§ Whalesong hyperbolic space in detail

We can build a toy model of a space where velocity increases with depth. Let the x-y axis be: left-to-right (→) is positive x, top-to-bottom (↓) is positive y. Now let the velocity at a given location $(x^\star, y^\star)$ be $(y^\star+1, 1)$. That is, velocity along $y$ is constant; the velocity along $x$ is an increasing function of the current depth. The velocity along $x$ increases linearly with depth. Under such a model, our shortest paths will be 'curved' paths.