Dürer's conchoid

 

Let there be two points Q(q,0) and R(0,r) where the sum of q and r is constant (a). Now construct points P on line l through RQ, so that PQ is equal to a constant (b). Then the locus of P defines Dürer's conchoid.
The definition is a variation on the (normal) conchoid.
Sometimes the curve is called Dürer's shell curve.

The curve appeared in a work of Albrecht Dürer (1471 - 1528), named 'Instruction in measurement with compasses and straight edge' (1525). He only found one of the two branches of the curve. Albrecht called the curve 'ein muschellini' ¹⁾. Dürer looked at the situation that a=13, for points P and P' where PRQ and RQP' both have a length of 16 units.

It can be shown that the envelope of the line P'QRP is a parabola.

notes

1) Muschellini = conchoid = shell.