Conchoid of Dürer


The conchoid of Dürer, also called Dürer's shell curve, is a variant of a conchoid or plane algebraic curve, named after Albrecht Dürer and introduced in 1525. It is not a true conchoid.

Construction

Suppose two perpendicular lines are given, with intersection point O. For concreteness we may assume that these are the coordinate axes and that O is the origin, that is. Let points and move on the axes in such a way that, a constant. On the line, extended as necessary, mark points and at a fixed distance from. The locus of the points and is Dürer's conchoid.

Equation

The equation of the conchoid in Cartesian form is
In parametric form the equation is given by
where the parameter is measured in radians.

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.
Special cases include:
The envelope of straight lines used in the construction form a parabola and therefore the curve is a point-glissette formed by a line and one of its points sliding respectively against a parabola and one of its tangents.

History

It was first described by the German painter and mathematician Albrecht Dürer in his book Underweysung der Messung, calling it Ein muschellini. Dürer only drew one branch of the curve.