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The two of you who have downloaded my RoundEdge module from the Object
Collection may have noticed a gap in the sample images. This gap was for an
oblong toroid. However, to fit the theme of the module, the toroid would have
to be able to serve as a rounded edge or join for scaled cylinders. To do
this, the central curve, the extreme inner curve, and the extreme outer curve
of the toroid must all be ellipses. A sphere sweep does not satisfy this
condition (except for the boundary case of the circular torus).
So towards the end of August, I drew a diagram of an elliptical torus, and
stared at it. A pattern emerged almost immediately, but it just sat there
looking pretty. Hrummph. I continued to stare at the diagram.
Finally, a couple of weeks ago, the pattern whispered to me from the graph
paper:
"t^4 - 2*z*t^3 + (x^2+z^2-d^2)*t^2 + 2*d^2*z*t - d^2*z^2 = 0
Solve for t."
:-O
No, no, no, it can't be! Let me recheck my derivation. It checked out. But
I'm prone to mixing up signs, dropping coefficients, and getting exponents
wrong, so let me plug this into a spreadsheet just to make sure. The
spreadsheet responded, "Your numbers are perfect. Good luck with that formula.
Bwa ha ha ha!"
So here I am, having to (re?)learn how to solve a quartic equation, and a
messier one than you'll find on any algebra exam. I entered the "depressed
quartic" stage into the spreadsheet, and it checks out. But after that, I'm
flying blind until I drag a "solution" out kicking and screaming. I look
forward to rendering the result. I *hope* it will be an elliptical torus.
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