In the course of finding I needed a lens cone for my monster enlarger, I had to learn how to make a frustum (the similarity to “frustration” is no accident), or a truncated cone. I found a few pages that purported to explain it, but as with so many things mathematical, the people who know it don’t always understand why the rest of us don’t.
Anyway, I made use of the good volunteers at Drexel University in the Dr. Math project. And it worked. This page gives the same result but was not as clear to me. If I have just used it and stopped thinking, I would have been done yesterday. But alas, I doubted.
I also looked at and tried to understand this approach. But I found an arithmetic error which put me off (can you find it?) and some other infelicities (numbers called by the wrong names, bad news for people like me). Plus the mysterious appearance of theta in the final stage was baffling. I didn’t see how it was derived or how it came into the picture. I just showed up. I’m clueless like that.
And the “I solved for |PT|” was maddening. I even asked my much more numerate Partner to take a whack at it and she gave up. (Dr Math has a much better explanation of that on another page.)
Anyway, my lens cone/frustum is drying. And if I need to make another, I feel pretty confident I can do it.