Christmas Tree Light: Investigating the Math Behind Its Curve

[Livethefire]

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:
$$x^2+y^2=const.$$
And Hyperbola:
$$x^2-y^2=const.$$
And subsitution for 90 degrees rotation? :
$$y\rightarrow iy$$

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

 

[tiny-tim]

Hi Livethefire!

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 ​

 

[Livethefire]

Ah yes!

But is there any relevance or justified motivation to present such a thing by substituting "iy" in a circular equation?

 

[tiny-tim]

not following you

 

[Livethefire]

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.

 

[tiny-tim]

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)