Pseudosphere in Mathworld

*To*: mathgroup at smc.vnet.net*Subject*: [mg81450] Pseudosphere in Mathworld*From*: Narasimham <mathma18 at hotmail.com>*Date*: Sun, 23 Sep 2007 21:17:39 -0400 (EDT)

Line parameterization for a single asymptotic line on the classical central pseudosphere is {x,y,z} = { sech(u) cos(u), sech(u) sin(u), u - tanh(u) }. Here u is the angle reckoned from the cuspidal equator towards axis in cylindrical coordinates {r,u,z}. To depict other rotated asymptotic lines around axis of symmetry the command ought to be {x,y,z} = { sech(u) cos(u + v),sech(u) sin(u + v), u - tanh(u) }.The set of positive and negative helical lines can be Shown together as {x,y,z} and {x,-y,z}, if so desired. However,in http://mathworld.wolfram.com/Pseudosphere.html we have surface parameterization {x,y,z} = {sech(u) cos(v),sech(u) sin(v), u -tanh(u)}.While the meridian is that of a pseudosphere OK, we lost sight of the physical significance literally,we are not able to "see" what the angle u is on the 3D plot that we have made/ obtained, Java applet given cannot identify or demonstrate it diretly, if want to see or label or mark azimuth angle u for any one curve of the set on the 3D figure by turning around with the mouse.It (u) is just one parameter in a confused context, as some sort of a pseudo parameter. The problem has arisen IMHO as the same ParametricPlot3D command of Mathematica serves for line and surface plots, with single and double argument domains.[ I had suggested here earlier FWIW that separate plot commands like ParametricPlot31D and ParametricPlot32D followed by confirming single/double arguments for lines and surfaces respectively could serve such purposes]. Regards, Narasimham