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SPHERO-CYLINDRICAL CURVE


Other name: cyclo-cylindrical curve.

The sphero-cylindrical curves are the intersections between a sphere and a cylinder of revolution.
 
For a sphere centered on O with radius a, a cylinder with radius b, and axis at distance c from O:
System of Cartesian equations: .
Biquadratic.
Cartesian parametrization: .

Case a £b:
The curve is not empty iff a - b  £ c £ a + b and, in this case, it has only one component.

Opposite, the case a = b = c.

Case b < a;
 
b + c < a: the curve has two components

b + c = a: eight-like curve called hippopede.

a - b  < c £a + b: curve with one component

 
 
It can be generalized by the intersection of the sphere and an elliptic cylinder, sometimes called spherical ellipse.
 
 

 
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© Robert FERRÉOL  2018