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I am trying to make a blob like the one shown below, but in 3D so I can rotate it. I'm trying to use ParametricPlot3D or SphericalPlot3D. I can make a sphere easily. But my knowledge of how to modify the sphere to make it bulge out in a few places in limited. Anyone have any suggestions? I'd like to make it non symmetric so it look like a random blob like the one below. If I can create the outside blob I'm sure I can create an inside one. I can than place the vectors in the appropriate places. I am a teacher trying to illustrate how to use the divergence theorem on a region like this. Any suggestions would be appreciated. Thank you.

Blob in 3 dimensions

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    $\begingroup$ Consider modelling an electrostatic potential isosurface, for instance. $\endgroup$ Commented Nov 18, 2015 at 5:48
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    $\begingroup$ There was somewhere similar post about marking infinitesimal parts of such surfaces, can't find it :/ $\endgroup$ Commented Nov 18, 2015 at 7:25

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This can also be made a little lumpier with spherical harmonics.

realization[r_] := Assuming[{0 <= θ <= π, 0 <= φ <= 2 π}, Simplify[r + Abs[ComplexExpand[Plus @@ Flatten[ Table[ RandomReal[{-1, 1}] 1/(l^2 + m^2) SphericalHarmonicY[l, m, θ, φ], {l, 1, 4}, {m, 0, l}] ]]]]] Block[{inner, outer}, outer = realization[1]; inner = realization[1/2]; Show[ SphericalPlot3D[outer, {θ, 0, π}, {φ, 0, 2 π}, PlotStyle -> Directive[Orange, Opacity[0.2], Specularity[White, 10]], Mesh -> None, PlotPoints -> 50], SphericalPlot3D[inner, {θ, 0, π}, {φ, 0, 2 π}, PlotStyle -> Directive[GrayLevel[0.4], Opacity[0.2], Specularity[White, 10]], Mesh -> None, PlotPoints -> 50], Axes -> False, Boxed -> False ]]

enter image description here

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  • $\begingroup$ Use the "Image" button on the editing toolbar (the one that looks like a picture of a landscape). I've done it for you this time. $\endgroup$ Commented Nov 18, 2015 at 20:41
  • $\begingroup$ @Rahul : I haven't found a way to drag images from Mathematica to the dialog that pops up... $\endgroup$ Commented Nov 18, 2015 at 23:39
  • $\begingroup$ Oh, I just right-click, "Save Graphic As...", save it as whatever.png, and upload the image file. Not the most luxurious of techniques, I know... $\endgroup$ Commented Nov 19, 2015 at 0:17
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Here's a function to create a random scalar field:

randomFunction3D[xrange_, yrange_, zrange_] := Interpolation[ Flatten[Table[{{x, y, z}, RandomReal[{-1, 1}]}, Evaluate@{x, Sequence @@ xrange}, Evaluate@{y, Sequence @@ yrange}, Evaluate@{z, Sequence @@ zrange}], 2], Method -> "Spline"]

Now instead of drawing a sphere with constant radius $x^2+y^2+z^2=r^2$, let's make the "radius" vary randomly over space, so we get an irregular blobby shape:

SeedRandom[0]; f = randomFunction3D[{-3, 3}, {-3, 3}, {-3, 3}]; ContourPlot3D[ x^2 + y^2 + z^2 == (1 + 0.4 f[x, y, z])^2, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, Mesh -> None, PlotRange -> All, BoxRatios -> Automatic, Boxed -> False, Axes -> False]

enter image description here

You can also change the grid spacing to control the size of the bumps:

SeedRandom[0]; f = randomFunction3D[{-3, 3, 0.25}, {-3, 3, 0.25}, {-3, 3, 0.25}]; ContourPlot3D[ x^2 + y^2 + z^2 == (1 + 0.06 f[x, y, z])^2, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, Mesh -> None, PlotRange -> All, BoxRatios -> Automatic, Boxed -> False, Axes -> False]

enter image description here

You can have a lot of fun adding a bunch of different random fields with different scalings to create interesting effects, but I'll leave that as an exercise. For inspiration, see Ken Perlin's classic Making Noise talk.

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    $\begingroup$ Congratulations on the Populist badge. $\endgroup$ Commented Nov 22, 2015 at 11:33
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I swear I've seen potatoes like this:

realSphericalHarmonic[ℓ_Integer?NonNegative, 0, θ_, φ_] := SphericalHarmonicY[ℓ, 0, φ, θ]; realSphericalHarmonic[ℓ_Integer?NonNegative, m_Integer, θ_, φ_] /; -ℓ <= m <= ℓ := I^Boole[m < 0] (SphericalHarmonicY[ℓ, -Abs[m], φ, θ] + (-1)^(m + Boole[m < 0]) SphericalHarmonicY[ℓ, Abs[m], φ, θ])/Sqrt[2] BlockRandom[SeedRandom[42, Method -> "Rule50025CA"]; (* for reproducibility *) n = 3; ρ[θ_, φ_] = 1 + Sum[RandomVariate[NormalDistribution[]] realSphericalHarmonic[k, j, θ, φ]/k!, {k, 0, n}, {j, -k, k}, Method -> "Procedural"] // FunctionExpand; ParametricPlot3D[ρ[θ, φ] {Sin[φ] Cos[θ], Sin[φ] Sin[θ], Cos[φ]}, {θ, -π, π}, {φ, 0, π}, Axes -> None, Boxed -> False, Evaluated -> True, Mesh -> False, PlotPoints -> 55, ViewPoint -> {-1.3, -2.4, 2.}]]

