Polar equation: . Cartesian parametrization: . Complex parametrization: . Cartesian equation: . Rational sextic.

Given a circle (C) with centre O and radius a and a line (D) passing by O ((D) is here Ox), the Ceva trisectrix is the locus of the point M such that OP = PQ = QM with P on (C), Q on (D) and such that O, P and M are aligned. The angle xOM is the third of the angle xQM, hence the name of trisectrix. Compare to the construction of the Maclaurin trisectrix.

The complex expression , equivalent to the previous construction, shows that the Ceva trisectrix is a polytrochoid, as the composition of 3 uniform circular motions.
Like all tritrochoid, the Ceva trisectrix is the locus of the gravity centre of 3 circular motions.

This curve is also a conchoid of the quatrefoil (therefore, a conchoid of a rose).
 

The construction above can be continued, as shown in the figure below:

The curve of order n, with polar equation , is a (2n+1)-sectrix, and is called "Ceva sectrix".

the 5-sectrix of Ceva

© Robert FERRÉOL 2017