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By Mumford.

ISBN-10: 0387112901

ISBN-13: 9780387112909

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The last step depicted above affects the doubling of the last midpoint that is shared by the two control polygons consisting of the first four and last four points of the control polygon above. So we see that the basic step in the de Casteljau algorithm is corner cutting from both ends. This has the feature of adding one more control point at each step and leaving the ends fixed. Algebraically, it is given by the equations The numbers z/, are assumed to lie in the interval (0,1). In the de Casteljau algorithm //, always is chosen to be \.

93) holds. Therefore, because A\ is TP the induction hypothesis gives us the inequality Similarly, if | is the right endpoint of [tj,t r ] we use the equation Just as above, we choose integers j\,•••,$-! G { z i , - - . ) V } » 1 £ ,?? < • • • < Jr-i ^ n ~ 1' such that 50 CHAPTER 1 and so by the induction hypothesis it follows that The case when | e (ti,tr) is more involved. ,. ,jr} fill) • • • > n}\ -> ^ then, by taking linear combinations of its first fc rows the matrix we conclude that it has a zero determinant.

K and Mi = 1 Fig' 1-15). 40 CHAPTER 1 FlG. 14. Cutting two corners from the right: n = 5, fc = 3. FlG. 15. Cutting three corners from the left: n = 5, fc = 4. Note that, unlike cutting corners from both ends, cutting corners from the left or right does not increase the number of control points. 5. The reason we have chosen exactly n - 1 corner cuts from the left and the same from the right is that there is a converse to our remarks above. 84). We refer the interested reader to the appropriate section of Micchelli and Pinkus [MPi] (see also Goodman and Micchelli [GM]).

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Abelian varieties by Mumford.

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