Alain <kua### [at] videotronca> wrote:

> > "JimT" <nomail@nomail> wrote:
> >>> Is it possible to create a revolution plot directly from a parametric function?
> >>>
> >>> For example, I have a 2d parametric function
> >>>
> >>> x[u]=Cos[u]*(1 + 2 Cos[2 u])
> >>> y[u]=Sin[u]*(1+2Cos[2 u])
> >>>
> >> In this case, yes.
> >>
> >> Since x = rcos(theta) and y = rsin(theta) in standard polars, your parametric
> >> equations are just saying
> >> r   = 1+2cos(2theta)
> >>      = 1+2cos^2(theta)-2sin^2(2theta)
> >>      = 3cos^2(theta) - sin^2(2theta)
> >>      = (3x^2 - y^2)/r^2
> >> r^3 = (3x^2 - y^2)
> >> Square to pick up the negative r lobes
> >> r^6 = (3x^2 - y^2)^2
> >> In 2D
> >> (x^2+y^2)^3 = (3x^2 - y^2)^2
> >> To go to 3D, to be axi-symmetric around the x axis, replace y^2 with y^2+z^2.
> >> Replacing x^2 by x^2+z^2 would be axi-symmetric around the y axis. A similar
> >> shape, but the large lobe rotates to a "disk".
> >>
> >> #include "colors.inc"
> >>
> >> #declare surftest =
> >> isosurface {
> >>   function { pow(x*x+y*y+z*z,3)-pow(3*x*x-y*y-z*z,2)}
> >> // evaluate 356*Min_factor,  sqrt(356/(356*Min_factor)),  0.7
> >>   max_gradient 1500
> >>   all_intersections
> >>   contained_by { sphere { 0 3}}
> >>   clipped_by {box {-3 3}}
> >>   pigment {Red}
> >>   finish { phong 0.6 reflection 0.2 }
> >> }
> >> object{surftest }
> >> background{White}
> >> light_source { <20,20,20> color 1}
> >> camera{location <0, 0, 20> look_at <0, 0, 0> angle 40}
> >>
> >> Produces the surface.Note the rather large max_gradient. For a while, with
> >> smaller max_gradient, I wasn't getting anything.
> >>
> >> Thanks,
> >>
> >> JimT
> >
> > Thanks. That's very helpful.
> > But can we get the most outer surface of the plot? I mean can we make the
> > surface into a single solid object without all the internal structures?
> > And what does this line mean "evaluate 356*Min_factor,
> > sqrt(356/(356*Min_factor)),  0.7" ?
> >
> >
>
> "evaluate 356*Min_factor, sqrt(356/(356*Min_factor)),  0.7" is an
> alternate manner to account for the max_gradient value of your function.
> In effect, it allow the used max_gradient do dynamicaly change during
> the trace within some limits.
>
> The first value is a starting gradient value. It should be somewhat
> smaller than the found max_gradient.
> The second value is a ovder-evaluation factor. It controll how fast the
> gradient used is allowed to increase. This value must be larger than 1.
> It can be simplified as sqrt(1/Min_factor)
> Lastly, is the under-evaluation factor controlling how fast the gradient
> is allowed to fall. "0.7" is a good starting value.
>
> After rendering without setting max_gradient, you take note of the found
> max_gradient from the message window (for the Windows and Mac versions,
> messages stream on Linux).
> You use that value in place of the "356" of that example.
>
> It's very usefull when you have several isosurfaces placed with a #while
> loop, using slight variations, variously scalled, rotated and translated
> and you have messages for both to large AND to small max_gradient values
> at the same time...
>
>
>
> Alain

In the rendered plot, we can see through the object and see multiple surfaces.
Can we make the iosurface into a solid object? I mean can we plot the out most
surface only?

Thanks.

Best,
xslittlegrass

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