Conchoid of Dürer

Conchoid of Dürer, constructed by him

The conchoid of Dürer, also called Dürer's shell curve, is a variant of a conchoid or plane algebraic curve, named after Albrecht Dürer. It is not a true conchoid.

Construction

Let Q and R be points moving on a pair of perpendicular lines which intersect at O in such a way that OQ + OR is constant. On any line QR mark point P at a fixed distance from Q. The locus of the points P is Dürer's conchoid.

Equation

The equation of the conchoid in Cartesian form is

Properties

The curve has two components, asymptotic to the lines . Each component is a rational curve. If a>b there is a loop, if a=b there is a cusp at (0,a).

Special cases include:

History

It was first described by the German painter and mathematician Albrecht Dürer (1471–1528) in his book Underweysung der Messung (S. 38), calling it Ein muschellini.

See also

References


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