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DÜRER'S SHELL CURVE

Curve studied by Albrecht Dürer in 1525.
See also: http://www-groups.dcs.st-and.ac.uk/%7Ehistory/Curves/Durers.html and  K. Fladt p. 236.

Given a fixed point A(a,0) and a constant b > 0, let a line (PQ) vary in such a way that P describes Ox and Q describes Oy, with ; Dürer's shell curve, that the engraver had designed from an articulated system, is the locus of the points M on the line (PQ) such that PM = b.
 
Parametric system:  ().
Cartesian parametrization 1: .
Cartesian parametrization 2:  ().
Cartesian equation: .
Rational bicircular quartic.

The parametric system above shows that Dürer's shell curve is the projection on xOy of the biquadratic, intersection of the elliptic cylinder  with the hyperbolic cylinder .
 
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© Robert FERRÉOL  2017