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Automation of 3D Paper Modeling

i am a programmer and not a mathematician, and my math knowledge are a little bit rusty. i have found this nice picture surfing on the net:

enter image description here

and i want to replicate it.

as i remember there is written an implicit function, and for plotting i need to transform it in a parametric one. is this right? how can i do?

then, once i have the parametric surface i can iterate over an axis and obtain the slices for replicate this nice paper-work.

i am code agnostic and i can use every opensource tool for mathematical computation (octave/matlab, wolfram alpha, python, java/c/etc).

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marked as duplicate by Raskolnikov, nkint, William, Rahul, Guess who it is. Sep 19 '12 at 9:28

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

To me, it looks like what is "plotted" is $z=sin(\sqrt{x^2+y^2})/\sqrt{x^2+y^2}$ and not the equation written down on the paper. Or maybe the colors imply the third dimension? –  Raskolnikov Sep 13 '12 at 13:35
@Raskolnikov: Seconded. –  Clive Newstead Sep 13 '12 at 13:39
yeah it seems a duplicate question –  nkint Sep 13 '12 at 13:52
buy the way, what about transformation to implicit to parametric? does it make sense? is it possible? –  nkint Sep 13 '12 at 13:53

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up vote 1 down vote accepted

what about transformation of implicit to parametric? is it possible?

Sometimes, but not always. Here is one example of a case where it works. Suppose you have an implicit equation $g(x,y,z)=0$. If you're lucky, then, through each point $(x,y)$ in the $xy$-plane, you can fire a vertical "ray" (parallel to the $z$-axis), and it will hit the surface once. In other words, for each given $(x,y)$, you can find a value of $z$ for which $g(x,y,z)=0$. For complicated functions, you'll have to use numerical methods to find $z$ given $x$ and $y$. Anyway, numerical or not, this process maps a given $(x,y)$ to a point on the surface -- in other words, it gives you a parametric equation of a portion of the surface.

This is a fairly narrow scenario, but broad enough to cover your paper model, I think.

Also, you should note that parameterisation is helpful, but it's not absolutely necessary. There are methods of drawing surfaces given by implicit equations. Mathematica has functions to do this, or you can write your own code http://www.unchainedgeometry.com/jbloom/pdf/impencyc.pdf

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