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Pseudosphere in Mathworld

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

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].


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