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BICYLINDRICAL CURVE
Other name : Steinmetz curve. 
The bicylindrical curves are the intersections between two cylinders of revolution.
First case : two orthogonal cylinders with radii a
and b, and axis at distance 2c.
System of Cartesian equations:.
Biquadratic. Cartesian parametrization: . Cartesian equation of the projection on xOy: (see the Alain curve). Area of the portion of cylinder delimited by each component, for and c = 0, given by an elliptic integral of the second kind: , that reduces to 4a^{2} when a = b. 
Case a = b:
The curve is invariant under the two halfturns
that swap the two cylinders.

Small c. 
c = a /2 
Case a < b:
(here c = 0): curve with two components. 
See the Alain curve. 

One can notice that the bicylindrical curve is traced
on the ellipsoid .
By scaling, we can turn the ellipsoid into a sphere, while the two cylinders become elliptic cylinders. The intersection obtained is one of the possible seam lines of a tennis ball. 
Coiling the isoenergy curves of the pendulum leads to bicylindrical curves that are the intersection between cylinders with perpendicular axes. 
Second case : two cylinders with secant axes, one of radius
a,
the other of radius b, forming an angle
with the plane orthogonal to the first.
System of Cartesian equations: .
Cartesian parametrization: , where . Case a = b : , union of two ellipses. 


The Swiss jeweler Philippe
Mingard uses bicylindrical curves for his creations (case a = b,
small c); he believes that this curve is "the manifestation of simplicity
and purity incarnate".
See also the case of the seam line of a tennis ball, or the pancake curve, other curves that are invariant under a halfturn. 

Beams of my chalet... 
Botzaris station, in the Parisian Metro. 
Lights in my staircase 
Many other examples on the mathourist's page!
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© Robert FERRÉOL 2022