Studying the hyperboloid model, what is represented by the conic sections?

I am trying to get my head around the hyperboloid model of hyperbolic geometry.

Hyperboloid_model

The article is much too technical for me, please improve.

And was thinking the hyperboloid $ x^2 +y^2 – z^2= -1: z >0 $
has as asymptotic cone $ x^2 +y^2 – z^2= 0: z >0 $

And with cones you have conic sections

ConicSection

Then I got puzzled what hyperbolic item is represented for each conic section, came to the following list:

Conic section -> Hyperbolic item

  • circle -> circle centered at (0,0,1)
  • ellipse -> circle not centered at (0,0,1 )
  • parabola -> horocycle
  • hyperbola (one half) plane through (0,0,0) (euclidean point) -> hyperbolic line
  • hyperbola (one half) plane not through (0,0,0) (euclidean point) -> hypercycle

Is this list correct? Are there possible additions?

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