CISSOID OF DIOCLES Curve studied by Diocles, 180 BC; Fermat; Huygens. From the Greek Kissos: ivy, probably in reference to the nervures... Diocles (2nd century BC): Greek mathematician. Polar equation: . Cartesian equation: (the curve is studied here).  Rational circular cubic with a cuspidal point. Rational Cartesian parametrization: , i.e. (where t = tanq). Cartesian tangential angle: . Curvilinear abscissa: , . Radius of curvature: . Area between the curve and its asymptote: .

The simplest construction of cissoid of Diocles is by double projection on two parallel lines: given two parallel lines (T) and (T') and a point O on (T'), a variable point P on (T) is projected on the point Q on (T'), which in turn is projected on M on (OP): the cissoid of Diocles is the locus of M. Like all rational circular cubics, the cissoid of Diocles can also be defined as:

 - the cissoid with pole O of a circle with diameter [OA] where A(a,0)...  ... and therefore also the medial curve of a circle and a straight line - the pedal of a parabola with respect to its vertex (here the parabola with vertex O and focus F, the symmetrical image of A about O)... .. and therfore also the envelope of circles, centered on a parabola, and passing through its vertex. - the inverse of a parabola with respect to its vertex (here, the parabola with vertex O and focus A, the circle of inversion being the circle with centre O passing by A)). Like all right rational circular cubic, the cissoid of Diocles can be constructed...
 ...by Newton's set square method: ...as a kieroid  It can also be defined as:

 - the locus of the vertex of a parabola rolling without slipping on an isometric parabola such that the two parabolas are outside of one another and their vertices eventually meet (see orthotomic). -  the locus of the focus of a variable parabola with fixed vertex passing by a fixed point (see glissette). - the orthocaustic of a cardioid with respect to its summit (here, the cardioid with cusp at (a, 0) and summit at (4a, 0)). - a Rosillo curve: given a diameter [BC] of a circle and a point P describing this circle, the cissoid is the locus of the intersection point of the perpendicular to [BC] passing through P and the parallel to (CP) passing by B. Finally, it is a special case of cubic hyperbola and of an ophiuride. The cissoid of Diocles is a duplicatrix: if B is the point with coordinates (0, 2a) and C the intersection point of (C) and (AB), the coordinates of the intersection point X of (OC) and (T) are (a, ). The evolute of the cissoid of Diocles is the polynomial quartic . This figure is not to be mistaken for the plot of the tractrix and its evolute, the catenary. Its polar curve with respect to the circle with centre O and radius a is a semicubical parabola: .