Dürer's Conchoid
These curves appear in Dürer's work Instruction in Measurement with Compasses and Straight Edge (1525) and arose ininvestigations of perspective. Dürer constructed the curve by drawing lines 
and 
of length 16 units through
and 
, where 
. The locus of 
and 
is the curve, although Dürer found only one of the twobranches of the curve.
The Envelope of the lines and is a Parabola, and the curve is thereforea Glissette of a point on a line segment sliding between a Parabola and one of its Tangents.
Dürer called the curve ``Muschellini,'' which means Conchoid. However, it is not a true Conchoid and so issometimes called Dürer's Shell Curve. The Cartesian equation is
The above curves are for
, 
, 
. There are a number of interesting special cases. If 
,the curve becomes two coincident straight lines 
. For 
, the curve becomes the line pair 
, 
, together with the Circle 
. If 
, the curve has a Cusp at 
.References
Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 157-159, 1972. Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, p. 163, 1967. MacTutor History of Mathematics Archive. ``Dürer's Shell Curves.''http://www-groups.dcs.st-and.ac.uk/~history/Curves/Durers.html.
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