In his book Essentail Topology, Martin Crossley deals with an example involving the closed disk D (see attachment)

$\displaystyle D^2 = \{ (x,y) \in \mathbb{R}^2 : \surd (x^2 + y^2) \leq 1 \} $

Crossley notes on page 78 (see attachment) that $\displaystyle D^2 - \partial D^2 $ is homeomorphic with $\displaystyle \mathbb{R}^2 $, by the mapping

$\displaystyle (r cos \ \theta, r sin \ \theta ) \longrightarrow ( tan \ \frac{\pi r}{2} \ cos \ \theta \ , \ tan \ \frac{\pi r}{2} \ sin \ \theta ) $

However, Crossley just states this homeomorphism and gives no sense by which it was derived or discovered. Can someone help me get a sense of why, intuitively speaking, one would come up with such a mapping - why it is intuitively reasonable! i.e. some motivation as to why this mapping would be a sensible or appropriate one to try.

I am working on proving that it is a homeomorphism - but this is not the same as seeing why it would be appropriate or understanding how and why someone came up with this mapping.

Can someone please help?

Peter