Cone as surface

1. Nov 14, 2011

paluskar

cone as surface....

In Barrett O'Neill's Elementary Differential Geometry book...he says that the cone
M:$x^{2}$ + $y^{2}$=$z^{2}$ is not a surface in that
there exists a point p in M such that there exists no proper patch in M which can cover a neighbourhood of p in M
Intuitively I realise that this is point is the apex...here $\left(0,0,0\right)$
but how would the patches for the rest of points be??...and what goes wrong with the patch at the apex of the cone??
Note: A PATCH IS A 1-1 REGULAR FUNCTION X:D→$ℝ^{3}$..D open in $ℝ^{2}$
This is an exercise problem.