Sadly, neither author manages entirely to avoid the trap of using the same term (line, etc) to refer indiscriminately to an element both of the hyperbolic space and its Euclidean model, sowing predictably impenetrable confusion in the mind of any reader not already familiar with the concepts concerned. While this kind of Jekyll and Hyde approach --- transforming the familiar into an exotic alter ego --- might have the advantage of tapping in to a student's imagination, hyperbolic geometry may also be investigated in the much more concrete guise of the geometry of circles or spheres [as the Poincare model illustrates]. So why not ditch the hocus-pocus aspect, and just stick to those? Fred Lunnon On 9/28/10, Henry Baker <hbaker1@pipeline.com> wrote:
One of the usual introductions to hyperbolic trig functions is through the analogy with circular functions. These introductions show that the hyperbolic "angle" can be identified with an _area_ under a hyperbola (y=1/x or one of its rotations), which then leads one to exponentials and logarithms.
http://en.wikipedia.org/wiki/Hyperbolic_angle#Imaginary_circular_angle
http://myyn.org/m/article/hyperbolic-angle/
Bjørn Felsager and Christina Sheets have also developed some wonderful teaching materials for hyperbolic trigonometry/geometry (do a Google search). These materials are simple enough to enable a high school student with only a traditional plane geometry background to start appreciating hyperbolic geometry; no advanced trig or even complex numbers are required.
Physicists in the early 20th century were uncomfortable enough with these concepts that they stumbled around for several decades without noticing that the usual trig functions with complex arguments work just fine for special relativity. (Google search for "rapidity"; also Scott Walter; John Rhodes and Mark Semon.)
Are there any other interpretations of "angle", other than the "area" interpretation and the "rapidity" interpretations?
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