Unimodal properties of B-spline and Bernstein-basis functions

P. J. Barry, J. C. Beatty, R. N. Goldman

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


A sequence of functions {fj(u)} is said to be unimodal on the interval [a,b] if and only if the sequence {fj(ǔ)} has only one local maximum for each ǔ ε{lunate} [a,b]. It is shown that thisunimodality 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.

Original languageEnglish (US)
Pages (from-to)627-636
Number of pages10
JournalComputer-Aided Design
Issue number12
StatePublished - Dec 1992

Bibliographical note

Funding Information:
The research described in this paper was supported by the National Engineering Research Council of Canada, the Province of Ontario's Information Technology Research Centre, and Digital Equipment Corporation of Canada. The use of the symbolic-algebra system Maple 6 is gratefully acknowledged.


  • Bernstein basis
  • Bézier curves
  • Schönberg-normalized B-splines
  • beta-splines
  • de Boor-normalized B-splines
  • tensor products
  • triangular Bernstein basis
  • unimodality


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