Bézier curves that are close to elastica

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

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review

Abstract

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
Volume104
Pages (from-to)36-44
Number of pages9
ISSN0010-4485
DOIs
Publication statusPublished - 2018

Keywords

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

Cite this

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title = "B{\'e}zier curves that are close to elastica",
abstract = "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",
keywords = "Cubic B{\'e}zier curves, Elastic curves, Splines, Approximation, Computer aided design, Physically-based modeling",
author = "David Brander and B{\ae}rentzen, {Jakob Andreas} and Ann-Sofie Fisker and Jens Gravesen",
year = "2018",
doi = "10.1016/j.cad.2018.05.003",
language = "English",
volume = "104",
pages = "36--44",
journal = "Computer-Aided Design",
issn = "0010-4485",
publisher = "Pergamon Press",

}

Bézier curves that are close to elastica. / Brander, David; Bærentzen, Jakob Andreas; Fisker, Ann-Sofie; Gravesen, Jens.

In: Computer-Aided Design, Vol. 104, 2018, p. 36-44.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

T1 - Bézier curves that are close to elastica

AU - Brander, David

AU - Bærentzen, Jakob Andreas

AU - Fisker, Ann-Sofie

AU - Gravesen, Jens

PY - 2018

Y1 - 2018

N2 - 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

AB - 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

KW - Cubic Bézier curves

KW - Elastic curves

KW - Splines

KW - Approximation

KW - Computer aided design

KW - Physically-based modeling

U2 - 10.1016/j.cad.2018.05.003

DO - 10.1016/j.cad.2018.05.003

M3 - Journal article

VL - 104

SP - 36

EP - 44

JO - Computer-Aided Design

JF - Computer-Aided Design

SN - 0010-4485

ER -