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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:.
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 4a2 when a = b.

Case a = b:
 The curve is invariant under the two half-turns 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.

: figure-eight 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 iso-energy 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 half-turn.


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