I know that parallel light beams hitting a parabola will be focused at the focus of the parabola (f = 1/4a) and a light source at the focus of the parabola will produce parallel light. What will happen if the light was not parallel but came from a light source shorter then the focus of the parabola and at an angle to the axis of symmetry? More general how will dispersement of the light from the parabola be affected with the light source at different angles and lengths from the parabola.


2 Answers 2


The behavior of the non-axial rays is illustrated on the picture below. Rays (red) falling in direction determined by the vector CD (in circle) reflect from the surface of a parabola (blue), forming an intersection at point J (red dot). The intersection point is obviously out-of original focus (yellow dot A). Tracking the direction vector shows the tracks of an intersection point (red and gray dots), which form a mustache-like pattern originating from A.

Non-axial rays into parabola

More interesting is the following picture - it shows that parallel rays do not even focus in a single point at all! One pair of rays intersect at J (red), while other pair intersect at N (green). Green and red tracks are different, so rays do not focus. They are smeared along a (probably linear) path consisting J-N.

Non-axial rays

  • $\begingroup$ Will more than two rays intersect pairwise at the same point? $\endgroup$ Oct 17, 2011 at 10:51
  • $\begingroup$ @Niel de Beaudrap: Added the picture describing that they won't. $\endgroup$
    – mbaitoff
    Oct 17, 2011 at 10:54
  • $\begingroup$ The parallel rays only concentrate on the focus if they are parallel with the symmetry axis, sorry that is what I meant should have phrased that better. $\endgroup$
    – Cornelius
    Oct 17, 2011 at 14:06

Not sure if this answers your question, but a parabola can be seen as a limit case of the ellipse. An ellipse has two focuses, and the light coming from one source focuses on another one. Take one of the focuses to infinity - and you've got a parabola.

As a zero-order approximation parabola focuses the light from a point light source, which is far enough. A first-order correction should be linear in the angle IMHO. The exact dispersion is probably complex.

  • $\begingroup$ ""he exact dispersion "" Dispersion is something which does not occur in reflection! $\endgroup$
    – Georg
    Oct 17, 2011 at 10:51
  • $\begingroup$ @ Georg: I think valdo meant the dispersion of a rays (displacement), not the dispersion of a light (wavelength-dependancy). $\endgroup$
    – mbaitoff
    Oct 17, 2011 at 10:56
  • 2
    $\begingroup$ @mbaitoff: I think Georg understood this as well (this is pretty clear from the context). It's just IMHO Georg is too much perfectionist, it's his duty to comment & downvote every questionable word :) $\endgroup$
    – valdo
    Oct 17, 2011 at 12:02

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