Today I’m going to do something that one should never do. I’m claiming that I possibly got the equations correct for the vector description of the theory of everything. For a very long time I’ve been hitting my head against the wall with the attempt to describe how accurate refraction takes place and before I realized there were no true ellipses involved, the mathematics just seemed impossibly hard.
However, as soon as I realized that there are only two circles involved in refraction and that the only ellipticity in in their two-dimensional projections, I also realized that the mathematics would have to be simple.
Well, simple is a relative term. I still had to draw the below image and write the equations describing the shape.
However, it took about a day for me from figuring out the shape to come up with the (probably) correct equations. All of these equations boiled down to the angle φ, as depicted in the above image. Rather surprisingly, to just get the location of the refracted dot, you don’t even need to know what all the other angles are. From the above image we see that with a tilt of zero degrees, the non-refracted dot (in green) is located r up along the (unnamed) z axis and by √2r horizontally along the x axis. And while this is an x-z projection, the non-refracted dot would be r deep along the (invisible) y-axis.
To make the mathematics a bit easier, I shift the x-z plane by φ and only depicted the shifted dots along this shifted plane. But if one wanted to describe the shifted dot along the original plane, one would have to add a couple of extra terms. Not difficult ones, but I’ll not include them, so that the equations are at least a bit easier.
So, if I haven’t botched my equations, the length of the shifted vector along the three axes are:
I’m only moderately confident that these are the right equations. I’ve been hunting for them for so long that I’d be pleasantly surprised if there still wasn’t a small or possibly even a large error hiding somewhere.
However, if these figures really are correct, I can use these to draw refraction vectors in Excel. And usually, Excel is the perfect tool to observe errors. If the equations don’t make any sense, then this will clearly result in nonsensical results in Excel. However, if Excel produces sensible results, I can be quite confident that I’m correct.
In the lucky occasion that Excel confirms my equations, I can start working on my second Theory of Everything -manuscript. But this time with much more rigorous mathematics.
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