if I have the unit ball in R^3, that is

{(x,y,z) | x^2+y^2+z^2 <= 1 } = B

and i have to find a chart, can then just use the chart

(B,f)

where f(x,y,z) = (x,y,z) for (x,y,z) in B. That is my first question. My next question is if I can just do this, what about if I identify (x,y,z) with (-x,-y,-z), then i gues my chart have to illustrate this by sending (x,y,z) and (-x,-y,-z) to the same point, ie.

f(-x,-y,-z) = f(x,y,z)

is that correct that the way to interpreted that to point are the same, that the chart have to do that. Ps. I don't need anyone to give me the charts, (i like to try and find them myself) but I just eanted to make sure I'm going the right way.

Thanks in advance.