By Bernard Mourrain, Scott Schaefer, Guoliang Xu
This publication constitutes the refereed lawsuits 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 conscientiously reviewed and chosen from a complete of 30 submissions. The papers disguise a large spectrum within the region of geometric modeling and processing and handle issues reminiscent of ideas 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
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 . At the same time, the y-component of the vector U V is positive for α ∈ (−α, ¯ α) ¯ and negative otherwise. For this reason ⎧ ⎪ for α ∈ (−π + β2 , −¯ α) ⎨arctan (F ) + π, ω = arctan (F ), for α ∈ (−¯ α, α ¯) ⎪ ⎩ arctan (F ) − π, for α ∈ (¯ α, π − β2 ) and we get (11) for β ∈ [π, 4π ).
The Tschirnhausen cubic represents all non-degenerate Pythagorean Hododgraph cubics. We determine its support function and represent it as a convolution of a centrally symmetrical curve and a curve with linear normals. We use the support function to parametrize the Tschirnhausen cubic by normals. This parametrization is then used to an elegant and complete solution of the G1 Hermite interpolation by Pythagorean Hodograph cubics. We apply the resulting algorithm to various examples and extend it to the interpolation by offsets of PH cubics.
It is simply obtained from the Figure 3 by keeping the part β < π corresponding to the first case of (26) and adding symmetrical images of the remaining parts corresponding to the second case of (26). In order to make our results compatible with , we transform the angles φ, ψ to the −−→ → u and angles θ1 , θ2 used therein. These are oriented angles between U V and vectors − → − v respectively, where θ1 is measured anticlockwise and θ2 clockwise. These angles are related to φ, ψ by a simple linear transformation and the results of the interpolation problem are displayed on the figure 6, right.
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