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SPHEROCYLINDRICAL CURVE
Other name: cyclocylindrical curve. 
The spherocylindrical 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: eightlike 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, that is a spherical ellipse when the axes of the cylinder goes through the center of the sphere. 
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© Robert FERRÉOL 2022