Distance in the $y$-axis of the hyperbolic plane

Question

I'm reading Stillwell's Geometry of Surfaces but I'm having a little bit of trouble because my background in calculus isn't great, I'm struggling with these problems:

• In the upper half-plane model, show that the $H_2$-length of the segment of the $y$-axis from $i$ to $iB$ is $|\log B|$. This is what I'm given as distance: $$ ds = \frac{\sqrt{dx^2 + dy^2}}{y} = \frac{|dz|}{\operatorname{Im}z}. $$

• In the Poincaré disk model, show that a circle of Euclidean radius $r < 1$ and center $0$ has hyperbolic radius twice the integral from $0$ to $r$ of $dx/(1 - x^2) = 2\tanh^{-1}r$. Deduce that the hyperbolic circumference of a circle of hyperbolic radius $\rho$ is $2\pi\sinh \rho$. This is what I'm given as distance: $$ ds = \frac{2\sqrt{dx^2 + dy^2}}{1 - (x^{2} + y^{2})} = \frac{2|dz|}{1 - |z|^{2}}. $$

Any help explaining how to work with this, an example, or any hints, would be most welcome.

Answer 1

Suppose $\gamma(t) = \bigl(u(t), v(t)\bigr)$, with $a \leq t \leq b$, is a smooth plane curve. In a geometry with "line element" $ds$, the arc length of $\gamma$ is the integral from $t = a$ to $t = b$ of the line element evaluated on $\gamma$; that is, compute the velocity, substitute $x = u(t)$, $y = v(t)$, $dx = u'(t)\, dt$, and $dy = v'(t)\, dt$ into the line element, and integrate.

For example, the segment of the $y$-axis from $i$ to $iB$ can be parametrized by $$ \gamma(t) = it = (0, t),\qquad 1 \leq t \leq B. $$ Since $u(t) = 0$, $v(t) = t$, $u'(t) = 0$, and $v'(t) = 1$, the line element along $\gamma$ is $$ \frac{\sqrt{dx^{2} + dy^{2}}}{y} = \frac{\sqrt{u'(t)^{2} + v'(t)^{2}}\, dt}{v(t)} = \frac{dt}{t}, $$ and the length of the segment is $$ \int_{1}^{B} \frac{dt}{t}. $$

Similarly, let $r < 1$ be real; the path $\gamma(t) = (t, 0)$, $0 \leq t \leq r$, traces a "hyperbolic radius" of the circle of Euclidean radius $r$ centered at the origin. By similar means you can set up an arc length integral to discover the hyperbolic radius $\rho$ of this circle.

The circle itself may be parametrized by $$ \gamma(t) = (r\cos t, r\sin t),\quad 0 \leq t \leq 2\pi. $$ Expressing the answer in terms of $\rho$ gives the desired circumference.

Incidentally, to integrate $2\, dt/(1 - t^{2})$ you'll need either a table of integrals or the "partial fractions decomposition" $$ \frac{2}{1 - t^{2}} = \frac{1}{1 - t} + \frac{1}{1+ t}. $$ I leave all these details to you because you may need fluency with the hyperbolic functions, and the only way to acquire fluency is practice.