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Re: Getting started with 3D cardioid

  • To: mathgroup at smc.vnet.net
  • Subject: [mg112056] Re: Getting started with 3D cardioid
  • From: Alexei Boulbitch <alexei.boulbitch at iee.lu>
  • Date: Fri, 27 Aug 2010 04:08:12 -0400 (EDT)

Hi, David,

To answer your first question:
I do not understand, where the figure 68 for the average radius comes 
from? Indeed, the radius of the 2D cardioid is r= Cos[\[Theta]/2]^2.
By averaging it over the angle theta one finds
1/\[Pi] Integrate[Cos[\[Theta]/2]^2, {\[Theta], 0, \[Pi]}]

1/2

If one instead averages it in 3D (e.g. over theta and phi) one gets

1/(4 \[Pi])
  Integrate[
  Cos[\[Theta]/2]^2, {\[Theta],
   0, \[Pi]}, {\[CurlyPhi], -\[Pi], \[Pi]}]

\[Pi]/4

So you probably make the average in some different way. Anyway, if you 
conclude that the result is 68 times larger than that you need,
why not to normalize the expression by 68? In other words
r = Cos[\[Theta]/2]^2/68;

Concerning your second question:

I am not aware of the horn shape and it would be helpful, if you  post 
the  corresponding formula in whatever coordinates.
Nevertheless, looking at you function I have a feeling that you (a) have 
written the first term of the product

\[Theta]^-1 Sin[\[Theta] + 1] 

in a misleading way so that Mathematica interpret it incorrectly  and 
(b) use wrong limits for the angle Theta
(which typically varies between 0 and Pi). Try this:

SphericalPlot3D[
 1/\[Theta] Sin[\[Theta] + 1], {\[Theta], 0, \[Pi]}, {\[CurlyPhi], 0,
  2 \[Pi]}, ViewPoint -> {1, 0, -2}]

Finally, you can combine the two graphs by using Show. Like this, for 
instance:

gr1 = SphericalPlot3D[
   1/\[Theta] Sin[\[Theta] + 1], {\[Theta], 0, \[Pi]}, {\[CurlyPhi],
    0, 2 \[Pi]}, ViewPoint -> {1, 0, -2},
   PlotStyle -> Directive[Opacity[0.5]]];
gr2 = SphericalPlot3D[
   Cos[\[Theta]/2]^2, {\[Theta], 0, \[Pi]}, {\[CurlyPhi], 0, 2 \[Pi]},
    ViewPoint -> {1, 0, -2}];
Show[{gr1, gr2}]

I add the Opacity option to one of them to make visible the carioid 
inside the pulse.

Have fun, Alexei


On Aug 24, 5:14 am, Alexei Boulbitch <alexei.boulbi... at iee.lu> wrote:
> Hi, Dave,
> try this:
>
> SphericalPlot3D[
>  Cos[\[Theta]/2]^2, {\[Theta], 0, \[Pi]}, {\[CurlyPhi], 0, 2 \[Pi]},
>  ViewPoint -> {1, 0, -2}]
>
> Have fun, Alexei

Thanks Alexei,

The equation produces the geometry I am looking for.  How would I
adjust the proportions such that the mean radius equals 1 instead of .
68?

Also, I tried to use this function with a different equation:

SphericalPlot3D[\[Theta]^-1 Sin[\[Theta] + 1], {\[Theta], 0,
  2 \[Pi]}, {\[CurlyPhi], 0, 2 \[Pi]}, ViewPoint -> {1, 0, -2}]

but it did not produce the geometry I was looking for.  I was hoping
to produce a horn shape (a 3D pulse).

Dave


-- 
Alexei Boulbitch, Dr. habil.
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