The conchoid of Dürer, also called Dürer's shell curve, is a variant of a conchoid

Conchoid (mathematics)

A conchoid is a curve derived from a fixed point O, another curve, and a length d. For every line through O that intersects the given curve at A the two points on the line which are d from A are on the conchoid. The conchoid is, therefore, the cissoid of a circle with center O and the given curve...

 or plane

Plane (mathematics)

In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

 algebraic curve

Algebraic curve

In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...

. 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:

• a=0: the line y=0;
• b=0: the line pair together with the circle ;

History

It was first described by the German painter

Painting

Painting is the practice of applying paint, pigment, color or other medium to a surface . The application of the medium is commonly applied to the base with a brush but other objects can be used. In art, the term painting describes both the act and the result of the action. However, painting is...

 and mathematician

Mathematician

A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

 Albrecht Dürer

Albrecht Dürer

Albrecht Dürer was a German painter, printmaker, engraver, mathematician, and theorist from Nuremberg. His prints established his reputation across Europe when he was still in his twenties, and he has been conventionally regarded as the greatest artist of the Northern Renaissance ever since...

(1471–1528) in his book Underweysung der Messung (S. 38), calling it Ein muschellini.