What Does Preserving Distance Mean in Metric Mapping?

[whattttt]

Can someone please exPlain to me what the phrase. Which metric do we have to impose in order that the mapping preserves distance means.
The example I have is
((-),phi)--->(x,y) =
(2a tan(theta/2)cos(phi) ,
2a tan(theta/2)sin(phi)) thanks

 

[Tinyboss]

It means you need to find a metric $$\delta$$ such that for all $$(\theta,\phi)$$ you have $$\delta(\theta,\phi)=\delta(2a\tan(\theta/2)\cos\phi,2a\tan(\theta/2)\sin\phi)$$.

 

[whattttt]

I guess it will be a diagonal metric but could you possibly give me a hint how to work out the other2 entries. Thanks