plotting - ListPlot point colors by list of colorfunctions

I would like to display the positions of a collection of particles as a function of time (animated) and color code them based on their velocities. For example, consider 10 particles moving in a 2D harmonic oscillator:

velocity = Flatten[(Sqrt[x'[t]^2 + y'[t]^2] /. position)];

Num = 10;
position0 = 
  Table[{RandomReal[{-5, 5}], RandomReal[{-5, 5}]}, {i, 1, Num}];

ωx = 1; ωy = 1.07;
position = Table[
   NDSolve[{x''[t] == -ωx^2 x[t], x[0] == position0[[i, 1]], x'[0] == 0, 
     y''[t] == -ωy^2 y[t], y[0] == position0[[i, 2]], 
     y'[0] == 0}, {x, y}, {t, 0, 10}], {i, 1, Num}];
velocity = Flatten[(Sqrt[x'[t]^2 + y'[t]^2] /. position)];

I can then plot a single particle's trajectory colored by its velocity:

ParametricPlot[Evaluate[{x[t], y[t]} /. position[[2]]], {t, 0, 10},
 ColorFunction -> 
  Function[{x, y, t}, 
   ColorData[{"ThermometerColors", {0, 5}}][velocity[[2]]]], 
 ColorFunctionScaling -> False
 ]

But what I would like to do is plot the points using ListPlot, and color code each point by its velocity as a function of time. I've started:

Animate[
 ListPlot[Table[{x[t], y[t]} /. position[[i, 1]], {i, 1, Num}],
   PlotRange -> {{-5, 5}, {-5, 5}}, Joined -> True, 
   ColorFunction -> 
    Function[{i, x, y, t}, 
     ColorData[{"ThermometerColors", {0, 5}}][velocity]]], 
   ColorFunctionScaling -> False] /. Line -> Point,
 {t, 0, 10}, AnimationRunning -> False]

I get the animation right but the colors do not appear -- I'm sure I'm not addressing the ColorFunction correctly but I'm not sure how to get it to index by point rather than by one of the continuous variables (eg. x or y).

Comments

plotting - Filling between two spheres in SphericalPlot3D

Manipulate[ SphericalPlot3D[{1, 2 - n}, {θ, 0, Pi}, {ϕ, 0, 1.5 Pi}, Mesh -> None, PlotPoints -> 15, PlotRange -> {-2.2, 2.2}], {n, 0, 1}] I cant' seem to be able to make a filling between two spheres. I've already tried the obvious Filling -> {1 -> {2}} but Mathematica doesn't seem to like that option. Is there any easy way around this or ... Answer There is no built-in filling in SphericalPlot3D . One option is to use ParametricPlot3D to draw the surfaces between the two shells: Manipulate[ Show[SphericalPlot3D[{1, 2 - n}, {θ, 0, Pi}, {ϕ, 0, 1.5 Pi}, PlotPoints -> 15, PlotRange -> {-2.2, 2.2}], ParametricPlot3D[{ r {Sin[t] Cos[1.5 Pi], Sin[t] Sin[1.5 Pi], Cos[t]}, r {Sin[t] Cos[0 Pi], Sin[t] Sin[0 Pi], Cos[t]}}, {r, 1, 2 - n}, {t, 0, Pi}, PlotStyle -> Yellow, Mesh -> {2, 15}]], {n, 0, 1}]

plotting - Plot 4D data with color as 4th dimension

I have a list of 4D data (x position, y position, amplitude, wavelength). I want to plot x, y, and amplitude on a 3D plot and have the color of the points correspond to the wavelength. I have seen many examples using functions to define color but my wavelength cannot be expressed by an analytic function. Is there a simple way to do this? Answer Here a another possible way to visualize 4D data: data = Flatten[Table[{x, y, x^2 + y^2, Sin[x - y]}, {x, -Pi, Pi,Pi/10}, {y,-Pi,Pi, Pi/10}], 1]; You can use the function Point along with VertexColors . Now the points are places using the first three elements and the color is determined by the fourth. In this case I used Hue, but you can use whatever you prefer. Graphics3D[ Point[data[[All, 1 ;; 3]], VertexColors -> Hue /@ data[[All, 4]]], Axes -> True, BoxRatios -> {1, 1, 1/GoldenRatio}]

plotting - Adding a thick curve to a regionplot

Suppose we have the following simple RegionPlot: f[x_] := 1 - x^2 g[x_] := 1 - 0.5 x^2 RegionPlot[{y < f[x], f[x] < y < g[x], y > g[x]}, {x, 0, 2}, {y, 0, 2}] Now I'm trying to change the curve defined by $y=g[x]$ into a thick black curve, while leaving all other boundaries in the plot unchanged. I've tried adding the region $y=g[x]$ and playing with the plotstyle, which didn't work, and I've tried BoundaryStyle, which changed all the boundaries in the plot. Now I'm kinda out of ideas... Any help would be appreciated! Answer With f[x_] := 1 - x^2 g[x_] := 1 - 0.5 x^2 You can use Epilog to add the thick line: RegionPlot[{y < f[x], f[x] < y < g[x], y > g[x]}, {x, 0, 2}, {y, 0, 2}, PlotPoints -> 50, Epilog -> (Plot[g[x], {x, 0, 2}, PlotStyle -> {Black, Thick}][[1]]), PlotStyle -> {Directive[Yellow, Opacity[0.4]], Directive[Pink, Opacity[0.4]],

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