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Conic Sections

Named curves

circle | ellipse | hyperbola | parabola | rectangular hyperbola

Example plots

Example plots Circle

Example plots Ellipse

Example plots Hyperbola

Equations

circle | x(t) = a cos(t)
y(t) = a sin(t)
ellipse | x(t) = a cos(t)
y(t) = b sin(t)
hyperbola | x(t) = a sec(t)
y(t) = b tan(t)
parabola | x(t) = 2 a t
y(t) = a t^2
rectangular hyperbola | x(t) = a sec(t)
y(t) = a tan(t)

circle | x^2 + y^2 = a^2
ellipse | x^2/a^2 + y^2/b^2 = 1
hyperbola | x^2/a^2 - y^2/b^2 = 1
parabola | y = x^2/(4 a)
rectangular hyperbola | x^2 - y^2 = a^2

circle | r(θ) = a
ellipse | r(θ) = (a b)/sqrt((b^2 - a^2) cos^2(θ) + a^2)
hyperbola | r(θ) = (a b)/sqrt(b^2 cos^2(θ) - a^2 sin^2(θ))
parabola | r(θ) = 4 a tan(θ) sec(θ)
rectangular hyperbola | r(θ) = a sqrt(sec(2 θ))

Common properties

algebraic | conic | parametric | quadratic

Basic properties

circle | r = a

circle | d = 2 a

circle | C = 2 π a

circle | A = π a^2
ellipse | A = π a b

circle | s = 2 π a
ellipse | s = 4 a E(1 - b^2/a^2)

circle | d = 2
ellipse | d = 2
hyperbola | d = 2
parabola | d = 2
rectangular hyperbola | d = 2

Conic properties

 | eccentricity | focal parameter | semilatus rectum
circle | e = 0 | | 
ellipse | e = sqrt(1 - b^2/a^2) | p = b^2/sqrt(a^2 - b^2) | L = b^2/a
hyperbola | e = sqrt(b^2/a^2 + 1) | p = b^2/sqrt(a^2 + b^2) | L = b^2/a
parabola | e = 1 | p = 2 a | L = 2 a
rectangular hyperbola | e = sqrt(2) | p = a/sqrt(2) | 
 | foci | asymptotes | directrix
ellipse | {(-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)
hyperbola | {(-sqrt(a^2 + b^2), 0), (sqrt(a^2 + b^2), 0)} | y = -(b x)/a ∨ y = (b x)/a | x = -a^2/sqrt(a^2 + b^2) ∨ x = a^2/sqrt(a^2 + b^2)
parabola | {(0, a)} | | y = -a
rectangular hyperbola | {(-sqrt(2) a, 0), (sqrt(2) a, 0)} | | x = -a/sqrt(2) ∨ x = a/sqrt(2)

Derived curves

 | evolute | involute
circle | point at origin | circle involute
ellipse | ellipse evolute | ellipse involute
parabola | semicubical parabola | parabola involute

Distance properties

 | mean line segment length
circle | s^_ = (4 a)/π

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