[I want to think that this question has an answer, but it may be more a "community wiki" discussion. Feel free to re-tag.]

What are trig classes like within a universe that's "noticeably"[*] hyperbolic?

Using an appropriate model to peek into such a universe from our Euclidean one shows us that all the fundamental trig relations (Law of Sines, Law of Cosines, etc) involve transcendental functions in both angle measures and lengths. But, then, in our earliest mathematical days, similar triangles allowed us to build a theory of (circular) trig functions based on ratios of lengths of sides, and eventually our mathematics became sophisticated enough to include hyperbolic functions and to be comfortable enough with them and how they might apply to our non-Euclidean models. (We also had the benefit of being able to interpret products as areas of rectangles whose side-lengths correspond to factors.) What if you don't --can't-- draw from our experience base?

How do you even get started developing trigonometry (or explaining it to your hyperbolic trig students[**]) without a concept of similar triangles?

Obviously(?), the Unit Circle is out.

It makes some sense that the Angle of Parallelism would be the fundamental bridge between angle-information and distance-information; one can imagine that hyperbolic people would be "aware" of the phenomenon on some level, and we know that it's a "universal" property. Even so, the most-concise representations of the AoP relationship are transcendental in both angle measure and length. How insightful would a hyperbolic mathematician have to be to discern the equations from tables of observed measurements? And is there a clear path from those equations to, say, (what we know as) the Law of Sines and the Laws of Cosines?

Or perhaps the fundamental figure in an "intrinsic" hyperbolic trig class is the Equilateral Triangle. This idea actually exploits non-similarity to set up a bridge between angle-information and distance-information (with area-information thrown in as a bonus); and it seems that it might be more likely to provide a path to the Laws of Sines and Cosines, since it already relates angles and sides of triangles. But, is it really a particularly helpful starting point? Can you get from there to the Angle of Parallelism relation?

Something else?

[*] For instance, anyone can pull out a protractor and easily see that triangles have an angle-sum less than 180 degrees.

[**] "hyperbolic" modifying both "trig" and "students". :)

  • 1
    $\begingroup$ You still get Euclidean trigonometry in the small-scale limit. If your universe is uniformly hyperbolic, you can also extract Euclidean subspaces of constant geodesic curvature from horospheres on the universal cover. $\endgroup$
    – S. Carnahan
    Commented May 5, 2010 at 1:19
  • $\begingroup$ See my answer to mathoverflow.net/questions/23253 $\endgroup$
    – Will Jagy
    Commented May 5, 2010 at 1:46
  • $\begingroup$ @Scott: That seems to suggest (to me) that a hyperbolic mathematician has to know some pretty sophisticated math (or have a remarkable instinct for abstraction) before she can understand the very basics of measurement in her own experience. I'm trying to get at a development of hyperbolic trig that doesn't require starting with Euclidean sensibilities. Extracting Euclidean subspaces of constant geodesic curvature from horospheres on the universal cover of one's own universe would appear to be a difficult first-day activity for hyperbolic high-schoolers. @Will: Contacting you off-site. $\endgroup$
    – Blue
    Commented May 5, 2010 at 2:29
  • $\begingroup$ Don, got your email, sent Marvin's article and my earlier one. I think you will enjoy Marvin's work, he has gotten increasingly concerned with philosophical notions over the years. $\endgroup$
    – Will Jagy
    Commented May 5, 2010 at 2:52
  • $\begingroup$ Let's assume your hyperbolians live on a large spherical planet. Unlike in Euclidean space, a large sphere in hyperbolic space is not approximately a hyperbolic plane, but rather approximately a horosphere (which is intrinsically a Euclidean plane). So the hyperbolians could simply draw their triangles, rectangles, and circles on the ground (not to mention navigating and building on the ground). They would be quite familiar with Euclidean geometry! $\endgroup$
    – mr_e_man
    Commented May 3, 2022 at 19:34

3 Answers 3


Chapter V of Harold E. Wolfe's book: Introduction to Non-Euclidean Geometry (Holt, Rinehart, and Winston), 1945 (and reprinted, 1966) is entitled: Hyperbolic Plane Trigonometry, and has a systematic treatment of this topic.

  • $\begingroup$ The book's available at the university library not too far away. I'll (ahem) check it out. $\endgroup$
    – Blue
    Commented May 6, 2010 at 1:25

I'm more comfortable with how trigonometry and hyperbolic geometry actually did happen than how they might happen in a hypothetical hyperbolic world. So, this may not be what you're looking for, but I hope it sheds some light.

First, similar triangles did not lead immediately to trigonometry. There are similar triangles in Euclid, but not trigonometry. In fact, the first trigonometry to develop was spherical trigonometry, for use in astronomy.

Second, hyperbolic functions were introduced by Lambert (around 1760), who observed that they were the trigonometric functions on a "sphere of imaginary radius" -- which we now realize can be viewed as a hyperbolic plane.

Third, around 1830 hyperbolic functions were found by Lobachevsky to describe the trigonometry of the non-Euclidean plane that he was studying axiomatically. He even attempted to claim that formulas he found were a "realization" of the non-Euclidean plane. About the same time, in ignorance of Lobachevsky's work, Minding found that hyperbolic functions describe the trigonometry of any surface of constant negative curvature.

Finally, Beltrami in 1868 showed that in hyperbolic space there are surfaces that realize both spherical and Euclidean geometry. So, even in a hyperbolic world, one has the option of using spherical or Euclidean geometry -- which are easier. So creatures in the hyperbolic world might still follow the path that we did. Students in trig class might use "horosphere paper" so as to take advantage of Euclidean trig.

  • $\begingroup$ @John: When I've discussed the question before with others, I've noted that spherical trig actually came first, and the fact that all the relations are transcendental in both angle-measure and distance didn't seem to be a hindrance there. :) So, I agree that it's inappropriate for me to imply that our trig was based on similar triangles from its inception. However, triangle similarity is what makes the Unit Circle the touchstone of modern introductions to the subject, and what makes the Unit Circle not-nearly-as-helpful to Hyperbolians; which leads me to wonder about the UC's counterpart. $\endgroup$
    – Blue
    Commented May 5, 2010 at 5:40

One model for the hyperbolic plane is Minkowski space. This is 2-space, say, with the caveat that the dot product is negative on the last coordinate, i.e. $(x_1,y_1)\cdot(x_2,y_2)=x_1x_2 -y_1y_2$. If we want to define the length of a vector to be the square root of a vector dotted with itself in this inner product, we need to require that $x^2\ge y^2$. One can then check that the point $(0,1)$ is distance $1$ from the curve $(t,1-\sqrt{t^2-1})$, and likewise for generic points so that the distribution of points equidistant from a fixed point is a hyperbola. Using calculus, one can compute the length of the arc along the "unit" hyperbola starting at $(0,1)$. This is how the Greeks defined angles, if I am not mistaken, though they did so without calculus. One can then define the hyperbolic cosine as the inverse function of the length of the above curve, and so forth.

I think this is probably how it would begin. Defining the hyperbolic trig functions as combinations of exponentials would likely come later. Perhaps there is a way to come up with the length of the hyperbolic arc without calculus?


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