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Locus of light falling on a plane surface

  1. Apr 13, 2012 #1
    When there is a lamp on a wall and the light from the lamp falls on the wall, we notice that the shape formed by the light is a hyperbola. I would like to know what the explanation for this is.
    Also, when you have a convex lens and you kee it flat on a horizontal surface, and sunlight falls at an angle on to the lens, you can see a curve of light on the surface that looks to me very much like a hyperbola.
    A hyperbola is the locus of a point moving such that the difference of its distances from two fixed points is constant. That makes sense when you're wondering why the locus of an interference pattern should be a hyperbola, because the path difference is constant. But I can't relate this to the phenomenon of light falling on a wall, or the lens thing.
    Thanks in advance!
     
  2. jcsd
  3. Apr 14, 2012 #2

    haruspex

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    I assume the light has a shade with a circular opening.
    The light passing through the opening projects a cone.
    The intersection of a cone with a plane produces various "conic sections" depending on the angle.
    If the axis of the shade's opening is parallel to the wall then you will see a hyperbola.
    If you tilt the shade down so as to direct the light more fully onto the wall, at a critical angle (i.e. when the light falling furthest from the wall is going straight down) it becomes a parabola.
    Tilt a fraction more and you have an ellipse.

    In general, conic sections are the curves that satisfy quadratic equations. The locus of intersection of a cone with a plane will also satisfy a quadratic equation. And the locus of a point that's always further from one fixed point than another by a constant amount is also a quadratic equation.

    Lenses are ground with spherical surfaces - more quadratics. So it wouldn't surprise me if that also gives you hyperbolae, though I haven't checked it in detail.

    Btw, have you noticed the pattern light from a point source forms at the bottom of a cylindrical cup? No conic this time.
     
  4. Apr 14, 2012 #3
    Thank you so much!! This explained things perfectly. I will try to check out the light source at the bottom of a spherical cup. :)
     
  5. Apr 14, 2012 #4

    haruspex

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    Actually I said cylindrical, the usual shape for a mug, and the light source is outside the mug. It's the pattern produced at the bottom that's interesting.
     
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