How do you plot random points in three dimensions?

Question

I've got a function that maps a 2D plane onto a sphere (I'm trying to learn about Geodesics).

f[u_,v_]:={X[u,v],Y[u,v],Z[u,v]};
X[u_,v_]:=Cos[v]Sin[u];
Y[u_,v_]:=Sin[v]Sin[u];
Z[u_,v_]:=Cos[u];

I want to plot this, but the Plot3D gives me only a single value for any given [X,Y] set of coordinates. At the least, I'd like something like a scatter chart (that is, just give it 3 coordinates and have it plot a point), but it would be nice to be able to generate a wire-frame so I could draw my solutions.

A related question. – J. M.'s missing motivation May 25 '20 at 14:53 — J. M.'s missing motivation, May 25 '20 at 14:53

Bob Hanlon · Answer 1 · 2019-12-17T16:04:40.037

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Clear["Global`*"]

f[u_, v_] := {X[u, v], Y[u, v], Z[u, v]};
X[u_, v_] := Cos[v] Sin[u];
Y[u_, v_] := Sin[v] Sin[u];
Z[u_, v_] := Cos[u];

SeedRandom[1234]

data = f @@@ RandomReal[{0, 2 Pi}, {5000, 2}];

Graphics3D[Point[data]]

However, you can get a smoother distribution with RandomPoint on a Sphere

SeedRandom[1234]

data2 = RandomPoint[Sphere[{0, 0, 0}], 5000];

Graphics3D[Point[data2]]

EDIT: Converting the points into a 3-D surface

ListSurfacePlot3D[data2, Axes -> False]

edited Dec 17 '19 at 16:04

answered Dec 17 '19 at 01:50

Bob Hanlon

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Your first approach is not correct. Pay your attention that the points flock near the poles. – user64494 Dec 17 '19 at 08:01
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@user64494 it's not incorrect — it's just a different distribution. – Ruslan Dec 17 '19 at 10:39
Great answer. Thank you. For extra points, can you make that into a wire-frame? – Quark Soup Dec 17 '19 at 15:22
@Ruslan: Exactly saying, the points produced by the first code are not uniformly distributed on the sphere so these can't be called random points on the sphere. – user64494 Dec 18 '19 at 08:15
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@user64494 random doesn't imply uniform. – Ruslan Dec 18 '19 at 08:16
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@user64494 - The OP did not specify that the points in the "scatter chart" should be uniformly distributed on the sphere. The first approach demonstrates that if you start with a uniform distribution in 2-D, the transformation does not produce a uniform distribution in 3-D. While RandomPoint[reg, n] gives a list of n pseudorandom points uniformly distributed in the region reg; if you were to transform these 3-D points on a sphere to 2-D, they would not be uniform in 2-D. – Bob Hanlon Dec 18 '19 at 14:09

score 0 · Answer 2 · answered Dec 18 '19 at 01:28

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Thanks to Bob Hanlon for his answer. Using that as a starting point, I was able to work out this, which is what I was really after:

ParametricPlot3D[{R[u, v]}, {u, 0, Pi}, {v, 0, 2*Pi}]

answered Dec 18 '19 at 01:28

Quark Soup

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How do you plot random points in three dimensions?

2 Answers2