This question is different as I seek to know a particular context of this subject i.e. wrt 3d printing. 
It is important for me to know what this means.
" So, projection onto the first coordinate is a continuous, and invertible, mapping from the upper arc to the open interval (−1,1):"
Why is it important ?
It's like doing a self motivated research on this subject. Bump into a doubt and ask here for clarification. How else could I do it ?
ref : http://en.wikipedia.org/wiki/Manifold#Motivational_examples
