The conchoid of Nicomedes –
the three cases:
a = b (blue curve); a > b (white curve); a < b (red curve).
A conchoid is a shell-shaped curve. Given a point A and a curve C, if we pick a point Q on C and draw a line L through A and Q and mark points P and P' on L at some fixed distance in either direction from Q, then the locus of P and P' as Q moves on C is a conchoid.
The conchoid of Nicomedes is a conchoid in which the given line is a straight line; i.e., given a line C and a point A we pick a point Q on C, draw a line L through A and Q, and mark P and P' on L at some fixed distance from Q. The conchoid of Nicomedes is the locus of P and P' as Q moves along C. It has the polar equation r = a sec θ + k.
The conchoid of de Sluze is the curve with the Cartesian equation a(x – a)(x 2 + y 2) = k 2x 2.