Metrics on upper half-plane and sphere: natural way In the Euclidean plane, the distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ is defined by considering a Pythagorean triangle in which the line segment joining $(x_1,y_1)$ and $(x_2,y_2)$ will become a diagonal; the other two sides of the triangle have length $(x_2-x_1)$ and $(y_2-y_1)$.
 
Thus, a natural way to define a metric on Euclidean plane is by Pythagorean triangles.
Question: What is a natural way to define metric on upper half pane (hyperbolic geometry) and on sphere (spherical geometry). 
I saw the formulae for the distance between two points in these two other geometries, which are in terms of $\log$ or triagonometric functions. I didn't find any natural way to understand how they arise. Can you help me?
 A: The most immediately obvious way to put a metric on a sphere is to inherit the Euclidean metric in $\mathbb R^3$. However, there are lots of ways to (smoothly) embed a sphere in $\mathbb R^3$ (or higher dimensions), not all of them immediately recognizable as a sphere, but each one will give rise to a different inherited metric (for instance, imagine an embedding where the north and south poles are quite close).
Ultimately, these metrics are not satisfactory because they are not intrinsic to the geometry of the sphere. They depend on how the sphere is embedded in its ambient space. To emphasize that I'm now thinking about the sphere as a space in its own right, separate from any embedding in Euclidean space, I will denote the sphere by $\mathbb S^2$.
To define an intrinsic metric, one that depends only on the internal geometry of $\mathbb S^2$, we use geodesic segments. Thinking about the sphere in the usual way (i.e., embedded in $\mathbb R^3$ in the usual way), geodesic segments are segments of great circles. In general, geodesics play the same role as straight line segments do in Euclidean geometry.
The distance between two points $a,b \in \mathbb S^2$ is then defined to be the infimum of the lengths of all geodesics beginning at $a$ and ending at $b$. In well-behaved spaces, this will give rise to a metric.

Now, hyperbolic geometry is a tad... different.
For starters, $\mathbb S^2$ is topologically distinct from $\mathbb R^2$ and the differences in the topologies plays a part in the geometric differences. However, the hyperbolic plane is homeomorphic (topologically identical) to $\mathbb R^2$.
Calling the upper half plane the hyperbolic plane is a bit disingenuous; properly, the upper half plane together with the right choice of things to call straight lines (i.e., geodesics) provides a model for hyperbolic geometry. You may or may not have seen other models of hyperbolic geometry such as the Klein and Poincare models.
