Transform flat surface into paraboloid

Is it possible to transform a flat surface into a paraboloid

$$z=x^2+y^2$$

such that there is no strain in the circular in the circular cross section (direction vector A)?

If the answer is yes, is it possible to calculate the shape of such a flat surface?

Where can I find more information to solve this kind of problems?

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Since it seems my solution is what the OP wanted...

Consider the parametric equations

\begin{align*} x&=cu\cos\,\theta\\ y&=cu\sin\,\theta\\ z&=h(1-u^2) \end{align*}

with the parameter ranges $0\leq u\leq 1$ and $0\leq\theta\leq2\pi$.

For $h=0$, you have a disk of radius $c$; from here, varying $h$ in either the positive or negative directions will yield a paraboloid of revolution:

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Like a heart beating at a high pulse rate! :-) –  Joseph O'Rourke Aug 9 '12 at 15:36
Maybe I should have chosen a more sedate frame rate... :D –  Ｊ. Ｍ. Aug 9 '12 at 15:44
Maybe this will help (and maybe not!). You can build a paraboloid by stacking circles. If you made circular rings out of paper, with a small thickness $\delta$, you could build a model of a paraboloid by stacking these rings. Each ring is itself developable, and is unstrained. The strain comes from trying to glue the rings to one another. –  Joseph O'Rourke Aug 9 '12 at 14:44