1911 Encyclopædia Britannica/Logocyclic Curve


LOGOCYCLIC CURVE, STROPHOID or FOLIATE, a cubic curve generated by increasing or diminishing the radius vector of a variable point Q on a straight line AB by the distance QC of the point from the foot of the perpendicular drawn from the origin to the fixed line. The polar equation is r cos θ = a(1 ± sinθ), the upper sign referring to the case when the vector is increased, the lower when it is diminished. Both branches are included in the Cartesian equation (x2 + y2)(2ax) = a2x, where a is the distance of the line from the origin. If we take for axes the fixed line and the perpendicular through the initial point, the equation takes the form y √(ax) = x √(a + x). The curve resembles the folium of Descartes, and has a node between x = 0, x = a, and two branches asymptotic to the line x = 2a.