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Pedal Curve

Named curves

astroid pedal curve | cardioid pedal curve | circle pedal curve | ellipse pedal curve | hyperbola pedal curve

Example plots

Example plots Astroid pedal curve

Example plots Cardioid pedal curve

Example plots Circle pedal curve

Equations

astroid pedal curve | x(t) = cos(t) (sin(t) (a sin(t) - y_0) + x_0 cos(t)) y(t) = 1/2 sin(t) (a cos(2 t) + a - 2 x_0 cos(t) + 2 y_0 sin(t)) cardioid pedal curve | x(t) = 1/4 ((a + 2 x_0) cos(3 t) + 3 a cos(t) - 3 a cos(2 t) - a + 2 y_0 sin(3 t) + 2 x_0) y(t) = 1/4 ((a + 2 x_0) sin(3 t) + 3 a sin(t) - 3 a sin(2 t) - 2 y_0 cos(3 t) + 2 y_0) circle pedal curve | x(t) = cos(t) (a - y_0 sin(t)) + x_0 sin^2(t) y(t) = 1/2 (2 a sin(t) - x_0 sin(2 t) + y_0 cos(2 t) + y_0) ellipse pedal curve | x(t) = (a (a x_0 sin^2(t) + b cos(t) (b - y_0 sin(t))))/(a^2 sin^2(t) + b^2 cos^2(t)) y(t) = (b (a sin(t) (a - x_0 cos(t)) + b y_0 cos^2(t)))/(a^2 sin^2(t) + b^2 cos^2(t)) hyperbola pedal curve | x(t) = (2 a (sin(t) (a x_0 sin(t) + b y_0) + b^2 cos(t)))/(a^2 (-cos(2 t)) + a^2 + 2 b^2) y(t) = (2 b (a sin(t) (x_0 - a cos(t)) + b y_0))/(a^2 (-cos(2 t)) + a^2 + 2 b^2)

