perspective of a conic section

[Richard Baker]

I am working on a TI Inspire video game, in which the player is in a dark room holding a flashlight.  The flashlight is located at the player's eye and is pointed in the look direction. I render the light by coloring in the conic sections formed by the intersection of each wall, floor, or ceiling and concentric cones of light starting from the innermost cone working outward. As it turns out whenever the conic section formed is an ellipse, it appears to transform into a perfect circle located at the dead center of the viewport after performing the perspective rotations and z divide.  This appears to always be the case regardless of the relative position and orientation of the wall. floor, or ceiling.  I am not certain if this is the case for hyperbolic conic sections. My question is what shape will the conic section appear to have when the conic section is a hyperbola?

Thank you.

[Richard Baker]

Correction on third line:  outermost cone working inward. 

Also the radius of each concentric circle is tan of θ .  Is that correct?