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By Brahmagupta

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Extra resources for Algebra, with Arithmetic and mensuration,: From the Sanscrit of Brahmegupta and BhaÌscara

Example text

Intersection of Fig. 2 with {t = t0 } Moreover we also have the following result, which shows that the Sa,q ’s, the Tb,j ’s and the Wc,k ’s “join properly”, in the sense that the adjacencies between them are kept invariant for t ∈ (ai , ai+1 ). The proof of this result is analogous to Phase 2 in [2] (see pp. 684-686). In this proof we will need the following subsets. e. Cyl(Yp,j ) = Yp,j × R; moreover, given [ta , tb ] ⊂ Ii , we denote Z = {(x, y, z, t)|t ∈ [ta , tb ]}; ﬁnally, Ti = Cyl(Yp,j ) ∩ Z ∩ Wc,k , where Wc,k stands for the topological closure of Wc,k .

In any case, the following result holds. Theorem 3. Assume that N = 0, and let ti , ti+1 ∈ R satisfy that (ti , ti+1 ) ∩ C = ∅. Then, the shapes of Sti , Sti +1 are the same (in the sense that there is a simplicial complex describing both the shapes of Sti , Sti +1 ). 3 Proof of the Main Result In the sequel, we will prove Theorem 1; the proofs of Theorem 2 and Theorem 3 are analogous and in fact simpler, and are omitted here. For this purpose, we write A = {a1 , . . , an } where a1 < · · · < an ; furthermore, we denote a0 = −∞, an+1 = ∞.

As a consequence, by applying known algorithms ([1], [6], [7], [11]) the diﬀerent shapes in the family can be computed. The algorithm is due to a generalization of the ideas in [2] to the surface case. 1 Introduction Families of surfaces depending on parameters are common in the context of CAGD. Examples like oﬀset surfaces (where the parameter is the oﬀsetting distance), canal or tubular surfaces (where the parameter is the “thickness” of the “tube”) are well-known; other examples include surfaces with shape parameters, that are chosen in order to produce results with certain topological features.