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Given F_1(x,y,z,u,v,w) = 0 (1) F_2(x,y,z,u,v,w) = 0 (2) F_3(x,y,z,u,v,w) = 0, (3) if the determinantof the Jacobian |JF(u,v,w)|=|(partial(F_1,F_2,F_3))/(partial(u,v,w))|!=0, ...

The pedal curve to the Tschirnhausen cubic for pedal point at the origin is the parabola x = 1-t^2 (1) y = 2t. (2)

For a unit circle with parametric equations x = cost (1) y = sint, (2) the negative pedal curve with respect to the pedal point (r,0) is x_n = (r-cost)/(rcost-1) (3) y_n = ...

Given a parabola with parametric equations x = at^2 (1) y = 2at, (2) the negative pedal curve for a pedal point (x_0,0) has equation x_n = (at^2[a(3t^2+4)-x_0])/(at^2+x_0) ...

The inverse curve of the Archimedean spiral r=atheta^(1/n) with inversion center at the origin and inversion radius k is the Archimedean spiral r=k/atheta^(-1/n).

The pedal curve of circle involute f = cost+tsint (1) g = sint-tcost (2) with the center as the pedal point is the Archimedes' spiral x = tsint (3) y = -tcost. (4)

The pedal curve of the cissoid, when the pedal point is on the axis beyond the asymptote at a distance from the cusp which is four times that of the asymptote is a cardioid.

The inverse curve of Fermat's spiral with the origin taken as the inversion center is the lituus.

The inverse curve of the logarithmic spiral r=e^(atheta) with inversion center at the origin and inversion radius k is the logarithmic spiral r=ke^(-atheta).

The inverse curve of a right strophoid with parametric equations x = (1-t^2)/(t^2+1) (1) y = (t(t^2-1))/(t^2+1) (2) for an inversion circle with radius 1 and center (1,0) is ...