Catalan Surface Saddle

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Jean-Marc Daviau-Williams <br/>MATH 401<br/><br/><br/>For this print, we were tasked with looking back at previous assignments and finding one to do again. I decided to redo saddle as described during the multivariable calculus assignment, but with one twist; I would make the saddle using parametric equations. Luckily, there was already a template to go off of - in particular, the Catalan surface. As described by Eric Weisstein from <a href="http://mathworld.wolfram.com/CatalansSurface.html" target="_blank">Wolfram</a>, a Catalan surface is "a minimal surface given by the parametric equations: x(u,v) = u-sin[u]cosh[v] (1) y(u,v) = 1-cos[u]cosh[v] (2) z(u,v) = 4sin[u/2]sinh[v/2] (3)" The result of this is a saddle-like shape when u is taken from 0 to 2*Pi*. However, when taken to 2*kPi*, where *k* is some integer greater than 0, the shape will generate *k* copies of that original shape, intersecting the tips of the saddles. Granted, this saddle shape if used as an actual horse saddle would be very uncomfortable, but is still a saddle nonetheless. As for the actual code for the surface, little was done differently from Eric's sample <a href="http://mathworld.wolfram.com/notebooks/Surfaces/CatalansSurface.nb" target="_blank">code</a>. I simply changed the range for *u* in line 6 from [0,6*Pi*] to [0,2*Pi*] so that I could export a single saddle and then used the `Extrude` argument to bring out every point on the shape by some amount. Then, it was ready to slice. This object was sliced on Makerbot Print and printed on the GMU Math Maker Lab's Makerbot 3D printer. It was printed on yellow filament and took approximately 5 hours to print, as it was scaled to around 8cm long, 8cm wide, and 8cm tall with about .75cm thickness. In addition, the shape of the object was not too sharp along the bottom, so his resulted in a very sturdy object that needed minimal supports. One thing that could have been done better about this print was the use of the unsafe `Extrude` argument. It caused a strange fold along the corners of the saddle, that make the saddle seem not continuous, even though the surface is. In the future, I'll do a better job of using more safe methods to add thickness for printing feasibility. Source: http://mathworld.wolfram.com/CatalansSurface.html

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