Definitions

# Conchoid of Dürer

The Conchoid of Dürer also called Dürer's shell curve is a variant of a conchoid or plane algebraic curve. It is not a true conchoid.

## Construction

Let Q and R be points moving on a pair of perpendicular lines which intersect at O in such a way that OQ + OR is constant. On any line QR mark point P at a fixed distance from Q. The locus of the points P is Dürer's conchoid.

## Equation

The equation of the conchoid in Cartesian form is

$2y^2\left(x^2+y^2\right) - 2by^2\left(x+y\right) + \left(b^2-3a^2\right)y^2 - a^2x^2 + 2a^2b\left(x+y\right) + a^2\left(a^2-b^2\right) = 0 . ,$

## Properties

The curve has two components, asymptotic to the lines $y = pm a / sqrt2$. Each component is a rational curve. If a>b there is a loop, if a=b there is a cusp at (0,a).

Special cases include:

• a=0: the line y=0;
• b=0: the line pair $y = pm x / sqrt2$ together with the circle $x^2+y^2=a^2$;

## History

It was first described by the German painter and mathematician Albrecht Dürer (1471–1528) in his book Underweysung der Messung (S. 38), calling it Ein muschellini.