Suppose I have a point $x \in \mathbb{R}^n$ on an n-sphere. Suppose I divide the n-sphere into 4 sections (I think this makes sense in $n$ dimensions), how do I know which section $x$ lies on?
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This is just a rephrasing of ShreevastasR's answer; no credit to me. It does make sense to divide an $n$-sphere into quadrants, as you explain in $\mathbb{R}^3$: partition by two coordinate planes. But then deciding which quadrant is, as ShreevastasR says, simply looking at the signs of the coordinates of $x$. If $x_1$ and $x_2$ are both positive, you are in the first, $++$, quadrant; if $x_1$ is negative and $x_2$ positive, you are in the second, $-+$, quadrant. And so on. If instead you partition the sphere into $2^n$ orthants, then you consider all the signs of the coordinates of $x$.
Joseph O'Rourke
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My understanding from you question is that you have $x$ in your hands, presumably in coordinate form. So you look at the first coordinate, the second, the third, etc. I guess I don't understand your question! – Joseph O'Rourke Aug 05 '10 at 21:20
$(n-1)$-> $(n-1)$. – kennytm Aug 05 '10 at 20:21$x$didn't work when it didn't appear on the preview. Thanks! – Jacob Aug 05 '10 at 20:24