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PANCAKE CURVE
Since this curve resembles the edge of a curved circular pancake (when it is being flipped), and it does not have an official name, I decided to call it "pancake curve".
Opposite, read a text by J.E. Mebius on this topic. 
As far as I know, this curve doesn't have any name of its own. However, it is closely related to a famous item of 19thcentury mathematics, the cylindroid surface, discovered by William Kingdon Clifford during his research into the theory of screws. The equation of the cylindroid in 3D Cartesian coordinates commonly reads z = (xx  yy) / (xx + yy). Turning the whole thing thru 90 deg about the Z axis yields z = 2xy / (xx + yy), and there you are: your curve is the intersection of this cylindroid and the unit cylinder about the Z axis. This is generic: cylindroid and cylinder with common axis always intersect in this kind of space curve. 
Cartesian parametrization:
form #1: ; form #2: (rotation by with respect to the previous form). Rational biquadratic (3D quartic of the first kind). 
The pancake curve can be obtained as the intersection between a cylinder of revolution () and:
 a hyperbolic paraboloid with the same axis ( with for form #1)
 a Plücker conoid with the same axis: ( for form #1)
 a parabolic cylinder with the line at the summit perpendicular to the axis of the cylinder ( for form #2).
Intersection with a hyperbolic paraboloid 
Intersection with a Plücker conoid 
Intersection with a parabolic cylinder 
All in all, there are 6 definitions as intersection between these 4 surfaces.
The pancake curve is a special case of cylindrical sine wave; therefore if we make it roll on a plane, the contact point describes a sinusoid:
The projection on xOy is a circle; the projections on xOz and yOz are isometric lemniscates of Gerono for form #1 and portions of parabolas for form #2.
The projections on the planes passing by Oz are the besaces (first animation).

See also at bicylindrical curve for a similar curve.
Despite the name I gave it, the curve must not be mistaken for another similar one: the curve described by the edge of a circular pancake with radius b placed on a cylinder with radius a, parametrized by: .  This curve is transcendental, contrary to the one studied here.
It develops into a circle when the cylinder is developed: it is a geodesic circle of the cylinder. Moreover, it has double points when . 


Compare to the intersection between a hyperbolic paraboloid and a sphere.
See also Hector Guimard's curve and a sine torus.
The edges of Pringles, which resemble hyperbolic paraboloids, look like the pancake curve...
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© Robert FERRÉOL 2018