**Cartesian equation: **

(*x*^{2} + *xy* + *ax* - *b*^{2})^{2} = (*b*^{2} - *x*^{2})(*x* - *y* + *a*)^{2}

These curves appear in Dürer's work

Dürer calls the curve 'ein muschellini' which means a conchoid, but since it is not a true conchoid we have called it Dürer's shell curve (muschellini = conchoid = shell).

This curve arose from Dürer's work on perspective. He constructed the curve in the following way. He drew lines *QRP* and *P*'*QR* of length 16 units through *Q* (*q*, 0) and *R* (0, *r*) where *q* + *r* = 13. The locus of *P* and *P*' as *Q* and *R* move on the axes is the curve. Dürer only found one of the two branches of the curve.

The envelope of the line *P*'*QRP* is a parabola and the curve is therefore a glissette of a point on a line segment sliding between a parabola and one of its tangents.

There are a number of interesting special cases:

In the above formula we have:

*b* = 0 : Curve becomes two coincident straight lines *x*^{2} = 0.

*a* = 0 : Curve becomes the line pair *x* = *b*/√2, *x* = -*b*/√2

together with the circle *x*^{2} + *y*^{2} = *b*^{2}.

*a* = *b*/2 : The curve has a cusp at (-2*a*, *a*).

JOC/EFR/BS January 1997

The URL of this page is:

http://www-history.mcs.st-andrews.ac.uk/Curves/Durers.html