## Adaptive subdivision and the length and energy of Bézier curves

Publication: Research - peer-review › Journal article – Annual report year: 1997

It is an often used fact that the control polygon of a Bézier curve approximates the curve and that the approximation gets better when the curve is subdivided. In particular, if a Bézier curve is subdivided into some number of pieces, then the
arc-length of the original curve is greater than the sum of the
chord-lengths of the pieces, and less than the sum of the
polygon-lengths of the pieces. Under repeated subdivisions, the
difference between this lower and upper bound gets arbitrarily
small.If $L_c$ denotes the total chord-length of the pieces and
$L_p$ denotes the total polygon-length of the pieces, the best
estimate of the true arc-length is $(2L_c+(n-1)L_p)/(n+1)$, where
$n$ is the degree of the Bézier curve. This convex combination of
$L_c$ and $L_p$ is best in the sense that the error goes to zero
under repeated subdivision asymptotically faster than the error of
any other convex combination, and it forms the basis for a fast
adaptive algorithm, which determines the arc-length of a Bézier
curve.The energy of a curve is half the square of the curvature
integrated with respect to arc-length. Like in the case of the
arc-length, it is possible to use the chord-length and
polygon-length of the pieces of a subdivided Bézier curve to
estimate the energy of the Bézier curve.

Original language | English |
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Journal | Computational Geometry |

Volume | 8 |

Issue number | 1 |

Pages (from-to) | 13-31 |

ISSN | 0925-7721 |

DOIs | |

State | Published - 1997 |

Citations | Web of Science® Times Cited: 14 |
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