## 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 and of length 16 units through and , where . The locus of and is the curve, although Dürer found only one of the two branches of the curve.

The Envelope of the lines and 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

The above curves are for , , . There are a number of interesting special cases. If , the curve becomes two coincident straight lines . For , the curve becomes the line pair , , together with the Circle . If , the curve has a Cusp at .

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.