Given two lines in a common plane in Euclidean space, there are
only two possible cases. Either they are parallel and have no
intersection or else they intersect at exactly one point. In the
first case, extending to projective space, the parallel lines
do intersect at a single point at infinity. Consider the locus of all
points equidistant from the two lines. In the first case the locus
is a line parallel to both and midway between them. In the second
case they are on two perpendicular lines passing through the common
intersection point. Pick any one of them. In both cases use the
line of equidistant points as a rotation axis to rotate the two
lines into a surface of revolution. In the first case it is an
infinite circular cylinder.
In the second case it is a circular
double cone. In the first case, for any plane that intersects the
cylinder, the points of intersection form a conic section --
either a circle, ellipse, or two parallel lines. For the second case,
every plane intersects the double cone in at least one point. The
points of intersection form a conic section -- either a single point,
a circle, an ellipse, a parabola, a hyperbola, a single line, or a pair
of intersecting lines.
In the case of intersecting lines, the plane
passes through the vertex of the double cone and the angle
of their intersection is less than or equal to the angle of the two
generating lines. The angles are only equal if the rotation axis lies
in the intersecting plane. As for visualization, try a web search for
conic section images.
