What is the geometrical meaning of the constant $k$ in the law of sines, $\frac{\sin A}{\sinh a} = \frac{\sin B}{\sinh b} = \frac{\sin C}{\sinh c}=k$ in hyperbolic geometry? I know the meaning of the constant only in Euclidean and spherical geometry.
[Math] The law of sines in hyperbolic geometry
hyperbolic-geometry
Related Solutions
Strictly speaking, there is just one approach to a uniform proof, which is the one given by Elementary Differential Geometry , Christian Bär, pages 201-209. This approach is based on Riemannian geometry.
The impossibility of coming up with a 'rule-and-compass' uniform proof is that the Pythagorean theorem is expressed in essential different ways:
Euclidean geometry: $a^2+b^2=c^2$
Spherical geometry: $\cos(a)\cos(b)=\cos(c)$
Hyperbolic geometry: $\cosh(a)\cosh(b)=\cosh(c)$
It is true that we may derive the following formulae for rectangular triangles:
Euclidean geometry: $\sin(\alpha)=\frac{a}{c}$
Spherical geometry: $\sin(\alpha)=\frac{\sin(a)}{\sin(c)}$
Hyperbolic geometry: $\sin(\alpha)=\frac{\sinh(a)}{\sinh(c)}$
and then the usual proof of the sine rule applies to the other two cases (just dividing a triangle into two rectagle ones by an altitude), but whereas $\sin(\alpha)=\frac{a}{c}$ is a definition, the other two expressions have to be found in a different way.
First, we should note that a very similar question has already been asked here, and several interesting answers were given. But because the question keeps coming up, I'm going to go out on a limb and suggest that there still might be room for a more complete list of reasons why hyperbolic geometry is important in its own right.
It's hard to know where to start, because there are so many good reasons to study hyperbolic geometry beyond its obvious historical importance in the development of geometry. I'm going to start this list with a few things I can think of off the top of my head; I'll make this answer community wiki so others can elaborate or add to it as they think of other things.
Why is hyperbolic geometry important?
- History of math: First, just for completeness, I'll mention the historical reason. The discovery of hyperbolic geometry was, in my opinion, the second most important event in the history of mathematics (the first being Euclid's introduction of the axiomatic method). Up until the nineteenth century, everyone thought of axioms as self-evident truths about the real world, which could be built upon to derive less self-evident truths. Since the discovery of hyperbolic geometry, axioms have been thought of as more or less arbitrary assumptions that could be used to get an axiomatic system started, and then everything proved within that system is exactly as valid as the axioms themselves. It's the foundation of the way we currently understand mathematical truth. (I've written more about this in the preface and Chapter 1 of my undergraduate textbook Axiomatic Geometry.)
- Complex analysis: The hyperbolic geometry of the unit disk (or, equivalently, the upper half-plane) plays a central role in complex analysis. For example, the Schwarz-Pick Lemma says that any holomorphic map from the unit disk to itself is either an isometry of the hyperbolic metric or a strict contraction (which decreases all distances). This has important consequences for understanding the nature of holomorphic maps.
- Geometry of Surfaces: Most connected surfaces (all but the plane, cylinder, torus, Möbius strip, Klein bottle, sphere, and projective plane) carry a Riemannian metric of constant negative curvature, which is therefore locally isometric to the hyperbolic plane. Moreover, all such surfaces can be realized as quotients of the hyperbolic plane modulo discrete groups of hyperbolic isometries.
- Geometry of Three-Manifolds: The geometrization theorem says that every closed 3-manifold can be cut along spheres and tori into finitely many pieces, each of which can be given one of eight possible highly symmetric geometric structures. By far the richest of these structures is hyperbolic geometry, which accounts for "most" three manifolds.
- Cosmology: The leading candidates for modeling the shape of the universe as a whole are the FLRW models, in which the spatial geometry of the universe is either flat, spherical, or hyperbolic. Which one depends on the average density of matter and energy and the value of the "cosmological constant."
- Fermat's Last Theorem: Wiles's proof of Fermat's last theorem made essential (and unexpected) use of modular forms, which are functions on the hyperbolic plane that satisfy a specific transformation property under a certain discrete group of hyperbolic isometries.
- Art: This list wouldn't be complete without mentioning the groundbreaking uses of hyperbolic geometry in the art of M. C. Escher.
Best Answer
"k" is the "distance scale," traditionally taken to be 1, therefore in a space of curvature $-1,$ as the curvature is $-1/k^2.$ In this and other ways, $k$ appears as a sort of imaginary radius. Note the curvature of the ordinary sphere of radius $r$ is $1/r^2.$
Oh, $k$ does NOT appear in the place you indicate in the Law of Sines. Erase it!
If you want to allow other $k,$ the correct Law is $$ \frac{\sin A}{\sinh(a/k)} = \frac{\sin B}{\sinh(b/k)} = \frac{\sin C}{\sinh(c/k)} $$
The actual meaning of $k$ is a relation between curves called horocycles. But, for something easier, the area of a geodesic triangle is its angular defect multiplied by $k^2.$
The easiest introduction I know to these matters is MY_ARTICLE
EDIT: evidently Apotema wanted some other geometric number associated with a triangle that gives the same number as the common value in the Law of Sines. I cannot imagine anything understandable that does that. See the article by Milnor on the first 150 years of hyperbolic geometry: MILNOR. There is no nice expression for the volume of a tetrahedron in $\mathbf H^3.$ I've got to think about whether I even know the volume of a geodesic sphere in $\mathbf H^3.$ Had to look it up, $$ V = 2 \, \pi \, ( \, \sinh r \; \cosh r \; \; - \; \; r ) = \pi \sinh(2r) - 2 \pi r, $$ and that the Taylor series of this around $r=0$ has first term $\frac{4}{3} \pi r^3,$ as is required in the small.