SRC Research Report 19
SRC Research Report 19
The standard explanations of the theory underlying the Bezier and B-spline curves and surfaces used in computer-aided geometric design are not as simple as they should be, because there is no easy way to tell, from the labels in the diagrams, what geometric relationships hold among the labeled points. This paper proposes a new labeling scheme, based on the work of P.~de~Casteljau. The key idea is a classical mathematical principle, which we christen the Blossoming Principle: a univariate polynomial of degree n is equivalent to a symmetric polynomial in n variables that is linear in each variable separately. Blossoming a Bezier curve or surface provides lucid labels both for its Bezier points and for all of the intermediate points that arise in the de Casteljau Algorithm. Blossoming a spline curve with parametric continuity provides lucid labels for its de Boor points and for the points that arise in the de Boor Algorithm. Spline curves with geometric continuity and spline surfaces with triangular patches present unsolved labeling challenges, however.
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