By Bernard Mourrain, Scott Schaefer, Guoliang Xu
This e-book 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 provided have been conscientiously reviewed and chosen from a complete of 30 submissions. The papers conceal a large spectrum within the sector of geometric modeling and processing and tackle issues similar to ideas of transcendental equations; quantity parameterization; gentle curves and surfaces; isogeometric research; implicit surfaces; and computational geometry.
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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
E. we are given two points U, V along with → → unit tangent vectors − u and − v and we want to find a PH cubic which interpolates these data. Up to similarities these data are described by two angles φ ∈ (−π, π), ψ ∈ [−π, π], −−→ → → where φ is the oriented angle between − u and − v and ψ the oriented angle between U V → → → → and the bisector (− u +− v )/2 (see Fig. 1). For simplicity we exclude the case − u = −− v which allows for no interpolants. Suppose we have an interpolant of the data by a segment of the Tschirnhausen cubic.
Moreover the y-component of the vec−−→ tor U V is in this case always positive and therefore ω = arctan(F ) yielding (11) for β ∈ (0, π). 36 ˇ ˇ ır E. Cernohorsk´ a and Z. S´ Case β ∈ [π, 4π ): 3 In this case ω is again decreasing function of α, see Fig. 4, lower row. In this case, α ¯ ∈ (−π + β2 , π − β2 ) and F has two discontinuities. However, α0 ∈ (−π + β2 , π − β2 ) and F has negative derivative everywhere. The values of F cover all R. Figure 4, upper −−→ row, shows typical behavior of F .
For more implementation details . An example of the Dual Contouring polygonization is shown in Figure 7. 3 Set Operations on Multi-material Contours One of the primary attractions of implicit modeling is the ease with which it can model Boolean operations from constructive solid geometry . To enable interactive construction of multi-material models in our representation, we next develop analogs of the set operations Union and Intersection for multi-material contours. These operations can be applied in several context with regards to interactive segmentation.