## Faking marbles

February 29, 2012

Still on the matter of Perlin noise, I finally managed to figure out how to wrap Perlin noise on a sphere (that is, as a Texture[]). It was a bit inconvenient that it seems (at least, to me) that not one of the usual references on Perlin noise mentioned the explicit formulae needed for a proper spherical embedding of Perlin noise. Making use of the functions already defined in the doc page for Compile[], here are two Mathematica functions you can use for generating textures that can be wrapped on a sphere:

sphericalPerlin1 = With[{octaves = 4}, Compile[{{theta, _Real}, {phi, _Real}, {amplitude, _Real}, {frequency, _Real}, {gain, _Real}, {lacunarity, _Real}}, Module[{noiseVal = 0.0, x, y, z, sf, freq = frequency, amp = amplitude}, sf = Sin[phi]; x = sf*Cos[theta] + 1; y = sf*Sin[theta] + 1; z = Cos[phi] + 1; Do[ noiseVal += amp*signedNoise[x*freq, y*freq, z*freq]; freq *= lacunarity; amp *= gain, {octaves} ]; noiseVal ], CompilationOptions -> {"InlineExternalDefinitions" -> True} ]]

sphericalPerlin2 = With[{octaves = 4}, Compile[{{theta, _Real}, {phi, _Real}, {amplitude, _Real}, {frequency, _Real}, {gain, _Real}, {lacunarity, _Real}}, Module[{noiseVal = 0.0, x, y, z, sf, freq = frequency, amp = amplitude}, sf = Sin[phi]; x = sf*Cos[theta] + 1; y = sf*Sin[theta] + 1; z = Cos[phi] + 1; Do[ noiseVal += amp*Abs[signedNoise[x*freq, y*freq, z*freq]]; freq *= lacunarity; amp *= gain, {octaves} ]; noiseVal ], CompilationOptions -> {"InlineExternalDefinitions" -> True} ]]

As an example, here is how to use sphericalPerlin2[] along with DensityPlot[] to generate a texture:

DensityPlot[sphericalPerlin2[th, ph, 0.5, 2.0, 0.5, 2.0], {th, 0, 2 Pi}, {ph, 0, Pi}, AspectRatio -> Automatic, Frame -> None, PerformanceGoal -> "Quality", PlotPoints -> 200, PlotRange -> All]

and here is the result, after adding appropriate coloring (with thanks to Rob Collyer for the color gradient):

In fact, this is the texture I used for upper leftmost “marble” in the first image.

## On faking a speckled paint finish

February 15, 2012

I’ve been experimenting a lot with Perlin noise recently in Mathematica, thanks to the example implementation given in the documentation for Compile[].

(I must however note that the method implemented in the Mathematica help file is not actually the “classic” Perlin noise. In particular, the compiled fade[] routine is the Hermite quintic produced by InterpolatingPolynomial[{{{0}, 0, 0, 0}, {{1/2}, 1/2}, {{1}, 1, 0, 0}}, x], not the classical one that makes use of InterpolatingPolynomial[{{{0}, 0, 0}, {{1/2}, 1/2}, {{1}, 1, 0}}, x]. The Mathematica code also makes use of the centers of the edges of a cube as the gradients, stored in the grad variable within the function dot[], as proposed in Perlin’s update.)

The grid of swatches depicted above is just one sample of Perlin noise, colored differently with 49 of the gradient color schemes available in ColorData["Gradients"], and rendered with ReliefPlot[]. The choice of coloring scheme makes (at least to me) a heap of difference in the visual effect of Perlin noise. I like how the outputs look like samples of the fancy speckled paint I sometimes see in hardware stores.

I still haven’t fully digested the theory behind Perlin noise (though Gustavson’s notes seem to be one of the better treatments I’ve seen), but my experiments with the routines thus far have been quite encouraging.

## On emulating the Texas Instruments random number generator

February 12, 2012

I recently needed to emulate the random number generator internally used by Texas Instruments calculators in Mathematica. Thanks to this United-TI forum post, I found that the algorithm being used is a combined multiplicative linear congruential generator due to Pierre L’Ecuyer. The particular generator used is listed on page 747 of the article; as noted there, the combined MLCG has a period of $2305842648436451838\approx 2.30584\times10^{18}$.

