An exact Paraxial equation derivation, 100% Cartesian

[difalcojr]

TL;DR Summary

The equation for the focal point is valid for all refractions through a spherical convex surface.
There are no angle or other length approximations used in the derivation. See what you think of this.

 

[renormalize]

You obviously spent significant time and effort in preparing your post. Can you spend some more of that on learning LaTeX and posting everything, including your exposition, in nice, readable text format?

 

[difalcojr]

Years. Been a hobby, trigonometric ray tracing, for a long time.

I probably can reproduce the above in LaTex if I have to. But I don't know it at all and have limited time to give to learn it, if just to reproduce the above in that format. Am still working at a job is one reason. I thought that a neat, handwritten copy would probably suffice. Hope that doesn't prevent an analysis of it in this form, but, if needed, I can comply. @berkeman advised me on that recently too.

 

[Ibix]

As with your thread on the same topic in the Optics forum, I don't really understand what you think you are doing. You aren't doing paraxial optics, despite your thread titles. You appear to be doing exact ray tracing for a spherical surface, which is fine, then doing a paraxial approximation at the end, which is fine, but for a reason I don't understand. You usually use the paraxial approximation to compute image locations, not where arbitrary rays from the axis cross. Knowing image locations is a great tool for roughing out an optical system. Then you either use aberration theory to understand and correct for how bad the system is, or just jump straight to getting a computer to throw hundreds of rays through it using exact ray tracing and optimisation algorithms to improve it. This doesn't really seem to help that, so I don't understand what you are trying to achieve.

 

[difalcojr]

Yes, you have summarized it exactly, thanks.

I had some things from years of a hobby study that I just wanted to show and tell to others, that's all.
I don't have problems to solve, just a few items of study to show to others I thought might be of interest.

I'd never seen an exact derivation, so I thought mathematicians might like to see that.

Had a simple, trig ray trace program and wanted to show off what it could do. For all the paraxial points, and all the other areas of a spherical surface too. Wasn't sure if an existing program could do all of that.

Found a perfect lens model and rules for monochromatic, monocentric lenses. Other members proved it valid too. That didn't get much interest.

I have no problem to solve here, just had some results to show. That's all.

 

[difalcojr]

If I write any more equations in any post, I will comply and write them in LaTex. There. It's on the record. Luckily, though, I don't think I have any more equations to write down, as I told @berkeman too. So, that's all I have for this thread or any others, I hope.

Yes, I just wanted to show an exact, Cartesian-referenced derivation of that old, well-known paraxial equation. Because every textbook author had only used approximations in the derivation. That's it.

Third thing I still have to show was what the simple trig, ray trace program can do. But I will start a 2nd OP in the Optics section called "Simple trig trace program" or something like that. And show a few more diagrams and possibilities. I plotted a lot of stuff over the years. Hope this is OK.