By Bernard Mourrain, Scott Schaefer, Guoliang Xu
This booklet 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 awarded have been rigorously reviewed and chosen from a complete of 30 submissions. The papers conceal a large spectrum within the sector of geometric modeling and processing and handle subject matters resembling options 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
Under Dual Contouring, we create a vertex within each cell that exhibits a label change, and then form a polygon for each grid edge that exhibits a material label change by connecting the vertices created within the cells sharing that edge. The vertex in a cell should be located closest to the intersection of all the pairwise tri-linear surfaces within the cell. More formally, let M be the set of materials in the cell and let x be a point inside the cell. Consider the function (tk (x) − tj (x))2 E(x) = (4) j=k∈M The minimum of this function describes a point that is close to the intersection of all the surfaces that satisfy tk (x) = tj (x).
E) shows the bi-linear function deﬁned by treating the two-labeled scalars at cell corners as signed scalars, and the function’s zero contour. 1 Defining contours To deﬁne the contour surfaces that partition the space into regions with diﬀerent materials, we ﬁrst consider the dual problem of classifying the material of an arbitrary point in space. Given a grid cell whose corners have associated nonnegative scalars si and material labels mi , the following method can be used to determine the material label of a point x inside the cell.
The values at endpoints are ±β/2 2 sin β 2 and we obtain (12). The extreme values of F can be computed as ± √1+2 cos and we β obtain boundary curves of Ω in (13). Let us denote α = arctan β/2 the left point of the interval where F cease to be proper. The loop condition value αl = β2 − 2π , see (20) will always fall in the interval 3 [α, π − β/2]. For this reason the segments of (12) will always have a loop. The position of αl with respect to α0 leads to various cases of presence of loops for segments in Ω.
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