Cours de geometrie by D'Ocagne M.

By D'Ocagne M.

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Un+s run along the lateral side of the tetrahedron while the parameters t i n , . . , w s+ i run along the triangle parallel to the base. This structure allows us to take advantage of overlapping computations by reusing again and again the computations running up the lateral side of the tetrahedron. , w n ) , . . , U2n) emerge at the apexes of the triangles parallel to the base—that is, along the edge of the tetrahedron which is skew to the edge with the input control points. Given some fixed progressive values B(t\,...

B™(t). 3. Homogenization We have shown that there is a rich collection of algorithms for progressive curves. The purpose of this section is to enlarge the category of progressive curves by introducing the notions of homogeneous, affine, and vector-valued 30 Algorithms for Progressive Curves knots. We shall also develop differentiation formulas for progressive curves and integration algorithms for progressive bases. Our guiding principle here is the technique of homogenization. 1 by introducing a homogeneous version of the de Boor algorithm.

N, then the recursive evaluation algorithm for the symmetric multiaffine polynomial B(u\,... ,u n ) specializes to a recursive evaluation algorithm for the diagonal univariate polynomial B(t, . . , £ ) . , t) is precisely the de Boor algorithm for the progressive curve with knots t\,. . , *2n)- Thus There is a 1-1 correspondence between univariate polynomials of degree n and symmetric multiaffine polynomials in n variables. To find the univariate degree n polynomial p ( t ) corresponding to the symmetric multiaffine polynomial B(u\,..

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