Jason Sutton wrote: > > Mike - > The ugly maths are in the exact 3-D description. However, you can > approximate the solution using spherical geometry and solving in small > increments. Converting from 3-D to 2-D is just algebra based on local > triangular elements and straight-line approximations, just like a surveyor > does. OK, I'll have to do some poking at those ideas.
> Even computer solutions do that, although with much smaller > increments than a mere human would ever have time for. It just depends on > what level of accuracy you want. If the errors are smaller thatn your > printer can print, who cares? (And frankly, that's a lot smaller error than > my ability to cut to the line anyway) Yeah, I have that problem with cutting too... ;) > BTW, just for the sake of language purity, the loxodromic curve is not > actually a spiral, even though it looks like one. A true spiral eventually > arrives (or originates) at the center (pole) point. Since a vector from any > point on the L-curve toward the pole is always at the same constant angle > away from the local path of the curve, the path can never actually reach the > pole, only get infinitesimally closer as it becomes infinitely longer. And > that brings up another quaint paradox of mathematics, which all fractal > freaks know: The surface area betweeen the lines is finite (it is the > surface of the sphere). but the length of the edges is infinite. So you > could paint the surface but never have enough paint to cover the edges. > (Don't ask me what happens if you dip it into a bucket of paint!) My favorite in that vein is the one where you take some kind of asymptotic curve (I forget exactly which one it was), and rotate it around the coordinate axis to form a more-or-less funnel-shaped shell. The area of the shell is infinite, but the volume inside it comes out finite, so, like yours, you'll never finish painting the thing... unless you just fill it up with paint! :) Math gets just a bit weird at times. Anyway, thanks much for the pointers on this! Mike > > Jayo > ----- Original Message ----- > From: "Michael G. Henders" <[email protected]> > To: <[email protected]> > Sent: Wednesday, May 13, 2009 7:19 AM > Subject: [Papermodels II 36484] Re: "Apple peel" sphere patterns > > >> Thanks Jason, I had looked at the loxodrome curve in >> my own investigations; it looked like it should be at >> least related, but I wasn't sure that it was what I >> was after. I can readily see how that describes the >> centreline of the "peel", but I guess what I should >> have asked is, how do you go from that 3D spiral to a >> flat development pattern? Is that where the calculus >> comes in? >> >> Mike > -snip-> > > > > > > --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Papermodels II" group. To post to this group, send email to [email protected] To unsubscribe from this group, send email to [email protected] For more options, visit this group at http://groups.google.com/group/Papermodels?hl=en -~----------~----~----~----~------~----~------~--~---
