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. Senior Scientist Material Development IEE S.A. ZAE Weiergewan 11, rue Edmond Reuter L-5326 CONTERN Luxembourg Tel: +352 2454 2566 Fax: +352 2454 3566 Mobile: +49 (0) 151 52 40 66 44 e-mail: alexei.boulbitch at iee.lu www.iee.lu -- 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. http://www.eset.com