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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 radius R passing through O
and a line parallel to the tangent at O at distance 2R from
the circle (here (C) is the circle with diameter [OA] where
A(a,0)).
construction as the medial curve of a circle and a straight line 
equivalent cissoidal construction 
 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).
 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)).




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, ). 
This figure is not to be mistaken for the plot of the tractrix and its evolute, the catenary. 

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© Robert FERRÉOL 2017