MathGroup Archive 2010

[Date Index] [Thread Index] [Author Index]

Search the Archive

Re: Getting started with 3D cardioid

  • To: mathgroup at
  • Subject: [mg112056] Re: Getting started with 3D cardioid
  • From: Alexei Boulbitch <alexei.boulbitch at>
  • 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]}]


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

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


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:

 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 

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> 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 .

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).


Alexei Boulbitch, Dr. habil.
Senior Scientist
Material Development

ZAE Weiergewan
11, rue Edmond Reuter

Tel: +352 2454 2566
Fax: +352 2454 3566
Mobile: +49 (0) 151 52 40 66 44

e-mail: alexei.boulbitch at


This e-mail may contain trade secrets or privileged, undisclosed or
otherwise confidential information. If you are not the intended
recipient and have received this e-mail in error, you are hereby
notified that any review, copying or distribution of it is strictly
prohibited. Please inform us immediately and destroy the original
transmittal from your system. Thank you for your co-operation.

__________ Information from ESET NOD32 Antivirus, version of virus signature database 5400 (20100826) __________

The message was checked by ESET NOD32 Antivirus.

  • Prev by Date: Washington DC Area Mathematica Special Interest Group
  • Next by Date: window frame
  • Previous by thread: Re: Getting started with 3D cardioid
  • Next by thread: Re: Getting started with 3D cardioid