The justification for calling all of the following
* points (tip of the cone)
* pairs of intersecting straight lines (point of intersection: tip of cone)
* circles (green plane of intersection perpendicular to cone axis)
* parabolas (green plane paralell to generating straight line of cone)
"conics" is seen in this animation. The animation demonstrates the case for point, circle, ellipse, parabola and hyperbola. The case of the pair of intersecting straight lines would need the green plane contain not only the tip of the cone, but also part of the mantle (pairs of generating straight lines).
The spherical view below shows what happens at "infinity" (=blue equatorial boundary of the sphere). One can thus see the transition from a two branched hyperbola (each branch has two distince intersections with infinity) to one parabola (just "tangent" to infinity) and then to the single ellipse, which is completely in the finite realm.
Compare the interactive version.
Soli Deo Gloria. Created with Cinderella by Eckhard Hitzer (Fukui).