By Bernard Mourrain, Scott Schaefer, Guoliang Xu
This ebook constitutes the refereed complaints 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 awarded have been rigorously reviewed and chosen from a complete of 30 submissions. The papers hide a large spectrum within the quarter of geometric modeling and processing and deal with issues corresponding to suggestions of transcendental equations; quantity parameterization; soft 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
Proposition 3. Any RCL surface P(u, v) has a rational unit normal field along the reference circle. On the other hand, any rational unit normal field along the reference circle can be extended to an RCL surface. Finally, two RCL surfaces given by (10) with functions q1 , q2 have the same normals along the reference circle if and only if q1 − q2 = (1 − u2 − v2 )f, (20) where f (u, v) is a rational function. Proof. Under the condition u2 + v2 = 1, the unit normal of P can be computed from (10) as 2qu 2qv 1 − q 2 , , .
In the two material case, our method would reproduce the standard trilinear contours deﬁned by signed scalars. In the case of three or more materials, the contouring method would generate piecewise tri-linear contours that form a continuous surface. Our method adds small overhead over the traditional signed volumes, and allows eﬃcient hardware-accelerated rendering. In the following, we ﬁrst introduce the deﬁnition of contours in our volume representation. We next present a number of properties of such contours.
Consider the two equations n · T (t) = −n1 t2 + n2 (t − t3 ) = h and n · T (t) = −n1 2t + n2 (1 − t2 ) = 0, (6) 3 where the former express that T (t) lies on a line and the latter forces its normal vector n to be perpendicular to the tangent vector at the same point T (t). After elimination 32 ˇ ˇ ır E. Cernohorsk´ a and Z. S´ = h= 2 3n2 2 h= n3 1 −3n1 3n2 2 h= n3 1 −3n1 +2 3n2 2 Fig. 2. Tschirnhausen cubic as a convolution of a LN curve and a curve with an even rational support function of t from (6) we get one quadratic equation for h.