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    Conic Section

    Named curves

    circle | ellipse | hyperbola | parabola | rectangular hyperbola

    Example plots

    Example plots Circle

    Example plots Ellipse

    Example plots Hyperbola

    Alternate names

    equilateral hyperbola | right 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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