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

Publication: Research - peer-reviewJournal article – Annual report year: 1997

### DOI

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 Computational Geometry 8 1 13-31 0925-7721 http://dx.doi.org/10.1016/0925-7721(95)00054-2 Published - 1997
Citations Web of Science® Times Cited: 13