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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











 

See also this German website: www.fh-lueneburg.de/u1/gym03/expo/jonatur/wissen/mathe/kurven/konchoid.htm#oben
 
 
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© Robert FERRÉOL, Jacques MANDONNET 2017