Plot the Whitney umbrella using Plot3D

Questions
,
1

Good day dear community,

I tried to plot the Whitney umbrella with the equation x^2 - y^2 z = 0.
Unfortunately this is only possible with the command:

ContourPlot3D[x^2 - y^2 z, {x, -1, 1}, {y, -1, 1}, {z, 0, 1}, Axes -> True, AxesLabel -> Automatic]

I have also already tried solving the equation according to the different variables and using the individual solutions to create the graph, which unfortunately has not given me any results so far.

Now my questions:

  1. is it possible to plot only the Whitney umbrella without the additional artifacts shown in ContourPlot3D.

  2. is it possible to plot it with Plot3D and the equation x^2 - y^2 z = 0? Unfortunately, with Plot3D I only get a strange image, which does not look anything like the Whitney umbrella.

I would greatly appreciate any Help.

Greetings Christian

some of my results .JPG
some of my results .JPG1065×1125 101 KB

2

Oh I forgot to say, I want to do this with Mathematica and I use Mathematica version 13.0

3

You only want to show the contour for the solution f == 0 which can be done with ContourPlot. For the Plot3D all you have to do is set the proper plot range (and remove the boundary and clipping styles). Or you can use the parametric equation and use parametric plot.

ContourPlot3D[x^2 - y^2 z == 0, {x, -1, 1}, {y, -1, 1}, {z, 0, 1}]
ContourPlot3D[x^2 - y^2 z, {x, -1, 1}, {y, -1, 1}, {z, 0, 1}, 
 Contours -> {0}]
ParametricPlot3D[{u v, u, v^2}, {u, -1, 1}, {v, -1, 1}]
Plot3D[x^2/y^2, {x, -1, 1}, {y, -1, 1}, PlotRange -> {0, 1}, 
 ClippingStyle -> None, BoundaryStyle -> None]

These all work

4

You can use ImplicitRegion to define the umbrella. And then Region to visualize:

reg = ImplicitRegion[x^2 - y^2 z == 0, {{x, -1, 1}, {y, -1, 1}, {z, 0, 1}}]

Region[reg, Boxed -> True, Axes -> True]
5

Thank you very much to both of you :slight_smile:

I did not think that there are so many ways to get the same result.

The solution with the Plot3D is my favorite solution.
But the others I will keep in mind.

Wish you a nice day