Conchoid of Dürer
In geometry, the conchoid of Dürer, also called Dürer's shell curve, is a 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 isIn 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:a = 0: the line y = 0;b = 0: the line pair together with the circle ;
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.