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MathGroup Archive 2000

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RE: Equation of a "potato"

  • To: mathgroup at smc.vnet.net
  • Subject: [mg24676] RE: [mg24416] Equation of a "potato"
  • From: Wolf Hartmut <hwolf at debis.com>
  • Date: Fri, 4 Aug 2000 01:19:01 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com


> -----Original Message-----
> From:	Kevin J. McCann [SMTP:Kevin.McCann at jhuapl.edu]
To: mathgroup at smc.vnet.net
> Sent:	Tuesday, July 18, 2000 6:58 AM
> To:	mathgroup at smc.vnet.net
> Subject:	[mg24416] Equation of a "potato"
> 
> I am doing some illustrations for class notes on vector calculus. I
> would be nice to have some drawings for a "random" 3d shape, i.e.
> something that is fairly rounded and regular like a potato, but not as
> simple as a sphere. Any ideas for the an equation that would draw
> something like this?
> 
[Hartmut Wolf]  

Dear Kevin,

the question is: how much of potato must it be? Thinking of a computational
model of a grown-up potato seems to be rather complicated; perhaps it may
useful to look into Benoit B. Mandelbrot's book "Fractal Geometry of
Nature". So part of the geometry of a potato may be fractal, but part of it
appears smooth.

Here I'll give you just a less ambitious, primitive means to generate random
objects which might be considered as looking like potatoes (randomly
harvested). As for an equation, they are specified implicitly as F[x,y,z] ==
const

So if you only search for an illustration ContourPlot3D will give you that.

<< Graphics`ContourPlot3D`


The idea is simple: we just compose objects that may be ellipsoids (which
alone however are too regular for a potato).

We need several random generators:

cRan := (-1)^Random[Integer, {0, 1}]Log[1 - Random[Real, {0, 1}]]

this gives a random coordinate (near the origin)

<< Statistics`ContinuousDistributions`

ndis = GammaDistribution[5, 1];

(The Ceiling of) this will give a random number of primitive components
(about 5 here).

sdis = ExponentialDistribution[0.25];

This will give us the relative "size" of the components: larger and smaller
ones.

excdis = BetaDistribution[3, 2];

And this will give us (conservative) random ellipsoidal deformations of the
primitives.


To be reproducible we

SeedRandom[Prime[1000000]]

Now we do:

n = Ceiling[First[RandomArray[ndis, 1]]]   (* no of primitives *)

pts = Table[{cRan, cRan, cRan}, {n}]       (* centers of primitives *)

str = RandomArray[sdis, n]                 (* "strengths" of primintives *)

exc = Table[RandomArray[excdis, 3], {n}]   (* deformations of primitives *)

Now the function for the potato is

potato = Plus @@ (1/(str Plus @@@ ((({x, y, z} - #)^2 &) /@ pts exc)));


We may look at the behaviour along a line

Plot[With[{y = 0.2, z = 0.3}, Evaluate[potato]], {x, -n, n}, PlotRange ->
All]

so we can see where we may place the contour. (N.B. these are the components

Plot[Evaluate[With[{y = 0.2, z = 0.3}, Evaluate[List @@ potato]], {x, -n,
n}, 
    PlotRange -> All]]


). 


Now look at the potato:

ContourPlot3D[
    potato, {x, -(n + 1), n + 1}, {y, -(n + 1), n + 1}, {z, -(n + 1), n +
2}, 
    PlotPoints -> {7, 3}, MaxRecursion -> 2, Contours -> {0.2}] // Timing

{52.966 Second, Graphics3D[]}


You may also use {0.15} or {0.25} for Contours (If the value is too great,
the potato will go to pieces). The bounds given here are ad hoc, and
certainly not senseful for n very different from 5.

You may collect these steps to a procedure, and then repeat to get as much
potato as you like. Also play with the parameters of the "theory".

Kind regards, 
Hartmut Wolf



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