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

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

algebraic | closed | loopy | pedal

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