By Bernard Mourrain, Scott Schaefer, Guoliang Xu
This e-book constitutes the refereed court cases of the sixth foreign 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 sector of geometric modeling and processing and handle issues resembling recommendations of transcendental equations; quantity parameterization; soft curves and surfaces; isogeometric research; implicit surfaces; and computational geometry.
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Additional info for Advances in Geometric Modeling and Processing: 6th International Conference, GMP 2010, Castro Urdiales, Spain, June 16-18, 2010, Proceedings
We propose a contouring method that can be eﬃciently implemented on the GPU to reduce the artifacts and jaggedness along the material boundaries. Our method extends naturally from the standard tri-linear contouring in a signed volume, and further provides sub-voxel accuracy for representing three or more materials. 1 Introduction Many scientiﬁc modeling applications require the ability to model objects composed of multiple materials. Probably the most common examples are found in bio-medicine, where researchers are often interested in the decomposition of a biological structure, obtained by imaging techniques like MRI or EM, into individual function units.
Each solution corresponds to one orientation of T and we choose one of them by fixing the normal (1, 0) at the point (0, 0) . After simplification of the corresponding solution using n21 + n22 = 1 we obtain the result (5). See  for more details about support functions obtained as restrictions of rational functions. For later reference purposes we substitute the parameterization n = (cos θ, sin θ) of the unit circle into (5) obtaining h(θ) = (1 − cos θ)(2 + cos θ) . e. the normal at T (θ) is simply n(θ) = (cos θ, sin θ) .
If u2 + v2 = 1, then P(u, v) lies on the reference circle C and α is the angle between the xy-plane and the tangent plane of the surface P at this point. Proof. We consider a surface (10). In the ﬁrst case, the unique sphere which passes through the reference circle and through the point P(u, v) has the center C = (0, 0, (q 2 − 1)/(2q)) and the radius r = (q 2 + 1)/(2|q|). The oriented angle α between the sphere and the xy-plane is equal to the angle between the vectors 2q (C−A1 ) and (0, 0, 1) , which gives tan α = 1−q 2 .