Dürer's Conchoid

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 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


There are a number of interesting special cases. For b=0, the curve becomes the line pair x=+/-a/2 together with the circle x^2+y^2=a^2. If a=0, the curve becomes two coincident straight lines x=0. If a=b, the curve has a cusp at (0,a).

Explore with Wolfram|Alpha


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."

Cite this as:

Weisstein, Eric W. "Dürer's Conchoid." From MathWorld--A Wolfram Web Resource.

Subject classifications