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From setting up a hyperbolic triangle with hyperbolic side length $a,b,c$ and corresponding angles $A,B,C$, it is not hard to prove the following law of cosine:

$$\cos A= \frac{\cosh b \cosh c -\cosh a}{\sinh a \sinh b}$$

Since we have $3$ formula for angles $A,B,C$, with brute force, it is possible to show the following identity:

$$\cosh a =\frac{\cos B \cos C + \cos A}{\sin B \sin C}$$

Comparing the formula above, it seems like there is a duality between hyperbolic distance and angles. Are there any deeper geometric implication in phenomenon?

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Yes, this is indeed a form of duality. The following outline follows the approach from Perspectives on Projective Geometry by J. Richter-Gebert.

Cayley-Klein metric

Probably the best way to understand this is probably using Cayley-Klein metrics. That's a very general way to measure distances and lengths in projective geometry. You start by fixing one fundamental conic. If you choose the unit circle at this point, you obtain the Beltrami-Klein model of hyperbolic geometry. To compute distances between two points $P$ and $Q$, you join them by a line, intersect that line with the fundamental conic to obtain $X$ and $Y$, and then define the distance as

$$c_{\text{dist}}\ln(P,Q;X,Y)$$

For hyperbolic geometry one conventionally chooses $c_{\text{dist}}=\tfrac12$ to obtain a curvature of $-1$. The term $(P,Q;X,Y)$ denotes a cross ratio, which you could express using lengths, but which I'd rather compute from the homogeneous coordinates of the points. The order of $X$ and $Y$ is not fixed by the instructions, and swapping them will reverse the sign. So this describes a distance only up to sign, which makes sense. I'm not drawing absolute value bars since we'll have to deal with complex numbers shortly, and there taking the absolute value is not the same as just ignoring a sign.

So now on to the dual measurement of angles. You have that same fundamental conic, and two lines $p$ and $q$. From the point where they intersect (dual to the line joining $P$ and $Q$) you draw tangents to the conic (dual to intersections) and call them $x$ and $y$. Then the angle is defined as

$$c_{\text{ang}}\ln(p,q;x,y)$$

If $p$ and $q$ intersect inside the hyperbolic plane, the tangents $x$ and $y$ will be a complex conjugate pair of lines. For this reason one conventionally chooses $c_{\text{ang}}=\tfrac1{2i}$ to obtain a real angle, measured in radians.

So at this point you should see that

  1. distance measurement and angle measurement is dual to one another and
  2. one (at this point rather “cosmetic”) difference is that one of them has a real constant and the other has an imaginary constant for hyperbolic geometry.

Excursus: other geometries

Cayley-Klein metrics can be used to describe other geometries as well, simply by changing the fundamental conic and the constants. If instead of the unit circle $x^2+y^2-z^2=0$ (in homogeneous coordinates) one takes the completely complex conic $x^2+y^2+z^2=0$ the result is elliptic geometry.

For Euclidean geometry the fundamental conic would be $z^2=0$, i.e. the line at infinity taken with multiplicity two, but that's just half the story: if the primal conic is a double line as in this case, one needs to know the dual conic explicitely since it cannot be computed from the primal. In this case the dual conic would be $x^2+y^2=0$ or equivalently $(x+iy)(x-iy)=0$. Which means that the step “draw tangents from the point of intersection to the fundamental conic” translates into “connect the point of intersection with the ideal circle points $(1,\pm i, 0)$”, resulting in Laguerre's formula for angles. Distances in Euclidean geometry will all be zero but it's still possible to compare one distance to another in a meaningful way.

With other fundamental conics one can obtain pseudo-Euclidean geometry or relativistic space-time geometry.

Trigonometric functions

Now one can start investigating the relation between the logarithm used in the above formulas and trigonometric functions. To give you a rough idea, you should know that trigonometric functions as well as hyperbolic functions can be defined using exponential functions:

\begin{align*} \sin x &= \frac{e^{ix}-e^{-ix}}{2i} & \sinh x &= \frac{e^{x}-e^{-x}}{2} \\ \cos x &= \frac{e^{ix}+e^{-ix}}{2} & \cosh x &= \frac{e^{x}+e^{-x}}{2} \end{align*}

So they re already quite similar. Both sign choices ($x$ and $-x$) have to be included in a symmetric way, which relates to the fact that the above formulas give their results only up to sign. Some occurrences of the imaginary unit indicate that choosing constants as real or imaginary may switch between both sets of formulas, leading to hyperbolic functions for lengths but trigonometric functions for angles. And there is also that factor $2$ in the denominator which we see in our scaling constants. So this should convey a gut feeling that these are all essentially the same thing.

The book I'm referring to has a section 21.4 where it goes into more depth establishing this relationship. It starts by expressing the value of the cross ratio without actually computing $X$ and $Y$ (resp. $x$ and $y$) first. It then applies a transformation $\alpha\mapsto\frac{\alpha+1}{2\sqrt\alpha}$ to that cross ratio which on the one hand drops the distinction between $X$ and $Y$ (or $\alpha$ and $\alpha^{-1}$ at this point), and on the other hand allows relating that to trigonometric functions via $\ln\alpha=2i\arccos\left(\frac{a+1}{2\sqrt\alpha}\right)$.

So on this level it would be possible to either transform one of the cosine law forms into the other by systematically considering things as exponential functions and replacing constants. Or one could obtain both laws from a more general equation expressed in terms of Cayley-Klein metrics. I haven't done either of these in a rigorous fashion. The book does the law of sines in section 22.5 (which combines measurements of both kinds), bit I can't find a cosine law there right now either. So I hope you get the general idea of this duality, and either trust that all the details work out fine, or get down and do it yourself following the pointers I provided.

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I explain this phenomen with “complex hexagonometry”. Let $z_{1},z_{2},z_{3},w_{1},w_{2},w_{3}$ be complex numbers. Let (i,j,k) be the cyclic permutation of (1,2,3). Complex hexagonometry is the involutory transformation of 3 functions of 3 variables defined as follows:

Cosine formulas: $cosh(w_{i})=(cosh(z_{i})+cosh(z_{j})cosh(z_{k}))/(sinh(z_{j})sinh(z_{k}))$. Its inverse cosine formulas: $cosh(z_{i})=(cosh(w_{i})+cosh(w_{j})cosh(w_{k}))/(sinh(w_{j})sinh(w_{k}))$. Sine formulas: $sinh(w_{1})/sinh(z_{2})=sinh(w_{2})/sinh(z_{2})=sinh(w_{3})/sinh(z_{3})$.

We can deduce above 2nd and 3rd formulas from 1st formulas.

We can specialize complex hexagonometry to hyperbolic trigonometry. We set $z_{1}=a_{1}+\pi i,z_{2}=a_{2}+\pi i,z_{3}=a_{3}+\pi i,w_{1}=-A_{1}i,w_{2}=-A_{2}i,w_{3}=-A_{3}i$,where $a _{1},a_{2},a_{3} $ are positive real numbers and $A_{1},A_{2},A_{3}$ are inner angles.

We can specialize complex hexagonometry to spherical trigonometry. We set $z_{1}=(a_{1}+\pi)i,z_{2}=(a_{2}+\pi)i,z_{3}=(a_{3}+\pi)i,w_{1}=-A_{1}i,w_{2}=-A_{2}i,w_{3}=-A_{3}i$ where$ a_{1},a_{2},a_{3}$ are positive real numbers and are $A_{1},A_{2},A_{3}$ are inner angles.

Complex hexagonometry does not discriminate distances and angles.

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