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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 19th-century 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).
The projections on the planes passing by Oy (form #2), give a portion of parabola and an ovoid quartic (second animation).
The projections on the planes containing Oy (form #1), give a circle and the piriform quartic (third 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