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 ).

With the notations of this page, we have: .
Therefore, in the frame centred on (a, 0), the base is the parabola: , the rolling curve the Kampyle of Eudoxus with polar equation and the roulettes are the curves: which give conchoids for . Therefore, the conchoids of straight lines are the loci of points on the transerval axis of a Kampyle rolling without sliding on a parabola. The other roulettes of this movement are the loci of the point M forming a constant angle when describes the line (D). Opposite are drawn a conchoid of a line and another roulette, obtained for ; the latter curve was named "orthoconchoid of a line" by Neuberg in 1904, and any roulette could be called "isoconchoid of a line". Loria p. 274.

The movement obtained when swapping the base and the rolling curve is the Kappa. See here the reason why portions of conchoids of Nicomedes appear in a conical anamorphosis. Conchoid drawer