By Eisenhart L. P.
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P n +i. Again let us assume for simplicity that each knot has multiplicity 1. 1 Knot Insertion and Deletion Algorithms 41 FlG. 10(b). The second step of Boehm's derivative algorithm (unnormalized) for cubic progressive curves: inserting a triple knot at a. The values at the base of the triangle are taken from the left edge of the triangle computed in the first stage of the algorithm. The values which emerge along the right edge of the triangle computed in this second stage are, up to constant multiples, the derivatives of the original progressive curve at t = a.
T$, w&}- For the homogeneous de Boor algorithm to work, none of the denominators on the right-hand side of the algorithm are allowed to vanish. , (/2n, W2n) progressive if none of the denominators in this recursive evaluation algorithm are zero. , (/2n? W2n) 'IS a progressive sequence if and only if del The curves p(t, w] defined by the homogeneous de Boor algorithm are again called progressive curves. Notice that the progressive curve p(t, 1) is a univariate polynomial with homogeneous knots.
B-spline curves are extremely popular and have been studied extensively , , , . One of our main motives for investigating progressive curves is to find techniques for transforming ^-spline curves to and from other piecewise polynomial forms. 2. Bernstein Polynomials. The Bernstein basis is a special case of the B-spline basis where the knots are a , . . , a, 6 , . . , 6, 6 ^ a. Explicitly the Bernstein basis functions #Q ( £ ) , . . , B^(i) are given by These basis functions satisfy the recurrence relationship By the de Boor-Fix formula, the Bernstein basis is self dual.