Arc length, curvature and area

“Metes and bounds”

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

$s=\int {\sqrt {dx^{2}+dy^{2}}}=\int {\sqrt {1+{\big (}f'(x){\big )}^{2}}}dx$

and the curvature as

$\kappa ={\frac {{\frac {d}{dx}}\arctan f'(x)}{1+{\big (}f'(x){\big )}^{2}}}={\frac {f''(x)}{\left(1+{\big (}f'(x){\big )}^{2}\right)^{2}}}$

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

To be continued ...