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I have been told that a disk with center (a,b) and radius r can be mapped to 3D point. And the 3D point is $(a,b,a^2+b^2-r^2)$. However i do not know what is the idea behind it. How do you calculate this point and how do you prove it? Inversely if you have a 3D point can you create a disk from that point?

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    You can map whatever set you want to whatever point you want. The mapping is $f(x) = p$ where $p$ is the point. That's probably not what you mean, so please tell us what you do mean.2012-09-30
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    @RobertIsrael, I mean how do we define the point in 3D? Specially why $a^2+b^2-r^2$ for the third coordinate of the point?2012-09-30

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