CONCHOID OF NICOMEDES

 From the Greek Kogkhoeidês: similar to a shell. Nicomedes (2nd century BC): Greek mathematician.

 Polar equation:  . Cartesian equation:  or . Rational Cartesian parametrization:  (with  ). Rational circular quartic.

The conchoids of Nicomedes are the conchoids of straight lines (here, the line is (D) and its equation is x = a; b can be considered to be positive without loss of generality).

They have two infinite branches for which the line (D) is an asymptote, the left one being ordinary for 0 < b < a, with a cuspidal point for b = a and with a loop for b > a.

 0 < b < a a = b b > a

The conchoids of Nicomedes are also the cissoids of the circle with centre O and radius b and the line (D) with respect to the centre of the circle.

 The conchoids of Nicomedes are the trisectrix. Opposite, we see the trisecting of an angle of 30° (); note that to each angle  to be trisected corresponds a different conchoid (). Method: draw a triangle OHI with a right angle in H, such that OIH is the angle to trisect. Draw the conchoid of the line (IH) with pole O and modulus OI. The circle with centre I and radius OI cuts the conchoid at M, symmetrical image of O about I and a second point N, the construction of which can only be approximative. The trisected angle is NIJ.

For b = 2a, the conchoid of Nicomedes is also a duplicatrix (see [GomesTexeira] page 266, or [Carrega] page 72).

One can also take an interest, more generally, to the movement of a plane over a fixed plane, called linear conchoidal movement, the moving plane being linked to the line (O),describing the line (D) (and  being fixed in the moving plane): indeed, the conchoids of Nicomedes are the roulettes of this movement, for tracing points located on the line (O).