Abstract
A sequence of functions {f j(u)} is said to be unimodal on the interval [a,b] if and only if the sequence {f j(ǔ)} has only one local maximum for each ǔ ϵ [a,b]. It is shown that this unimodality property holds for the Bernstein-basis functions, uniform B-splines, and degree-3 or lower B-splines over arbitrary knot vectors, but that it does not hold for general B-splines of degree 6 or greater; nonuniform B-splines of degrees 4 and 5 are left as an open question. It is also shown that all Schönberg-normalized B-splines are unimodal, and that the results extend to tensor-product B-spline and Bernstein-basis functions and triangular Bernstein-basis functions, and to some special geometrically continous basis functions, as well as to certain special nonuniform rational B-splines. The practical significance of this abstract algebraic property for geometric-modelling applications is also explained.
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