In differential geometry of R^3, real asymptotic lines are directly not
realizable on surfaces of positive Gauss curvature (K>0).
I like to be able see imaginary asymptotes really with a suitable new
ambience brought in. In hyperbolic trigonometry we get Sine/Cosine Rules
for a spherical/hyperbolic triangle by substitution of imaginary
arguments serving as a switch between elliptic/hyperbolic geometries.
Simply, the sign of K changes. Is such an interpretation/re-validation
possible along these lines? There are two ways to do that I know.
Loxodromes or rhumb lines of a sphere with constant 45 degrees initial
angle at equator are the supposed asymptotic lines with an imaginary
longitude/ polar angle. [Reference: Art(2-8),Eq(8-3),and
Art(2-2),Lectures in Classical Differential Geometry by D.J. Struik,
Dover,1988].
To understand imaginaries in surface theory more tangibly with
non-Euclidean geometry,I also made the substitution [u,v] -> [( u+v)/2,
(u-v)/2] , (which procedure given in Alfred Gray's book is valid to
get real asymptotic lines on K<0 surfaces ). The attempt is to see under
what new circumstances an imaginary asymptotic line (of zero normal
curvature) could be real and viewed again as real on a positive K
surface .
Mathematica graphics below depicts Gray's substitution resultant lines
along with loxodromes on a spherical surface.
x=Cos[(u+v)/2] Sin[(u-v)/2];
y=Cos[(u+v)/2] Cos[(u-v)/2];
z=.02 + Sin[(u+v)/2 ];
sphasym=ParametricPlot3D[{x,-y,z},{u,0,3},{v,0,Pi/2},PlotPoints->{11,11}
,ViewPoint->{0,0,2}];
xl=Sech[vl] Cos[vl+al] ; yl=Sech[vl] Sin[vl+al] ; zl= Tanh[vl];
loxo=ParametricPlot3D[ {xl,yl,zl}, {vl,0,Pi},{al,0,2
Pi},PlotPoints->{25,20}]
Show[sphasym,loxo];
But the two curves do not match, they are different, and the
investigation increases in complexity!
Is there a hope of realizing imaginary asymptotic lines on any other
appropriate surface or perhaps using any other type of line curvature?
Thanks in advance for any light thrown/research discussions on the
topic. After wondering about it, if someone also left it just there,
those views could be pointers to a solution.
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