By Bernard Mourrain, Scott Schaefer, Guoliang Xu
This ebook 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 conscientiously reviewed and chosen from a complete of 30 submissions. The papers hide a large spectrum within the sector of geometric modeling and processing and handle subject matters corresponding to options of transcendental equations; quantity parameterization; soft curves and surfaces; isogeometric research; implicit surfaces; and computational geometry.
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This publication 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 offered have been rigorously reviewed and chosen from a complete of 30 submissions. The papers conceal a large spectrum within the quarter of geometric modeling and processing and handle subject matters corresponding to ideas of transcendental equations; quantity parameterization; tender curves and surfaces; isogeometric research; implicit surfaces; and computational geometry.
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Additional resources for Advances in Geometric Modeling and Processing: 6th International Conference, GMP 2010, Castro Urdiales, Spain, June 16-18, 2010, Proceedings
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 ). 3 Case β = 4π : 3 This is the limit situation of the previous case. 0 = α ¯ ∈ (−π + β2 , π − β2 ), but α0 ∈ β β (−π+ 2 , π− 2 ). Therefore F has one discontinuity and its derivative is always negative. The behavior can be seen in Fig.
Transform the control points by the unique direct Euclidean similarity transformation which maps the first control point to U and the last one to V . Let us comment on some steps of this procedure to argue, that they are extremely simple and fast. Steps 1 and 2 are trivial. Step 3 might seem too complicated, but (16) with unknown α is essentially of the form A sin(α) + B cos(α) = C, (30) ˇ ˇ ır E. Cernohorsk´ a and Z. S´ 40 V U U V U V V U V U Fig. 7. Hermite data and PH cubic interpolants, see Table 1 for numerical values.
For fixed β we display F (upper row) and ω (lower row) as functions of α ∈ (−π + π − β2 ) β , 2 which is zero for α = ± arccos 2 + cos β 3 cos β2 . (19) α0 A segment on Tschirnhausen cubic cubic T (θ) has a loop if and only if the both values θ = ±2π/3 are contained in its domain. This gives following loop condition for the rotation angle α |α| ≥ β 2π − , 2 3 (20) which can be satisfied only for β ≥ 4π/3. Case β ∈ (0, π) : In this case ω is strictly decreasing function of α, see Fig. 4, lower row.