Bézier curves that are close to elastica

David Brander*, Jakob Andreas Bærentzen, Ann-Sofie Fisker, Jens Gravesen

*Corresponding author for this work

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We study the problem of identifying those cubic B´ezier curves that are close in the L2 norm to planar elastic curves. The problem arises in design situations where the manufacturing process produces elastic curves; these are difficult to work with in a digital environment. We seek a sub-class of special B´ezier curves as a proxy. We identify an easily computable quantity, which we call the λ-residual eλ, that accurately predicts a small L2 distance. We then identify geometric criteria on the control polygon that guarantee that a B´ezier curve has λ-residual below 0.4, which effectivelyimpliesthatthecurveiswithin1%ofitsarc-lengthtoanelasticcurveinthe L2 norm. Finally wegive two projection algorithms that take an input B´ezier curve and adjust its length and shape, whilst keeping the end-points and end-tangent angles fixed, until it is close to an elastic curve
Original languageEnglish
JournalComputer-Aided Design
Pages (from-to)36-44
Number of pages9
Publication statusPublished - 2018


  • Cubic Bézier curves
  • Elastic curves
  • Splines
  • Approximation
  • Computer aided design
  • Physically-based modeling

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