How to make a density plot for an isotropic function written in cylindrical coordinates?

Question

$f(\rho,z)=2\rho^2+\frac{2}{3}z^2 \quad$ is a function in the cylindrical coordinate system $(\rho,\theta,z)$

The function $f(\rho,z)$ is isotropic because it does not depend on the angle $\theta$. Therefore, a plot of the density distribution of the function $f(\rho,z)$ can be obtained. I would like to get a density plot of the function $f$, where the function's dependence on $z$ is shown by the density (in color).

How could this graph be plot?

Perhaps DensityPlot or ContourPlot are suitable for this type of plot.

Answer 1

You just need to define the function you want and use one of the following Mathematica functions. If I understand what you are asking for

f[\[Rho]_, z_] := 2 \[Rho]^2 + (2/3) z^2

DensityPlot[f[\[Rho], z], {\[Rho], 0, 5}, {z, -5, 5}, 
 PlotRange -> All, PlotPoints -> 100, 
 ColorFunction -> "TemperatureMap", 
 FrameLabel -> {"\[Rho]", "z", "Density"}, PlotLegends -> Automatic]

or

f[\[Rho]_, z_] := 2 \[Rho]^2 + (2/3) z^2

ContourPlot[f[\[Rho], z], {\[Rho], 0, 5}, {z, -5, 5}, Contours -> 20, 
 ColorFunction -> "TemperatureMap", ContourLabels -> True, 
 FrameLabel -> {"\[Rho]", "z"}, PlotLegends -> Automatic]

Updated

To plot the projection of a sphere onto a plane while showing the distance from a specific point on the sphere to the plane as a density distribution (color), here's how you can do it:

• Define the parameters of the sphere and the point:

(* Sphere parameters *)
sphereRadius = 2;
sphereCenter = {0, 0, 3}; (* Center of the sphere *)

(* Point parameters *)
pointOnSphere = {0, 0, 5}; (* Coordinates of a point on the sphere *)

• Create a function that calculates the distance from the point on the sphere to the plane for a given point on the plane:

distanceToPlane[pointOnPlane_, pointOnSphere_] := 
  EuclideanDistance[pointOnSphere, pointOnPlane]

• Create a density distribution plot where the x and y axes represent the coordinates of the plane, and the color represents the distance from the point on the sphere to the plane:

DensityPlot[
  distanceToPlane[{x, y, 0}, pointOnSphere], 
  {x, -sphereRadius, sphereRadius}, {y, -sphereRadius, sphereRadius}, 
  PlotRange -> All, ColorFunction -> "Rainbow", 
  PlotLegends -> Automatic, FrameLabel -> {"x", "y"}, 
  PlotLabel -> "Distance from Sphere Point to Plane"]