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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. 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
\( 2y^2(x^2+y^2) - 2by^2(x+y) + (b^2-3a^2)y^2 - a^2x^2 + 2a^2b(x+y) + a^2(a^2-b^2) = 0 . \, \)
Properties
The curve has two components, asymptotic to the lines \( y = \pm a / \sqrt2 \). 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 \( y = \pm x / \sqrt2 \) together with the circle \( x^2+y^2=a^2 \);
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
Conchoid of de Sluze
List of curves
References
J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 157–159. ISBN 0-486-60288-5.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
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