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Viewing Light Polarization as a Ruled Surface

Have you ever been looking at your phone with sunglasses on and turned your phone? At one orienation, the light goes straight through the lenses, but at the other orientation the screen goes completely black. This a result of the light from your phone being polarized from the sunglasses. Light can be modeled as a sinusoidal wave that moves through space. When polarized, this light wave is rotated 90 degrees moving forward in the same direction but lies parallel with a different plane. Even cooler, light can also be represented as a particle, classic duality stuff. So, if the wave is rotated, how would the particle move in terms of this polarization?

What is a Ruled Surfaces

A ruled surface is a surface such that every point on the surface has a straight line that goes through it that lies on the surface. Another way to think about these surfaces is to consider two functions that can have lines drawn between corresponding points on each function. That is to say that for any f(x), there is a corresponding g(x) where a straight line can be drawn between the two points. The bounds of these lines f(x) and g(x) exist only to help create the surface. In actuallity, these lines would extend off to infinity to create the ruled surface.

Imagining how Light Moves

Earlier, I mentioned how light exists both as a particle and a wave. Polarization results in the wave being rotated 90 degrees and it makes sense to imagine the particle being translated along a path. If the wave function of a proton is modeled as: \[ f(t) = (0, t/10, 6*\sin(t)+20) \]. This is not meant to be accurate to the actual shape of a proton wave, but exists as an example for visualization. If this function were to be polarized, the wave function could look like: \[ g(t) = (t/10, 6\sin(t)+20,0) \]. Here is a picture of the two functions in 3-space:
For every t, a line could be drawn from f(t) to g(t) and create a ruled surface. The following picture shows that ruled surface:

Significane of the Surface

The mapping of the light wave from one function to another seems pretty inutuitve. A rotated sinusoidal curve is still a sinusoidal curve. What is more interesting is the surface that is created between the two curves. Not only is this curved surface composed of straight lines, but the straight lines can also represent how light moves when viewed as a particle when polarized. The lines between the function show the path of the particle as it moves through a polarized lense. What makes this more interesting is that light always moves in the most efficient way and in this case, that would be the straight line created between the two functions. The polarization of light seemed like an excellent connection showing how curved surfaces can be formed entirely straight lines. The ruled surface is not the most accurate explanation for the behavior of the particle-wave form of a proton, but I thought it would be interesting to investiage this phenomenon in the scope of ruled surfaces.

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