Advances in Geometric Modeling and Processing: 6th by Bernard Mourrain, Scott Schaefer, Guoliang Xu

By Bernard Mourrain, Scott Schaefer, Guoliang Xu

This ebook 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 awarded 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 handle subject matters reminiscent of recommendations of transcendental equations; quantity parameterization; delicate 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

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4. From (9) compute values of t1 , t2 corresponding to θ = α − β/2 and θ = α + β/2. 5. Compute control points for the segment t ∈ [t1 , t2 ] of (3). 6. 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.

This new pair can be compute using the case look-up given in Table 1. Note that the rule for computing k is straightforward.

Fig. 1. If X belongs to the circumcircle of the vertex triangle, then any two of the three spheres (9) touch each other at this point and the circle degenerates into a single point. Corollary 1. If P is a rational chord length parameterization, then its restriction to the circumcircle of the reference triangle is the identity. Moreover, the surface is a rational chord length parameterization with respect to any reference triangle which possesses the same circumcircle. Proof. The surface P is a RCL surface if and only if any point P(X) lies on the circle described in Lemma 2.

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