were
k is a constant.
Abstract: The purpose of this project is to gain more insight into the Conchoid of Nicomedes.
Derivation of Equations:
Let 0 be a fixed point, and let L be a line through O intersecting curve AQC at a point Q. The Locus of points P1 and P2 such that
were
k is a constant.
Parametric Equation of Conchoid:
Now let t be the variable of the parametric equations.
L has equation
Or equivalent to
If P (x,y) represents either P1 or P2; then, since P is on L,
However, since
Or, using
Therefore,
Now Using Eq.3 and EQ.4 the resultant Parametric Equations of the Conchoid are
Polar Equation of Conchoid:
If 
is
the origin, then L has polar equation
Constant. 
Hence
So the polar equation for the Conchoid is
Examples:
The following figures show the curve family of a common line. Their pole is at the origin, and directrix y==1. The figure on the top has constants k from -2 to 2. The figure below has constants k from -100 to 100.
A Sinusoid:
The point O is called the pole. In the following figure, the red dots are the conchoid of a sinusoid with pole at {-3,3} and k:=2.
Brief History:
The conchoid was a favorite of the 17th century mathematicians.
According to modern records, the conchoid of Nicomedes
was first derived around 200 B.C. by Nicomedes. Its purpose was to solve
the angle trisection problem.
Bibliography:
Special Plane Curves, Xah Lee, http://www.best.com/~xah/SpecialPlaneCurves_dir/specialPlaneCurves.html
Index of Curves, jan wassenaar, http://www.mirajan.demon.nl/curves/QuarticCurve/QuarticCurveCo.htm
A Book of Curves, E.H. Lockwood, Cambridge University Press
A Catalog of Special Plane Curves, J. Dennis Lawrence, Dover Publications, Inc.