# conchoid

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*^{ 2}*x*^{ 2}.