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2013-11-27T23:57:34Z

72.214.61.187: /* Calculation */

[[Image:DuererMuschellinie.png|thumb|220px|right|Conchoid of Dürer, constructed by him]]

The '''conchoid of Dürer''', also called Dürer's shell curve, is a variant of a [[Conchoid (mathematics)|conchoid]] or [[Plane (mathematics)|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

:::<math>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 . \,</math>

==Properties==
The curve has two components, asymptotic to the lines <math>y = \pm a / \sqrt2</math>. 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 x / \sqrt2</math> together with the circle <math>x^2+y^2=a^2</math>;

==History==
It was first described by the German [[Painting|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==
* {{cite book | author=J. Dennis Lawrence | title=A catalog of special plane curves | publisher=Dover Publications | year=1972 | isbn=0-486-60288-5 | pages=157–159 }}

{{DEFAULTSORT:Conchoid of Durer}}
[[Category:Algebraic curves]]
[[Category:Albrecht Dürer]]

{{geometry-stub}}

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