astroid pedal curve | -16 a^2 x^2 y^2 - 256 a^2 x^2 y_0^2 + 128 a^2 x^2 y y_0 + 128 a^2 x_0 x y^2 - 256 a^2 x_0^2 y^2 + 2048 a^2 x_0 x y_0^2 - 1024 a^2 x_0 x y y_0 - 4096 a^2 x_0^2 y_0^2 + 2048 a^2 x_0^2 y y_0 + x^6 - 16 x_0 x^5 + 3 x^4 y^2 + 16 x^4 y_0^2 - 16 x^4 y y_0 + 96 x_0^2 x^4 - 32 x_0 x^3 y^2 - 128 x_0 x^3 y_0^2 + 160 x_0 x^3 y y_0 - 256 x_0^3 x^3 + 3 x^2 y^4 - 32 x^2 y^3 y_0 + 112 x_0^2 x^2 y^2 + 112 x^2 y^2 y_0^2 - 128 x^2 y y_0^3 + 256 x_0^2 x^2 y_0^2 - 512 x_0^2 x^2 y y_0 + 256 x_0^4 x^2 - 16 x_0 x y^4 + 16 x_0^2 y^4 + 160 x_0 x y^3 y_0 - 128 x_0^2 y^3 y_0 - 128 x_0^3 x y^2 - 512 x_0 x y^2 y_0^2 + 256 x_0^2 y^2 y_0^2 + 512 x_0 x y y_0^3 + 512 x_0^3 x y y_0 + y^6 - 16 y^5 y_0 + 96 y^4 y_0^2 - 256 y^3 y_0^3 + 256 y^2 y_0^4 = 0 cardioid pedal curve | x^6 + 6 a x^5 - 12 x_0 x^5 + 3 y^2 x^4 - 15 a^2 x^4 + 48 x_0^2 x^4 - 72 a x_0 x^4 - 12 y y_0 x^4 + 8 a^3 x^3 - 64 x_0^3 x^3 + 288 a x_0^2 x^3 + 12 y^2 a x^3 - 24 y^2 x_0 x^3 + 180 a^2 x_0 x^3 - 48 y a y_0 x^3 + 96 y x_0 y_0 x^3 + 3 y^4 x^2 - 384 a x_0^3 x^2 - 42 y^2 a^2 x^2 + 48 y^2 x_0^2 x^2 - 720 a^2 x_0^2 x^2 + 48 y^2 y_0^2 x^2 - 432 a^2 y_0^2 x^2 - 96 a^3 x_0 x^2 - 96 y^2 a x_0 x^2 - 24 y^3 y_0 x^2 + 276 y a^2 y_0 x^2 - 192 y x_0^2 y_0 x^2 + 384 y a x_0 y_0 x^2 + 960 a^2 x_0^3 x + 384 a^3 x_0^2 x + 192 y^2 a x_0^2 x + 96 y^2 a y_0^2 x - 192 y^2 x_0 y_0^2 x + 1728 a^2 x_0 y_0^2 x + 6 y^4 a x - 12 y^4 x_0 x + 228 y^2 a^2 x_0 x - 768 y a x_0^2 y_0 x - 48 y^3 a y_0 x + 96 y^3 x_0 y_0 x - 1344 y a^2 x_0 y_0 x + y^6 - 512 a^3 x_0^3 - 64 y^3 y_0^3 + 1728 y a^2 y_0^3 - 27 y^4 a^2 - 240 y^2 a^2 x_0^2 + 48 y^4 y_0^2 - 1296 y^2 a^2 y_0^2 - 384 y^2 a x_0 y_0^2 - 24 y^4 a x_0 - 12 y^5 y_0 + 324 y^3 a^2 y_0 + 960 y a^2 x_0^2 y_0 + 192 y^3 a x_0 y_0 = 0 circle pedal curve | (x^2 - x_0 x + y (y - y_0))^2 = a^2 ((x - x_0)^2 + (y - y_0)^2) ellipse pedal curve | (a^2 (x - x_0)^2 - (-(y - y_0) (b - y) + x^2 - x_0 x) ((y - y_0) (b + y) + x^2 - x_0 x)) (2 y_0^2 (a^2 - b^2 + x_0^2) + (-a^2 + b^2 + x_0^2)^2 + y_0^4) = 0 hyperbola pedal curve | (-a^2 (x - x_0)^2 - 2 y y_0 (b^2 + x^2 - x_0 x + y^2) + b^2 y^2 + y_0^2 (b^2 + y^2) + (x^2 - x_0 x + y^2)^2) (2 y_0^2 (a^2 + b^2 + x_0^2) + (a^2 + b^2 - x_0^2)^2 + y_0^4) = 0

Common properties

algebraic | closed | loopy | pedal

Basic properties

astroid pedal curve | A = -1/8 π (a^2 + 4 (x_0^2 + y_0^2))

cardioid pedal curve | A^* = 3/8 π (5 a^2 + 2 a x_0 + 2 (x_0^2 + y_0^2)) circle pedal curve | A^* = 1/2 π (2 a^2 + x_0^2 + y_0^2) ellipse pedal curve | A^* = 1/2 π (a^2 + b^2 + x_0^2 + y_0^2)

circle pedal curve | s = 4 sqrt(a^2 - 2 a sqrt(x_0^2 + y_0^2) + x_0^2 + y_0^2) E(-(4 a sqrt(x_0^2 + y_0^2))/(a^2 - 2 sqrt(x_0^2 + y_0^2) a + x_0^2 + y_0^2))

astroid pedal curve | d = 6 cardioid pedal curve | d = 6 circle pedal curve | d = 4 ellipse pedal curve | d = 4 hyperbola pedal curve | d = 4

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Vestibulum vitae aliquam nunc. Suspendisse mollis metus ac tellus egestas pharetra. Suspendisse at viverra purus. Pellentesque nec posuere ligula, eu congue leo. Integer vulputate tempor arcu. Vestibulum vulputate Vestibulum vitae aliquam nunc. Suspendisse mollis metus ac tellus egestas pharetra. Suspendisse at viverra purus. Pellentesque nec posuere ligula, eu congue leo. Integer vulputate tempor arcu. Vestibulum vulputate

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