Adaptive subdivision and the length and energy of Bézier curves

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.