Intuitively understanding the Poincare metric I was reading an expository paper on hyperbolic geometry and it said the following :

A model of $\Bbb{H}$, the hyperbolic plane is the upper half of
  $\Bbb{R}^{2}$ equipped with the metric
  $ds^{2}=\frac{1}{y^{2}}(dx^2+dy^2)$, called the Poincare metric which emerges out of complex
  analysis.

Is there a way to understand the metric without complex analysis, say, by looking on $\Bbb{H}$ as a disc, $\Bbb{D} \subseteq \Bbb{C}$ with the edge of the disc representing $\infty$? Or some simpler explanation of why the metric is how it is?
 A: The simplest way to move between the disc and half-plane models is using complex analysis:
The Mobius transformation $z \mapsto \frac{z+i}{iz+1}$ maps the unit disk to the upper half-plane conformally, inducing an isometry between the Poincare metrics of the half-plane and the unit disk.
Another, more explicit way to think about it is as follows:
Consider the stereographic projection $\varphi: (x,y,z) \mapsto (\frac{x}{1-z}, \frac{y}{1-z})$ from the unit sphere $S^2$ to the plane.
The preimage of the upper half-plane $\{(x,y) : y > 0\}$ under $\varphi$ is the "northern" half-sphere with $y > 0$.
The preimage of the unit disc $\{(x,y) : x^2+y^2<1\}$ is the "bottom" half-sphere with $z < 0$.
Now consider the map $r: (x,y,z) \mapsto (x,-z,y)$ that rotates the unit sphere by $90^\circ$. It maps the bottom half-sphere to the northern half-sphere.
The composition $\varphi \circ r \circ \varphi^{-1}$ maps the Poincare disk to the Poincare half-plane, inducing an isometry of the Poincare metrics.
The relation between these two explanations is the Riemann sphere: The stereographic projection maps complex numbers to their corresponding points on the Riemann sphere, and the particular Mobius transformation $z \mapsto \frac{z+i}{iz+1}$ translates to a rotation of the Riemann sphere by $90^\circ$.
