Unimodal properties of B-spline and Bernstein-basis functions

• Published in: Computer aided design Vol. 24; no. 12; pp. 627 - 636
• Main Authors: Barry, P.J., Beatty, J.C., Goldman, R.N.
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier Ltd 01.12.1992; Elsevier Science
• Subjects: Applied sciences; Bernstein basis; beta-splines; Bézier curves; Computer aided design; Computer science; control theory; systems; de Boor-normalized B-splines; Exact sciences and technology; Schönberg-normalized B-splines; Software; tensor products; triangular Bernstein basis; unimodality; beta-splines; Bernstein basis; tensor products; Bézier curves; unimodality; Schönberg-normalized B-splines; de Boor-normalized B-splines; triangular Bernstein basis; Unimodality; Computational geometry; Bézier curve; Tensor product; Computer graphics; Computer aided design; Spline; Bernstein polynomial
• ISSN: 0010-4485; 1879-2685
• DOI: 10.1016/0010-4485(92)90017-5

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.