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Penrose introduces hyperbolic geometry in the second chapter of his book "The Road to Reality". While discussing the conformal representation he introduces a formula to find the hyperbolic distance between two points A and B, which is

$$\log{\frac{QA\cdot PB}{QB\cdot PA}},$$

where Q and P are the points in which the hyperbolic straight line connecting A and B intersect the bounding circle at right angles. Further, he mentions that you can multiply the above expression by $C^{-1/2}$ if you want. My question is, what does he means by want? Are there some cases in which it is necessary to include this constant? Or is it just some type of normalization?

I understand that $C$ is a measure of curvature and depends on $R$, the radius of the hyperbolic plane which is an imaginary number. What is the formula that relates this two values? And how can the radius of the hyperbolic plane be an imaginary number?

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  • $\begingroup$ "If you want" probably means "if you have to consider surfaces of constant negative curvature $\ne -1$." Namely, consider the effect on curvature and on distances of multiplying the Riemannian metric on the hyperbolic plane by a constant factor." $\endgroup$ Commented Jan 8, 2021 at 17:27
  • $\begingroup$ @MoisheKohan thanks, that clears things up a little bit. If it's not much to ask, can you explain to me how the curvature is measured? Or maybe redirect me to some place I can read more about it. Thanks! $\endgroup$
    – epelaez
    Commented Jan 8, 2021 at 17:33
  • $\begingroup$ See this question. $\endgroup$ Commented Jan 8, 2021 at 17:36
  • $\begingroup$ @MoisheKohan will do, thank you. $\endgroup$
    – epelaez
    Commented Jan 8, 2021 at 17:37

1 Answer 1

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Side note: the conformal representation of hyperbolic geometry is most commonly reffered to as the Poincaré disk model.

My question is, what does he means by want? Are there some cases in which it is necessary to include this constant? Or is it just some type of normalization?

The constant $C^{-1/2}$ will always be equal to $1$ in the two dimensional representation of the the Poincaré disk model (which is what I was working with in the original question) since the curvature of the hyperbolic disk is always equal to $-1$ (source). Whenever the curvature of the hyperbolic disk is not equal to $-1$, the constant $C^{-1/2}$ needs to be introduced into the equation. However, Penrose points out that we can always scale things such that $C=1$.

What is the formula that relates this two values? And how can the radius of the hyperbolic plane be an imaginary number?

The formula that relates $C$ and $R$ is obtained from Albert Girard's equation for the area of a spherical triangle. Which is the following for a sphere of radius $R$

$$\Delta = R^2(\alpha + \beta + \gamma - \pi)$$

What Lambert noticed is "that certain formulae of hyperbolic geometry can be obtained by replacing, in certain formulae of spherical geometry, distances by the same distances multiplied by the imaginary number $\sqrt{−1}$, and by keeping angles untouched" (source). So, replacing $R$ with the imaginary radius of $\sqrt{-1}R$, we obtain the following formula

$$\Delta = -R^2(\alpha + \beta + \gamma - \pi) = R^2(\pi - \alpha - \beta - \gamma),$$

which is the same as the formula for the area of a triangle with angles $\alpha, \beta, \gamma$ in the hyperbolic plane of constant curvature $-\frac{1}{R^2}$. Therefore, we obtain the formula

$$C = -\frac{1}{R^2},$$

so that $C\Delta = \pi - \alpha - \beta - \gamma$, which is Lambert's formula for the area of a hyperbolic triangle.

From this derivation of the formula for $C$, we can see that the fact that $R$ is an imaginary number is a necessary condition.

Further reading: Hyperbolic geometry in the work of Johann Heinrich Lambert

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