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I delete my file which I used to produce this graph. Does anybody have some idea how to produce it again?

enter image description here

Thanks for a while.

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    This might be easier to answer if your picture were bigger. Do you have a bigger picture?2012-07-30
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    Unfortunately I don't have. Maybe if you zoom it. I will try to find a bigger one.2012-07-30
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    As it happens, $r^2\sin8\theta$ can be written as a rational function of $x$ and $y$, to wit: $8xy\big(x^6-7x^4y^2+7x^2y^4-y^6\big)\big/\big(x^2+y^2\big)^3$.2012-07-30
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    Have a look at the [generalized monkey saddle](http://www.math.hmc.edu/~gu/curves_and_surfaces/surfaces/genmonkey.html).2012-08-01
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    @J.M. Thanks for the msg, but I guess that my graph is not a saddle like this.2012-08-05

2 Answers 2

9

For example : $$f(x,y)=\sqrt{x^2+y^2}\sin(8 \arctan(y/x))$$

enter image description here

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    Isn't this roughly what Steven just posted? $\rho=\sqrt{x^2+y^2}$ and $\theta =\arctan \frac y x $.2012-07-30
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    @PeterTamaroff: don't know I was working! :-) Yes seems the same (coeff 1 at the end, his could be better!)2012-07-30
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    OK, I guess that you got the answer. Thanks so much for all comments. For me, this is enough. Nice plot. Thanks.2012-07-30
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    @Sigur: Nice if it corresponds! I too thought that they were 7 oscillations at the start but it didn't work (it's better to count the white parts under the curves at the front).2012-07-30
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    @StevenStadnicki: glad you liked this MuPad plot. My +1 was too for doing all the detailed pedagogic work (radial things and so on...), thanks for that!2012-07-30
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    @RaymondManzoni I'm having problems to plot it smoothly. How to plot it using polar coordinates? I'd like to export as PDF and the file is too huge.2014-11-19
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    @Sigur: Your file size problem is unclear but to obtain the picture with MuPAD (in Matlab now) I used $$-$$ plotfunc3d(sqrt(x^2+y^2)*sin(8*arctan(y/x)),x=-4.01..4,y=-4.01..4,Submesh=[4,4])$$-$$ In polar coordinates you may use $\;\displaystyle z:=\rho\,\sin(8\,\phi)\;$ and something like $$-$$plot(plot::Cylindrical([r, a, r*sin(8*a)], r= 0..4, a= 0..2*PI, VMesh=400, USubmesh=100)) //doc [here](http://fr.mathworks.com/help/symbolic/mupad_ref/plot-cylindrical.html) warning it's long! ![cylindrical result](http://i.stack.imgur.com/uGY2D.png). Cheers,2014-11-19
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    @RaymondManzoni, thanks. I supposed you had used *Mathematica*. I don't have *Matlab*. I'll try to use cylindrical coords.2014-11-19
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    You are welcome @Sigur! Btw what was the size problem?2014-11-19
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    Using rectangular coords in *Mathematica* the contour curves (from the mesh) are not smooth and the PDF exported is about 60Mb. Impossible to open it with Inkscape.2014-11-19
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    @Sigur: Ok I see... I'll let you play with the 'mesh' parameters or file format (I thought at .svg files or whatever). [Asymptote](http://asymptote.sourceforge.net/gallery/3D%20graphs) produces very neat pictures too (with $\LaTeX$ if needed). Excellent experiments !2014-11-19
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    *Mathematica* export the graph as SVG and when I open it with Inkscape, the object contains an embedded bitmap! Unbelievable! I'll take a look at Asymptote.2014-11-19
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    :-) @Sigur: perhaps that the mathematica SE forum will help you more. For example this [thread)(http://mathematica.stackexchange.com/questions/1542/exporting-graphics-to-pdf-huge-file). Cheers,2014-11-19
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A few key things that jump out looking at this graph:

  1. It's got a sort of 'radial symmetry' about it: if you look at a circular 'cross-section' centered around the origin then the shape looks roughly the same, just scaled. Similarly all of the 'radial' cross-sections along lines through the origin look roughly the same. This means that it's going to be best expressed as the plot of a function $z=f(r,\theta)$ where the 'base plane' is represented in polar coordinates. Moreover, the structure of it (a sequence of radial spires, of sorts) suggests that it's 'separable' in the sense that it can be written as $f(r,\theta) = g(r)\cdot h(\theta)$ for two individual functions $g$ and $h$.
  2. Going around the origin once seems (though it's unclear from the diagram) to encounter eight distinct peaks (possibly 7, but I think that's an artifact of the projection and there are actually 8), so the 'angular' part of the function, $h$, can be written as $h(\theta) = \sin(8\theta)$.
  3. Going out along any of the radial lines appears to be a straight line, so I'd say the radial portion of the function, $g$, can be written as $g(r) = c\cdot r$ for some small $c$, say $c=0.3$ or something on that order (but that's a function of the scale of the grap).

So, putting this together, the plot looks to be of an equation roughly like $z=.3r\sin(8\theta)$.

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    Looks more like $r^2$ should go there.2012-07-30
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    @PeterTamaroff That's possible - I'm looking at the ridge lines along the peaks of the radial sine waves, and to me they look straight but it's very possible they're a quadratic with relatively small constant.2012-07-30
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    I am trying to plot $z=r^2\sin(8\theta)$ with maple. But how to change to rectangular coordinates? I don't know how to use polar in maple. Thanks for the comments.2012-07-30
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    @Sigur As you've no doubt gleaned from Raymond's plot, you can change to rectangular coordinates via the substitution $r=\sqrt{x^2+y^2}$ and $\theta = \mathrm{atan}(y/x)$.2012-07-30
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    @Sigur, Maple is able to do cylindrical coordinates; see [here](http://www.maplesoft.com/support/help/Maple/view.aspx?path=plot3d/coords).2012-07-31
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    @J.M. Thanks for the link.2012-07-31