Re: Drawing Potatoes and a Problem with Piecewise

*To*: mathgroup at smc.vnet.net*Subject*: [mg80196] Re: [mg80121] Drawing Potatoes and a Problem with Piecewise*From*: Selwyn Hollis <sh2.7183 at earthlink.net>*Date*: Tue, 14 Aug 2007 07:14:49 -0400 (EDT)*References*: <200708130830.EAA27361@smc.vnet.net>

David, Use Exclusions -> None. - Selwyn On Aug 13, 2007, at 4:30 AM, 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]]}}] > > > > -- > David Park > djmpark at comcast.net > http://home.comcast.net/~djmpark/ > > >

**References**:**Drawing Potatoes and a Problem with Piecewise***From:*"David Park" <djmpark@comcast.net>