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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?

enter image description here

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2 Answers 2

up vote 2 down vote accepted

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:

paraboloid distortion

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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

If I understand correctly your phrase "no strain in the circular cross section," the answer to your question is No. The paraboloid is not a developable surface. A developable surface has zero Gaussian curvature everywhere, whereas the paraboloid has positive curvature. The analogy here is to a sphere, also not developable, which is why there is such a plethora of map projections aiming for minimal visual distortion of a non-developable surface.

Update. See the comments. I did not interpret the strain condition as the OP intended, apparently. The comments clarify.

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I added in the mean time a picture to my question. I mean no strain in the direction of vector A. Strain in the direction of vector B is allowed. –  wnvl Aug 9 '12 at 13:16
1  
"Strain in the direction of vector B is allowed." - like this? –  J. M. Aug 9 '12 at 13:42
    
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
    
@joriki when you transform a circle of radius 1 into a circle of radius 2 there will be strain in the direction of A although there is symmetry –  wnvl Aug 9 '12 at 14:52
    
@J.M.: yes....... –  wnvl Aug 9 '12 at 14:55

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