View source for Arc length, curvature and area

Computing [[arc length]]s on elliptic curves is in general very difficult and involves transcendental [[elliptic function]]s. If ''y''=''f''(''x''), then the arc length ''s'' of the function plot may be computed numerically as

:<math>s = \int \sqrt{dx^2+dy^2}
= \int \sqrt {1+\big(f'(x)\big)^2}dx</math>

and the curvature as

:<math>\kappa=\frac{\frac{d}{dx}\arctan f'(x)}{1+\big(f'(x)\big)^2}
=\frac{f''(x)}{\left(1+\big(f'(x)\big)^2\right)^2}</math>

where the radius of the osculating circle is ''r''=1/''κ'' when ''f’’''(''x'')≠0.