By Mohamed Elkadi (Editor), Bernard Mourrain (Editor), Ragni Piene (Editor)
This publication spans the space among algebraic descriptions of geometric gadgets and the rendering of electronic geometric shapes in line with algebraic types. those contrasting issues of view encourage a radical research of the major demanding situations and the way they're met. The articles specialize in vital sessions of difficulties: implicitization, type, and intersection. Combining illustrative pictures, computations and assessment articles this booklet is helping the reader achieve an organization sensible clutch of those matters.
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Additional resources for Algebraic Geometry and Geometric Modeling (Mathematics and Visualization)
There is a one-to-one correspondance between the subschemes of a projective space Pn−1 and the saturated homogeneous ideals of the polynomial ring R := k[X1 , . . , Xn ]. To see this notice that, by deﬁnition, two subschemes of Pn−1 are the same if they coincide on all the aﬃne charts Xi = 1. If φi is the specialization homomorphism Xi → 1 then the homogenization of φi (I) is the ideal I(i) := j I : (Xij ) . It follows that I and J deﬁne the same scheme if and only if I(i) = J(i) for all i, which is easily seen to be equivalent to the equality of their saturation as I sat = i I(i) .
The choice of the degree d depends on the singularities of the given curve/surface. Singularities can be reproduced by the algebraic approximation, provided that a suﬃciently high degree is employed in the algebraic approximation. For example, in 2D, in order to represent a double point, one has to use degree three or higher. 2, which addresses the conﬂict between pushing away unwanted branches and avoiding singularities. Due to the compact support of the B-splines, the implementation is relatively fast because of the sparsity of the resulting linear system of equations.
The Z-complex. [18, Ch. 3] — We consider fi ∈ R ⊂ S as elements of S and the two complexes K• (f ; S) and K• (T ; S) where T := (T0 , . . , Tn ). These p p n+1 complexes have the same modules Kp = S S (n+1) and diﬀerentials df• and dT• . 30 Marc Chardin • It directly follows from the deﬁnitions that dfp−1 ◦ dTp + dTp−1 ◦ dfp = 0, so that dTp Zp (f ; S) ⊂ Zp−1 (f ; S). The complex Z• := Z• (f ; S), dT• is the called Z-complex associated to the fi ’s. • Notice that Zp (f ; S) = S ⊗R Zp (f ; R) and — Z0 (f ; R) = R, — Z1 (f ; R) = SyzR (f0 , .