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Ellipse

Example plots

Equations

x(t) = a cos(t)
y(t) = b sin(t)

x^2/a^2 + y^2/b^2 = 1

r(θ) = (a b)/sqrt((b^2 - a^2) cos^2(θ) + a^2)

(for an ellipse with center at the origin, semimajor axis a parallel to the x-axis, and semiminor axis b parallel to the y-axis)

Properties

Basic properties

A = π a b

s = 4 a E(1 - b^2/a^2)

d = 2

Conic properties

e = sqrt(1 - b^2/a^2)

p = b^2/sqrt(a^2 - b^2)

L = b^2/a

${(-sqrt(a^2 - b^2), 0), (sqrt(a^2 - b^2), 0)}$

$piecewise | {x = -a^2/sqrt(a^2 - b^2) ∨ x = a^2/sqrt(a^2 - b^2)} | b<a {y = -b^2/sqrt(b^2 - a^2) ∨ y = b^2/sqrt(b^2 - a^2)} | b>a | (otherwise)$

Derived curves

evolute | ellipse evolute
involute | ellipse involute

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