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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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