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BICYLINDRICAL CURVE
The bicylindrical curves are the intersections between two cylinders of revolution.
In the case of 2 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 a £ b 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.
c = 0: it is the reunion of two ellipses with eccentricity that intersect at their secondary vertices with a right angle. 
small c. 
c = a /2 
Case a < b:
(here c = 0): curve with two components. 
: figureeight curve similar to the hippopede. See the Alain curve. 
: curve with one component. 
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. 
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 2018