isogeometric research; implicit surfaces; and computational geometry.
Read or Download Advances in Geometric Modeling and Processing: 6th International Conference, GMP 2010, Castro Urdiales, Spain, June 16-18, 2010, Proceedings PDF
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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
In fact, this curve is a typical representative of PH cubics as shown in the following lemma which is proved in . Lemma 2. Any (segment of) PH cubic can be obtained from (a segment of) the Tschirnhausen cubic via scaling, rotation, translation and linear reparameterization. For an (oriented) planar curve c we define its support function h as (possibly multivalued) function defined on the (subset of the) unit circle h : S 1 ⊃ U → R1 Support Function of Pythagorean Hodograph Cubics v 31 V w ω u β U Fig.
By Deﬁnition 1, the multimaterial contours deﬁned within each cell are piecewise contours of various trilinear functions. The contours are also continuous across neighboring cells. This fact follows from the observation that two cells sharing a common grid point, edge or face have the same scalars and material labels on that common grid element. Since the restriction of the tri-linear functions used in deﬁning our multi-material contour on each grid point, edge or face depend only the scalar and material labels on that grid element, the multi-material contours must agree across adjacent grid elements.
The interpolation algorithm can be simply extended to offsets of PH cubics. This problem naturally occurs when we want to produce certain shape with a circular tool and we want the center of the tool to follow a PH spline curve permitting to simply control its speed. 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.
Advances in Geometric Modeling and Processing: 6th International Conference, GMP 2010, Castro Urdiales, Spain, June 16-18, 2010, Proceedings by Bernard Mourrain, Scott Schaefer, Guoliang Xu