Arc length, curvature and area - Revision history

Rational Point: Metes and bounds

2024-12-30T15:45:15Z

Metes and bounds

← Older revision		Revision as of 15:45, 30 December 2024
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	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		== “Metes and bounds” ==

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

	:<math>s = \int \sqrt{dx^2+dy^2}		:<math>s = \int \sqrt{dx^2+dy^2}
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	where the radius of the osculating circle is ''r''=1/''κ'' when ''f’’''(''x'')≠0.		where the radius of the osculating circle is ''r''=1/''κ'' when ''f’’''(''x'')≠0.

			To be continued ...

Rational Point: Rational Point moved page Arc length and curvature to Arc length, curvature and area: mention area

2024-12-30T15:30:43Z

Rational Point moved page Arc length and curvature to Arc length, curvature and area: mention area

← Older revision	Revision as of 15:30, 30 December 2024
(No difference)

Rational Point: start

2024-12-30T13:12:23Z

start

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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.