Two quick questions on subspaces (intro to tensor calc)

Ed Quanta

So the parametric equations of a hypersurface in VN
are

x^1=acos(u^1)
x^2=asin(u^1)cos(u^2)
x^3=asin(u^1)sin(u^2)cos(u^3)
...
x^(N-1)=asin(u^1)sin(u^2)sin(u^3)...sin(u^(N-2))cos(u^(N-1))
x^N=asin(u^1)sin(u^2)sin(u^3)...sin(u^(N-2))sin(u^(N-1))

where a is a constant. How do I find the single equation of the hypersurface? And then from this how can it be determined whether the points (1/2a,0,0,...,0) and (0,0,0,...0,2a) are on the same or opposite sides of the surface?

Any insight will be appreciated. Thanks. I ask another question when this thing is resolved in my mind.

George Jones

Hint: What is the sum of the squares of all the coordinates?

Regards,
George