# How to draw $z =\rho^2+θ^2$ surface figure in cylindrical coordinate system The following code is from here:

g[ρ_, θ_, z_, c_] :=
c {Cos[θ], Sin[θ], 1} {ρ, ρ, z}
Show[Flatten[
Table[ParametricPlot3D[{g[ρ, θ, z, 1],
f[ρ, θ, z, -1]}, {θ, 0, 2.1 Pi}, {ρ, 0,
5}, PlotStyle -> Directive[Purple, Opacity[0.01]]], {z, -5, 5,
1.5}]]]


In addition, I also want to know how to adjust the display size of the picture when uploading the picture, which is too large.

I want to draw the graph by the variable {Rr, Ttheta, Zz} of column coordinate, but I can't find the corresponding built-in function. So I can only use the following methods:

CoordinateTransformData[
"Cartesian" -> "Cylindrical", "Mapping", {x, y, z}]
RegionPlot3D[
0 <= Sqrt[x^2 + y^2] <= 1 && -Pi <= ArcTan[x, y] <= Pi && -1 <= z <=
1, {x, -2, 2}, {y, -2, 2}, {z, -3, 3}, Mesh -> None,
PlotPoints -> 70, MaxRecursion -> 5,
PlotStyle -> Directive[Opacity[0.5]]]


Is there a better way?

And how to draw the surface graph of z = ρ^2+θ^2 in this cylindrical coordinate system?

• So you just want to graph the cylinder? By supplying the cylindrical coordinates? – CA Trevillian Feb 19 '20 at 0:12
• @CATrevillian Thank you for your prompt. I have added the surface function to be drawn as z = ρ^2+θ^2. – A little mouse on the pampas Feb 19 '20 at 0:17
• Don't use the abbreviation MMA, it migth be appropriate in comments, not in answers or questions, and it concerns even more the titles of questions. Moreover it's unnecessary , since this site is all about Mathematica. – Artes Feb 19 '20 at 1:49

I don't know exactly what you want but try this:

f[ρ_, θ_] = ρ^2 + θ^2

RevolutionPlot3D[
Evaluate[f[ρ, θ]], {ρ, 0, 5}, {θ, 0,
2.1 π}, BoxRatios -> {1, 1, .5}, ImageSize -> {300, 300}] You can adjust PlotStyle and ImageSize to whatever you want.

How about CountorPlot3D, however before utilization of which, the form of the equation should be transformed properly:

ContourPlot3D[{z == x^2 + y^2 + ArcTan[x, y]^2}, {x, 0, 2}, {y, -2, 2}, {z, 0, 2}] I don't know if the result is the correct graph of function z = ρ^2+θ^2 in cylindrical coordinate system:

  g[ρ_, θ_, z_, c_] :=
c {Cos[θ], Sin[θ], 1} {ρ, ρ, z}
ParametricPlot3D[
g[ρ, θ, ρ^2 + θ^2, 1], {θ, 0,
2.1 Pi}, {ρ, 0, 5},
PlotStyle -> Directive[Purple, Opacity[0.4]],
MeshStyle -> {Red, Blue}] g[ρ_, θ_, z_, c_] :=
c {Cos[θ], Sin[θ], 1} {ρ, ρ, z}
Show[{Flatten[
Table[ParametricPlot3D[{g[ρ, θ, z, 1],
g[ρ, θ, z, -1]}, {θ, 0, 2.1 Pi}, {ρ, 0,
5}, PlotStyle -> Directive[Purple, Opacity[0.01]],
MeshStyle -> {{Opacity[0.2], Red}, { Opacity[0.2], Blue}}], {z,
0, 60, 5}]],
ParametricPlot3D[{ρ*Cos[u], ρ*Sin[u], ρ^2 + u^2}, {u,
0, 2.1 Pi}, {ρ, 0, 5}, Mesh -> None,
PlotStyle -> Directive[Purple, Opacity[0.7]]]},
PlotRange -> {0, 50}] 