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MathGroup Archive 2007

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Drawing Potatoes and a Problem with Piecewise

  • To: mathgroup at smc.vnet.net
  • Subject: [mg80121] Drawing Potatoes and a Problem with Piecewise
  • From: "David Park" <djmpark at comcast.net>
  • Date: Mon, 13 Aug 2007 04:30:41 -0400 (EDT)

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]]}}]



-- 
David Park
djmpark at comcast.net
http://home.comcast.net/~djmpark/




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