I figured this might be useful to other people, so I’m posting the routines needed to get RandomReal[] and related functions to emulate the rand() function (and ilk) on TI calculators:

TexasInstruments /: RandomInitializeGenerator[TexasInstruments, ___] := TexasInstruments[12345, 67890]
 TexasInstruments[___]["GeneratesRealsQ"] := True; TexasInstruments[___]["GeneratesIntegersQ"] := True;

TexasInstruments[___]["SeedGenerator"[seed_]] := If[seed == 0, TexasInstruments[12345, 67890], TexasInstruments[Mod[40014 seed, 2147483563], Mod[seed, 2147483399]]];

TexasInstruments[s1_, s2_]["GenerateReals"[n_, {a_, b_}, prec_]] := Module[{p = s1, q = s2, temp}, {a + (b - a) Table[p = Mod[40014 p, 2147483563]; q = Mod[40692 q, 2147483399]; temp = (p - q)/2147483563; If[temp < 0, temp += 1]; temp, {n}], TexasInstruments[p, q]} ]

TexasInstruments[s1_, s2_]["GenerateIntegers"[n_, {a_, b_}]] := Module[{p = s1, q = s2, temp}, {a + Floor[(b - a + 1) Table[p = Mod[40014 p, 2147483563]; q = Mod[40692 q, 2147483399]; temp = (p - q)/2147483563; If[temp < 0, temp += 1]; temp, {n}]], TexasInstruments[p, q]} ]

This is based on the PRNG method plug-in framework in Mathematica that is described here. For instance, here is the Mathematica equivalent of executing 0→rand:rand(10) (for Zilog Z80 calculators like the TI-83 Plus) or RandSeed 0:rand() (for Motorola 68k calculators like the TI-89):

(* 0→rand:rand(10) *) BlockRandom[SeedRandom[0, Method -> TexasInstruments]; RandomReal[{0, 1}, 10, WorkingPrecision -> 20]]

Here’s corresponding code for randInt() and randM() on the Z80 calculators:

(* 0->rand:randInt(1,10,10) *) BlockRandom[SeedRandom[0, Method -> TexasInstruments]; RandomInteger[{1, 10}, 10]]

(* 0->rand:randM(3,3) *) BlockRandom[SeedRandom[0, Method -> TexasInstruments]; Reverse[Reverse /@ RandomInteger[{-9, 9}, {3, 3}]]]

Due to differences in precision, there is bound to be some discrepancy in the least significant figures of the results from the calculators and the results from the Mathematica emulation; still, I believe that this is a serviceable fake.

You can download a Mathematica notebook containing these definitions and tests here.

## Fly me to the moon

February 6, 2012

I had always wanted a nice animation depicting periodic solutions of the restricted three-body problem, and was always a skosh underwhelmed by the animations I’ve found. I then decided, on a quiet afternoon, to see if the new (to me) capabilities of Mathematica were up to the task, and I was not at all disappointed:

(click on the image to see a bigger version of the animation)

This is one of my favorite Arenstorf orbits (see also these slides); I went with the route taken in Hairer/Nørsett/Wanner and used the Bulirsch-Stoer extrapolation method for numerically solving the underlying differential equation (Method -> "Extrapolation" in NDSolve[]). (Note that for this animation, I used the Arenstorf orbit depicted in the Bulirsch-Stoer paper instead of the slightly more complicated path depicted in Hairer/Nørsett/Wanner; I can be persuaded to produce an animation for that version, though… ;) )

It wasn’t much trouble to depict the Earth, the Moon, and a spaceship. For ease of visualization, I (grudgingly) decided to exaggerate the sizes of the Moon and the spaceship; if I went with the actual scale relative to the Earth, you’d probably have a hard time seeing those. Mathematica‘s Texture[] function, with texture maps obtained from here, as well as ExampleData[{"Geometry3D", "SpaceShuttle"}] for the spaceship, were useful in this regard.

The one thing I did have trouble with is depicting the starry backdrop of outer space in Graphics3D[]`, so I elected to skip it. Still, the animation looks quite picturesque, no?