By Mohamed Elkadi (Editor), Bernard Mourrain (Editor), Ragni Piene (Editor)

This booklet spans the gap among algebraic descriptions of geometric items and the rendering of electronic geometric shapes in line with algebraic versions. those contrasting issues of view motivate an intensive research of the main demanding situations and the way they're met. The articles concentrate on vital periods of difficulties: implicitization, class, and intersection. Combining illustrative portraits, computations and overview articles this ebook is helping the reader achieve a company sensible seize of those matters.

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Since pushing away unwanted branches is the most important issue, we dedicate a separate subsection to this subject. 1 General criteria • Both PPL and PPS are able to handle general surfaces. Consequently, they can also be used to implicitize procedurally deﬁned surfaces, since they only need samples of points. In contrast, PS and ML can be applied only to parametrically deﬁned rational surfaces (patches of NURBS). • Both PPL and PPS are able to compute an approximate implicitization for all test cases with reasonable error.

According to the CPU benchmark tests [11], both systems have almost the same performance. 5 Results For implicitizing multi-patch parametric surfaces, the two piecewise approximate implicitization methods that are relevant are PPL and PPS. Even in the case of single patch parametric surfaces, which can be handled with single polynomial methods (ML, PL), these piecewise methods are able to produce an algebraic representation of lower degree. Both the PPL and PPS take as input the degree of the B-spline and information about the number of cells in the input data bounding box.

No hypothesis on the ﬁeld k is needed. When we compute for instance the image of a map, in the sense of algebraic k-schemes, it should be remembered that it corresponds geometrically to the image over the algebraic closure of the ﬁeld (the image is itself deﬁned over k if the scheme is). We let: • I := (f0 , . . , fn ) ⊂ R be the ideal generated by the fi ’s, • X := Proj(R/I) ⊂ Pn be the subscheme deﬁned by I. 2]: φ : P2 G P3 , given by f := (ac2 , b2 (a + c), ab(a + c), bc(a + c)) with RE := Q[a, b, c].