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).

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