Dürer's Conchoid

These curves appear in Dürer's work Instruction in Measurement with Compasses and Straight Edge (1525) and arose in investigations of perspective. Dürer constructed the curve by drawing 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, although Dürer found only 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.

Dürer called the curve ``Muschellini,'' which means Conchoid. However, it is not a true Conchoid and so is sometimes called Dürer's Shell Curve. The Cartesian equation is

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

The above curves are for $(a,b)=(3,1)$, $(3,3)$, $(3,5)$. There are a number of interesting special cases. If $b = 0$, the curve becomes two coincident straight lines $x = 0$. For $a = 0$, the curve becomes the line pair $x = b/2$, $x = -b/2$, together with the Circle $x+y = b$. If $a = b/2$, the curve has a Cusp at $(-2a,a)$.

References

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 157-159, 1972.

Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, p. 163, 1967.

MacTutor History of Mathematics Archive. ``Dürer's Shell Curves.'' http://www-groups.dcs.st-and.ac.uk/~history/Curves/Durers.html.