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Read e-book online Ancient Hindu Geometry: The Science Of The Sulba PDF

By Bibhutibhushan Datta

ISBN-10: 8170204607

ISBN-13: 9788170204602

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Extra resources for Ancient Hindu Geometry: The Science Of The Sulba

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Function f has an Ak or Dk singularity if, up to a diffeomorphism, it can be written as: Ak : f = ±x2 ± yk+1 , k ≥ 0, Dk : f = ±yx2 ± yk−1 , k ≥ 4. 12) ± The singularity is further denoted A± k or Dk if the product of the coefficients of the monomials is ±1. As subsumed by this definition, an Ak singularity precludes an Ak+1 singularity, and similarly for Dk . An important characteristic of these normal forms is their zero level set. Those of the Ak sequence are illustrated on Fig. 4, where the (branches of) curves are defined from x = ±y(k+1)/2.

Cet algorithme est le premier à calculer une approximation topologique certifiée de ridges. Une fois la topologie obtenue, une approximation géométrique plus précise peut facilement être déduite. Nous proposons également un autre algorithme qui calcule un tracé certifié à résolution fixée: les calculs sont moins coûteux, mais la topologie n’est plus garantie. Les résultats pour une surface paramétrée sont directement applicables aux surfaces de Bézier incontournable en CAO. Il est à noter que la méthode développée pour le calcul de la topologie des ridges peut être généralisée pour d’autres courbes algébriques, à partir du moment où le nombre de branches réelles passant par chaque singularité peut être calculé efficacement.

To see which ones, consider the contact function given by Eq. 11). Since r = 1/k1 = 1/k2 , it is dominated by the cubic terms. More precisely, the singularity is generically a D± 4 . The number of ridges passing through the umbilic is the number of curves in the zero level set of the contact function. Hence this number reads on the normal form, and is equal to one or three as mentioned in observation 2. This fact is not intuitive and it is neither obvious that ridges pass through umbilics. A way to explain these facts is to study the gradient field ∇k1 (the same holds for ∇k2 ) well defined at non umbilical points.

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Ancient Hindu Geometry: The Science Of The Sulba by Bibhutibhushan Datta

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