Dürer's Shell Curves

Cartesian equation: (x + xy +ax - b) = (b - x)(x-y+a)

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 a lines QRP and P'QR of length 16 units through Q (q,0) and R (r,0) where q+r=13. The locus of P and P' is the curve. Dürer only found one of the two branches of the curve.

The envelope of the lines QRP and P'QR 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 = 0.

a = 0 : Curve becomes the line pair
x = b/2, x = -b/2
together with the circle x + y = b.

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

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