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 (0, 0). Let points Шаблон:Math and Шаблон:Math move on the axes in such a way that Шаблон:Math, a constant. On the line Шаблон:Math, extended as necessary, mark points Шаблон:Math and Шаблон:Mvar at a fixed distance Шаблон:Mvar from Шаблон:Math. The locus of the points Шаблон:Math and Шаблон:Mvar is Dürer's conchoid.[1]
x &= \frac{b \cos(t)}{\cos(t) - \sin(t)} + a \cos(t),\\
y &= a \sin(t),
\end{align}</math>
where the parameter Шаблон:Mvar is measured in radians.[2]
Properties
The curve has two components, asymptotic to the lines <math>y = \pm a / \sqrt2</math>.[3] 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 <math>y = \pm a / \sqrt2</math> together with the circle <math>x^2+y^2=a^2</math>;
The envelope of straight lines used in the construction form a parabola (as seen in Durer's original diagram above) 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.[4]
History
It was first described by the German painter and mathematicianAlbrecht Dürer (1471–1528) in his book Underweysung der Messung (Instruction in Measurement with Compass and Straightedge p. 38), calling it Ein muschellini (Conchoid or Shell). Dürer only drew one branch of the curve.
