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MathGroup Archive 2007

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Re: Drawing Potatoes and a Problem with Piecewise

  • To: mathgroup at smc.vnet.net
  • Subject: [mg80190] Re: [mg80121] Drawing Potatoes and a Problem with Piecewise
  • From: Murray Eisenberg <murray at math.umass.edu>
  • Date: Tue, 14 Aug 2007 07:11:43 -0400 (EDT)
  • Organization: Mathematics & Statistics, Univ. of Mass./Amherst
  • References: <200708130830.EAA27361@smc.vnet.net>
  • Reply-to: murray at math.umass.edu

Rotating the first potato, w, makes quite obvious that it's in 4 pieces: 
you can see the slits of light between the pieces!

David Park wrote:
> Several years ago there was a question on MathGroup on how to draw a potato 
> in Mathematica.
> 
> The SphericalDraw3D documentation has an example of a lumpy sphere and I 
> decided to turn this into a potato by stretching it by a factor of two along 
> the z axis. However, to get a reasonable looking potato it is necessary to 
> damp down the variations in radius near the ends with a weighting function. 
> So I constructed the following w weighting function using Piecewise.
> 
> w[\[Theta]_] = Piecewise[
>   {{1/2 (\[Theta]/(\[Pi]/4))^3, 0 <= \[Theta] < \[Pi]/4},
>    {1 - 1/2 ((\[Pi]/2 - \[Theta])/(\[Pi]/4))^3, \[Pi]/
>       4 <= \[Theta] < \[Pi]/2},
>    {1 + 1/2 ((\[Pi]/2 - \[Theta])/(\[Pi]/4))^3, \[Pi]/2 <= \[Theta] <
>      3 \[Pi]/4},
>    {1/2 ((\[Pi] - \[Theta])/(\[Pi]/4))^3,
>     3 \[Pi]/4 <= \[Theta] <= \[Pi]}}]
> Plot[w[\[Theta]], {\[Theta], 0, \[Pi]}, PlotPoints -> 20,
>  Frame -> True]
> 
> The problem with this is that there are gaps in the function, which are more 
> or less visible depending on the number of PlotPoints. Am I defining 
> Piecewise incorrectly, or is there a bug with Piecewise in plotting 
> functions?
> 
> So I made another definition of a weighting function, w2, using Which, as 
> follows:
> 
> w2[\[Theta]_] =
>  Which[
>   0 <= \[Theta] < \[Pi]/4, (32 \[Theta]^3)/\[Pi]^3,
>   \[Pi]/4 <= \[Theta] < \[Pi]/2,
>   1 - (32 (\[Pi]/2 - \[Theta])^3)/\[Pi]^3,
>   \[Pi]/2 <= \[Theta] < (3 \[Pi])/4,
>   1 + (32 (\[Pi]/2 - \[Theta])^3)/\[Pi]^3,
>   (3 \[Pi])/4 <= \[Theta] <= \[Pi], (
>   32 (\[Pi] - \[Theta])^3)/\[Pi]^3]
> Plot[w2[\[Theta]], {\[Theta], 0, \[Pi]}, PlotPoints -> 20,
>  Frame -> True]
> 
> This definition causes no problem in plotting. Here is the potato with the 
> two weighting functions. You can see that the w function leaves the potato 
> sliced into four pieces.
> 
> Table[Show[
>   Graphics3D[
>    First[SphericalPlot3D[
>       1 + f Sin[5 \[Phi]] Sin[11 \[Theta]]/20, {\[Theta], 0,
>        Pi}, {\[Phi], 0, 2 Pi},
>       PlotPoints -> {16, 25},
>       PlotStyle -> Darker@Brown,
>       Mesh -> None]] // Scale[#, {1, 1, 2}, {0, 0, 0}] &],
>   ImageSize -> 250], {f, {w[\[Theta]], w2[\[Theta]]}}]
> 
> 
> 

-- 
Murray Eisenberg                     murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower      phone 413 549-1020 (H)
University of Massachusetts                413 545-2859 (W)
710 North Pleasant Street            fax   413 545-1801
Amherst, MA 01003-9305


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