BICYLINDRICAL CURVE

 Other name : Steinmetz 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 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!   © Robert FERRÉOL  2018