Strophoid

A strophoid, also known as a logocyclic curve or a foliate, is a cubic curve generated by increasing or diminishing the radius vector of a variable point P on a straight line by the distance PA of the point from the foot of the perpendicular drawn from the origin to the fixed line.

The polar equation is

:$r=a\; cos2\; hetasec\; heta$.

The Cartesian equation is

:$y^2\; =\; x^2(a-x)/(a+x)$,

where "a" is the distance of the line from the origin. The curve resembles the Folium of Descartes, and has a node between "x" = 0, "x" = "a", and two branches asymptotic to the line "x" = −"a".The curve has two more asymptotes, in the plane with complex coordinates, given by:$xpm\; iy\; =\; -a$.

A curve: $r=a\; sin(alpha-2\; heta)\; /\; sin(alpha-\; heta)$

References

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