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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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