I found this question recently in my booklet on hyperbolic geometry asking a very simple question but I could not answer it:

Why can we not define the area of a hyperbolic triangle as in the plane as half the product of the perpendicular and the base?

I know the half plane model and the Poincare disk models but I cannot find a satisfactory explanation. I thought it might have something to do with that there are no rectangles in hyperbolic geometry but I cannot proceed. Help appreciated.

  • 1
    $\begingroup$ I began to studied hyperbolic geometry three weeks ago, so I am not sure for the exact reason. But as far as I known, if we defined the area in hyperbolic plane as we did in Euclid plane, then at least the möbius transformations would not preserve the area as we except. I am really new at this topic, so I am not sure for my commenting. $\endgroup$ – KWSK Oct 2 '15 at 7:12
  • 1
    $\begingroup$ Nice question, but above my knowledge, i made a relatd question of a (tiny) bit simpler problem see math.stackexchange.com/questions/1462778/… $\endgroup$ – Willemien Oct 3 '15 at 18:07
  • 2
    $\begingroup$ Well, one reason is that in hyperbolic geometry $ah_a\neq bh_b$, so this 'area' would depend on the side we choose as the base — so in a sense it's not well-defined. $\endgroup$ – Grigory M Oct 3 '15 at 19:25
  • 1
    $\begingroup$ @KWSK Well, (hyperbolic) lengths of both the base and the (hyperbolic) perpedicular are manifestly invariant under hyperbolic motions. $\endgroup$ – Grigory M Oct 3 '15 at 19:27

What is the area? Well, at least we want it to be (1) a non-negative function of a polygon that is (2) additive: $S(A\cup B)=S(A)+S(B)$ if $A\cap B$ has no interior points. It turns out, these two simple properties define $S$ almost uniquely — it's unique up to multiplication by a constant.

Now in Euclidean geometry one can prove that $S(\Delta)=ah_a$ by considering a triangle inside the rectangle — but because 'there are no hyperbolic rectangle' the same proof doesn't work in hyperbolic geometry.

triangle inside rectangle

  • $\begingroup$ Thank you for a very clear well written answer $\endgroup$ – kroner Oct 3 '15 at 19:37
  • $\begingroup$ When scale factor for length is non-linear and varies point to point in either of these models what meaning can be there for an area, using a euclidean yardstick ? $\endgroup$ – Narasimham Oct 3 '15 at 19:50

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.