Measuring the length of an ellipse was hard enough to spawn new mathematics
Picture a curve as a piece of string and pull it straight: its length is the curve's arc length. For circles, parabolas and cycloids there is a tidy formula. For the humble ellipse there is none, and the struggle to measure one gave rise to a whole family of functions, the elliptic integrals.
The oldest approach is to cheat with straight lines. Pick points along the curve in order, join them with segments, and add up the segment lengths using Pythagoras. Adding more points can only keep that total the same or push it up, never down. If the totals never exceed some ceiling, the smallest such ceiling is the curve's length, and the curve is called rectifiable. The whole process goes by the old name of rectification.
Calculus gives a quicker route for smooth curves. Treat the curve as the path of a moving particle and integrate its speed over time. The answer does not depend on how the curve is parameterised, since the same path traced quickly or slowly has the same length.
Closed-form answers exist for a short list: the straight line, circle, parabola, semicubical parabola, catenary, cycloid and logarithmic spiral. Most curves, including quite simple ones, have none, so their lengths must be computed numerically. Differentiation in the formula costs one order of precision, but for very smooth curves numerical methods still work extremely well.
A quarter of a circle of radius one shows how well. Its true length is pi over two. A 15-point Gauss–Kronrod estimate of 1.570796326808177 misses by 1.3×10−11, and a 16-point Gaussian quadrature gives 1.570796326794727, off by only 1.7×10−13. Sixteen evaluations are enough to reach almost the full precision a computer can hold.
Source: Arc length
