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    Evolute

    Named curves

    catenary evolute | ellipse evolute | limaçon evolute

    Example plots

    Example plots Catenary evolute

    Example plots Ellipse evolute

    Example plots Limaçon evolute

    Alternate names

    Lamé curve

    Equations

    catenary evolute | x(t) = t - 1/2 a sinh((2 t)/a) y(t) = 2 a cosh(t/a) ellipse evolute | x(t) = ((a^2 - b^2) cos^3(t))/a y(t) = ((a^2 - b^2) sin^3(t))/b limaçon evolute | x(t) = (a (4 (a^2 + b^2) + 9 a b cos(t) - a b cos(3 t)))/(4 (2 a^2 + 3 a b cos(t) + b^2)) y(t) = (a^2 b sin^3(t))/(2 a^2 + 3 a b cos(t) + b^2)

    ellipse evolute | 3 b^2 y^2 (a^4 x^4 + 7 a^2 x^2 (a^2 - b^2)^2 + (a^2 - b^2)^4) - 3 b^4 y^4 (a^4 - a^2 (2 b^2 + x^2) + b^4) + (a^4 - a^2 (2 b^2 + x^2) + b^4)^3 + b^6 y^6 = 0 limaçon evolute | a^12 - 12 x a^11 + 60 x^2 a^10 + 12 y^2 a^10 - 3 b^2 a^10 - 160 x^3 a^9 - 96 x y^2 a^9 + 30 x b^2 a^9 + 240 x^4 a^8 + 48 y^4 a^8 + 3 b^4 a^8 + 288 x^2 y^2 a^8 - 123 x^2 b^2 a^8 - 27 y^2 b^2 a^8 - 192 x^5 a^7 - 192 x y^4 a^7 - 24 x b^4 a^7 - 384 x^3 y^2 a^7 + 264 x^3 b^2 a^7 + 168 x y^2 b^2 a^7 + 64 x^6 a^6 + 64 y^6 a^6 - b^6 a^6 + 192 x^2 y^4 a^6 + 78 x^2 b^4 a^6 + 18 y^2 b^4 a^6 + 192 x^4 y^2 a^6 - 312 x^4 b^2 a^6 - 72 y^4 b^2 a^6 - 384 x^2 y^2 b^2 a^6 + 6 x b^6 a^5 - 132 x^3 b^4 a^5 - 84 x y^2 b^4 a^5 + 192 x^5 b^2 a^5 + 192 x y^4 b^2 a^5 + 384 x^3 y^2 b^2 a^5 - 15 x^2 b^6 a^4 - 3 y^2 b^6 a^4 + 123 x^4 b^4 a^4 + 27 y^4 b^4 a^4 + 123 x^2 y^2 b^4 a^4 - 48 x^6 b^2 a^4 - 48 y^6 b^2 a^4 - 144 x^2 y^4 b^2 a^4 - 144 x^4 y^2 b^2 a^4 + 20 x^3 b^6 a^3 + 12 x y^2 b^6 a^3 - 60 x^5 b^4 a^3 - 6 x y^4 b^4 a^3 - 66 x^3 y^2 b^4 a^3 - 15 x^4 b^6 a^2 - 3 y^4 b^6 a^2 - 18 x^2 y^2 b^6 a^2 + 12 x^6 b^4 a^2 - 15 y^6 b^4 a^2 - 18 x^2 y^4 b^4 a^2 + 9 x^4 y^2 b^4 a^2 + 6 x^5 b^6 a + 6 x y^4 b^6 a + 12 x^3 y^2 b^6 a - x^6 b^6 - y^6 b^6 - 3 x^2 y^4 b^6 - 3 x^4 y^2 b^6 = 0

    Common properties

    evolute | parametric

    Basic properties

    ellipse evolute | A = (3 π (a^2 - b^2)^2)/(8 a b)

    limaçon evolute | A^* = (3 π a^4 b^2)/(2 (4 a^4 + a^2 (2 sqrt(4 a^4 - 5 a^2 b^2 + b^4) + 7 b^2) - 2 b^2 (sqrt(4 a^4 - 5 a^2 b^2 + b^4) + b^2))) for 2 a<b ∨ a>b

    ellipse evolute | s = (4 b^2)/a

    ellipse evolute | d = 6 limaçon evolute | d = 6

    Derived curves

    | catenary evolute | ellipse evolute | limaçon evolute evolute | point at origin | | involute | catenary | ellipse | limaçon

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