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CYLINDROCONICAL CURVE
Cylindroconical curves are the intersections between a cylinder of revolution and a cone of revolution. Therefore, they are biquadratics.
In the case of a cylinder with radius a the axis of which is perpendicular to that of a cone with halfangle at the vertex equal to 2α:
System of Cartesian equations: where and d is the distance from the vertex of the cone to the axis of the cylinder. Cartesian parametrization: . 
d = 0 
Here, d = a: the vertex of the cone is an isolated point of the curve. 


In the case of a cylinder with radius a the axis of which is parallel to that of a cone with halfangle at the vertex equal to 2α:
System of Cartesian equations: where and d is the distance from the vertex of the cone to the axis of the cylinder. Cartesian parametrization: . 
d = 0: two circles. 
0 < d < a 
d = a: Viviani curve. 
d > a 
Naturally, there are a lot of different cases.
Image sent by Julien Vorpe
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© Robert FERRÉOL 2018