If the center of the cone is in the plane, the intersection is a point, a straight line, or a pair of straight lines, depending on the angle of the axis of the cone.
If the center of the cone is not in the plane, the intersection is a conic section. Let
be the angle of the cone, that is, the angle between the axis and one of the generating lines of the cone. You get a circle if the angle is
or
, an ellipse if the angle is between
and
(or between
and
), a parabola if the angle is
, and a hyperbola if the angle is within
of
.
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