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 interactive graph. By dragging the four interactive points the cases for point, circle, ellipse, parabola and hyperbola can be seen. 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). You can change position of the plane, tilting of the plane, position and opening of the cone.
The spherical view below shows what happens at "infinity" (=blue equatorial boundary of the sphere). One can thus go through 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. You can use the interactive dragging of points also in this spherical view. The changes will be displayed simultaneously in the avobe Euclidean view.
Compare the animated version.
Soli Deo Gloria. Created with Cinderella by Eckhard Hitzer (Fukui).