Adaptive subdivision and the length and energy of Bézier curves

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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 Lc denotes the total chord-length of the pieces and Lp denotes the total polygon-length of the pieces, the best estimate of the true arc-length is (2Lc + (n - 1)Lp)/(n + 1), where n is the degree of the Bézier curve. This convex combination of Lc and Lp 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. © 1997 Elsevier Science B.V.

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Gravesen, J. (1997). Adaptive subdivision and the length and energy of Bézier curves. Computational Geometry: Theory and Applications, 8(1), 13–31.

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