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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: [mg80146] Re: Drawing Potatoes and a Problem with Piecewise
  • From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
  • Date: Tue, 14 Aug 2007 06:48:48 -0400 (EDT)
  • Organization: The Open University, Milton Keynes, UK
  • References: <f9p580$r58$1@smc.vnet.net>

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?
<snip>

David,

The definition of w is correct (w is continuous). However, the trouble 
seems to come from the adaptive sampling, as we can see by evaluating 
the following expressions:

  Plot[ w[\[Theta]], { \[Theta], 0, Pi}, PlotPoints -> 45, Frame -> True,
  MaxRecursion -> 15]

  Plot[ w[\[Theta]], { \[Theta], 0, Pi}, PlotPoints -> 20, Frame -> True,
  MaxRecursion -> 0]

  Plot[ w[\[Theta]], { \[Theta], 0, Pi}, PlotPoints -> 20, Frame -> True]

On my system, the first expression draws a smooth curve, while plot
rendered by the second expression is cut on the left (but not on the
right). (The last expression is the original one for comparison.)

-- 
Jean-Marc


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