TABLE OF CONTENTS B zier curves may be viewed as the backbone of all piecewise polynomial curve schemes; their rational counterparts enjoy a similar role for piecewise rational curves. Every piecewise polynomial or rational curve may be broken down into a collection of B zier curves, and thus their study is crucial to the understanding of the whole field of NURBS.
Historically, B zier curves in their polynomial, or integral, version are due to P. de Casteljau. In 1957, he invented what is now known as the de Casteljau algorithm, laid down in an internal report of the Citro n automotive company. His algorithm relies on the concept of constant ratios, and is thus tied in closely with afflne geometry. Rational B zier curves, however, may be evaluated using the concept of cross ratios, the fundamental invariant of projective geometry. [1]
[1]But when I sent de Casteljau a preprint of [50], he was not too enthusiastic about the new version of his algorithm: he just did not like the idea of using cross ratios...