surprise at the back

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  • $\begingroup$ Simply put...amazing! $\endgroup$ Commented Nov 19, 2015 at 11:10
  • $\begingroup$ Do you think we can use the code from my answer in order to generate a 3D graphic? $\endgroup$ Commented Nov 19, 2015 at 11:12
  • $\begingroup$ I'll need to think about it. $\endgroup$ Commented Nov 19, 2015 at 11:26
  • $\begingroup$ Couldn't your ParametricPlot3D be replaced with the slightly easier SphericalPlot3D? $\endgroup$ Commented Nov 19, 2015 at 18:01
  • $\begingroup$ @Rahul, it can, but the default convention always confuses and vexes me, since it's not the one I'm accustomed to (that is, $\theta$ is the longitude, and $\varphi$ is the colatitude). Note this bias in how I defined the real spherical harmonics as well. $\endgroup$ Commented Nov 19, 2015 at 18:05
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Not exactly what you want but a similar application (demonstration of divergence theorem). I guess it worths. I learnt the code eight years ago when still working with Mathematica 5.2. David Park was responsible for the code.

I tried as possible as I could in order to upgrade it so that it works with recent versions.

partitionfunction[d_][q_] := Piecewise[{{Sin[(Pi*q)/(2*d)]^2, Inequality[0, LessEqual, q, Less, d]}, {1, Inequality[d, LessEqual, q, Less, 2*Pi - d]}, {Sin[(Pi*(2*Pi - q))/(2*d)]^2, 2*Pi - d <= q <= 2*Pi}}] radius[d_][q_] := 1 + 1.5*partitionfunction[d][q]*BesselJ[5, (13/(2*Pi))*q + 5] curve[d_][q_] := radius[d][q]*{Cos[q], Sin[q]} tangent[t_] = N[curve[1][45*Degree] + t*Derivative[1][curve[1]][45*Degree]]; normal[t_] = N[curve[1][45*Degree] + t*Reverse[Derivative[1][curve[1]][45*Degree]]*{1, -1}]; n = {1.127382730502271, 1.037382730502271}; g = ParametricPlot[curve[1][q], {q, 0, 2*Pi}, Axes -> False, PlotPoints -> 50, PlotStyle -> Thickness[0.007], Exclusions -> None]; line = Cases[g, l_Line :> First@l, Infinity]; g1 = Graphics[{Opacity[0.4], Darker@Orange, EdgeForm[{Thick, Darker@Orange}], Polygon[line]}, Options[g]]; g2 = Graphics[{Thickness[0.007], Arrowheads[Large], Arrow[{normal[0], normal[0.3]}]}]; g3 = ParametricPlot[tangent[t], {t, -0.2, 0.2}, PlotStyle -> Thickness[0.006], PlotPoints -> 50]; cir = Graphics[{Circle[normal[0], 0.1, {3.3*(Pi/2), 2.15*Pi}]}]; po = Graphics[{PointSize[0.01], Point[n]}]; tex1 = Graphics[Text[Style["V", 17], {0.0532359, -0.0138103}]]; tex2 = Graphics[Text[Style["S", 17], {0.470751, -1.08655}]]; tex3 = Graphics[Text[Style["n", 17, Italic, Black, Bold], {1.5, 1.2}]]; Show[{g1, g2, g3, cir, po, tex1, tex2, tex3}, PlotRange -> All]

enter image description here

Just for fun: Here is the old good code for 5.2 (!), for anyone interested.

Block[{$DisplayFunction = Identity}, g = ParametricPlot[curve[1][o1], {o1, 0, 2*Pi}, Axes -> False, PlotPoints -> 50, PlotStyle -> Thickness[0.007]]; g1 = g /. Line[x_] -> {GrayLevel[0.8], Polygon[x]}; g2 = ParametricPlot[tangent[t], {t, -0.2, 0.2}, PlotStyle -> Thickness[0.006], PlotPoints -> 50]; g3 = Graphics[ {Thickness[0.007], Arrow[normal[0], normal[0.3], HeadLength -> 0.06, HeadCenter -> 0.7]}]; cir = Graphics[{Circle[normal[0], 0.1, {3.3*(Pi/2), 2.15*Pi}]}]; po = Graphics[{PointSize[0.01], Point[n]}]; tex1 = Graphics[Text["V", {0.0532359, -0.0138103}]]; tex2 = Graphics[Text["S", {0.470751, -1.08655}]]; tex3 = Graphics[ Text[StyleForm["n", FontSize -> 17, FontFamily -> "Times", FontColor -> Black, FontWeight -> "Bold"], {1.7, 1.2}]]; ] Show[ g, g1, g2, g3, tex1, tex2, tex3, cir, po, AspectRatio -> Automatic, TextStyle -> {FontSize -> 17, FontFamily -> "Times", FontWeight -> "Bold"} ];
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  • $\begingroup$ I guess a challenge is to use this approach in order to make 3D graphic. But this is beyond me:-)! $\endgroup$ Commented Nov 18, 2015 at 16:41

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