|Mar1-12, 01:24 PM||#1|
Christmas tree light
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:
And subsitution for 90 degrees rotation? :
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?
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|Mar1-12, 02:43 PM||#2|
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
|Mar1-12, 02:49 PM||#3|
But is there any relevance or justified motivation to present such a thing by substituting "iy" in a circular equation?
|Mar1-12, 02:54 PM||#4|
Christmas tree light
not following you
|Mar1-12, 02:58 PM||#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.
|Mar1-12, 03:21 PM||#6|
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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