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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 4a2
when a = b. |
Case a = b:
The curve is invariant under the two half-turns
that swap the two cylinders.
 
that intersect at their secondary vertices with a right angle. |
Small c. |
c = a /2 |
Case a < b:
|
  :
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. |  |
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 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 2022
(here
c
= 0): curve with two components.