The class of curve known as Dürer's conchoid appears 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
 |
There are a number of interesting special cases. For 
, the curve becomes the line pair 
together with the circle 
. If 
, the curve becomes two coincident straight lines 
. 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.
