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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. 

The Ulysse
Lacoste rulpidon, a Steinmetz
solid pierced with two full cylinders, reveals 10 edges which are 5
bicylindrical curves
one of which is formed by two ellipses.


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