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    Conic Sections Formulas

    Result

    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)

    Example plots

    Example plots Circle

    Example plots Ellipse

    Example plots Hyperbola

    Equations

    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))

    Parametric properties

    | arc length circle | s = 2 π a ellipse | s = 4 a E(1 - b^2/a^2) | arc length function circle | s(t) = a t ellipse | s(t) = b E(t|1 - a^2/b^2) hyperbola | s(t) = b F(t|-a^2/b^2) + b (-E(t|-a^2/b^2)) + (tan(t) sqrt(a^2 (-cos(2 t)) + a^2 + 2 b^2))/sqrt(2) parabola | s(t) = a (sqrt(t^2 + 1) t + sinh^(-1)(t)) rectangular hyperbola | s(t) = a (F(t|-1) - E(t|-1) + (sqrt(3 - cos(2 t)) tan(t))/sqrt(2)) | parametric curvature circle | κ(t) = 1/a ellipse | κ(t) = (a b)/(a^2 sin^2(t) + b^2 cos^2(t))^(3/2) hyperbola | κ(t) = -(a b cos(t) abs(sec(t)))/(a^2 tan^2(t) + b^2 sec^2(t))^(3/2) parabola | κ(t) = 1/(2 a (t^2 + 1)^(3/2)) rectangular hyperbola | κ(t) = -(sec^3(t))/(a ((sin^2(t) + 1) sec^4(t))^(3/2)) | parametric slope circle | m(t) = -cot(t) ellipse | m(t) = -(b cot(t))/a hyperbola | m(t) = (b csc(t))/a parabola | m(t) = t rectangular hyperbola | m(t) = csc(t) | tangential angle circle | ϕ(t) = t ellipse | ϕ(t) = t hyperbola | ϕ(t) = -tan^(-1)((a sin(t))/b) parabola | ϕ(t) = tan^(-1)(t) rectangular hyperbola | ϕ(t) = -tan^(-1)(sin(t)) | vector length circle | left double bracketing bar x(t) right double bracketing bar = a ellipse | left double bracketing bar x(t) right double bracketing bar = sqrt(a^2 cos^2(t) + b^2 sin^2(t)) hyperbola | left double bracketing bar x(t) right double bracketing bar = sqrt(a^2 sec^2(t) + b^2 tan^2(t)) parabola | left double bracketing bar x(t) right double bracketing bar = a t sqrt(t^2 + 4) rectangular hyperbola | left double bracketing bar x(t) right double bracketing bar = a sqrt(tan^2(t) + sec^2(t))

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