Durer's Shell Curves

Cartesian equation:
(x2+xy+axb2)2=(b2x2)(xy+a)2(x^{2} + xy + ax - b^{2})^{2} = (b^{2} - x^{2})(x - y + a)^{2}


These curves appear in Dürer's work Instruction in measurement with compasses and straight edge(1525).

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 QRPQRP and PQRP'QR of length 16 units through Q(q,0)Q (q, 0) and R(0,r)R (0, r) where q+r=13q + r = 13. The locus of PP and PP' as QQ and RR move on the axes is the curve. Dürer only found one of the two branches of the curve.

The envelope of the line PQRPP'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=0b = 0 : Curve becomes two coincident straight linesx2=0x^{2} = 0.

a=0a = 0 : Curve becomes the line pair x=b/2,x=b/2x = b/√2, x = -b/√2
together with the circle x2+y2=b2x^{2} + y^{2} = b^{2}.

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