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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.

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  • $\begingroup$ A belated welcome to Math.SE! It's preferable to ask only one question per post. (Yours are arguably similar enough that a single answer may suffice; I mention this because the second seems to have been added as an edit some time after the first.) Also, it's customary (and appreciated) if you format your answers. The site tour may be useful, as well. As to your questions: Do you know how to use $ds$ to set up integrals? And are you working in the upper half-plane, the unit disk, or (as it seems) both? $\endgroup$ May 24 '15 at 14:46
  • $\begingroup$ sorry I don't know hot to format them; no I don't, yes, both $\endgroup$
    – Edgar
    May 24 '15 at 14:54
  • $\begingroup$ I added additional formatting, reworded with the intent of clarifying, and added the (presumed) line element for the disk. If you click the "edit" button you can see what the code looks like. My previous comment also contains a couple of tutorial links. :) $\endgroup$ May 24 '15 at 16:05
  • $\begingroup$ You're too kind, thanks, I'll try to make my questions better. $\endgroup$
    – Edgar
    May 24 '15 at 16:20
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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.

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