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. -- Direct access to this group with http://web2news.com http://web2news.com/?sci.math To contact in private, remove noo+sp-0am