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Christmas tree light

  1. Mar 1, 2012 #1
    Forgive the sloppy use of math and inability to produce an image. I noticed this last christmas.

    If you have a fairy light ( or perhaps any LED etc), and shine it normal to a surface, you see a circle. If you place the light flat on the surface you see a curve - to me the fairy lights' curve looks like a hyperbola.

    Does this have any relation to the equation of a circle:
    [tex]x^2+y^2=const.[/tex]
    And Hyperbola:
    [tex]x^2-y^2=const.[/tex]
    And subsitution for 90 degrees rotation? :
    [tex]y\rightarrow iy[/tex]

    If so, how does this even work? The experiment is all in real space. If not, is this just sloppy use of math? Any significance?

    Thanks
     
  2. jcsd
  3. Mar 1, 2012 #2

    tiny-tim

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    Hi Livethefire! :smile:

    If the light comes out in a cone,

    then the shape will be the intersection of a cone with a plane …

    in other words, a conic section :wink:
     
  4. Mar 1, 2012 #3
    Ah yes!

    But is there any relevance or justified motivation to present such a thing by substituting "iy" in a circular equation?
     
  5. Mar 1, 2012 #4

    tiny-tim

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    not following you :confused:
     
  6. Mar 1, 2012 #5
    What I was saying in post #1 was to sub iy for y in the first equation you get the second. In other words, rotating the axis 90 degrees changes the view from a circle to a hyperbola.

    Sometimes i is used as a 90 degree operator, yet I think my reasoning is unsound, thus i am asking here for insight.
     
  7. Mar 1, 2012 #6

    tiny-tim

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    if the cone has semiangle λ along the z-axis, then its equation is

    z2 = (x2 + y2)tan2λ,

    so a plane z = xtanθ + c cuts it at x2(tan2λ - tan2θ) - 2cxtanθ + y2tan2λ = c2,

    which is an ellipse or hyperbola according to whether λ is greater or less than θ

    (but i don't see where i comes into it)
     
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