How to show values of points on surface plotted with ContourPlot3D

Question

The help for ContourPlot3D starts with this example

ContourPlot3D[x^3 + y^2 - z^2 == 0, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}]

This returns a Plots of the surface $x^2 + y^2 - z^2 = 0$:

.

Now I have a function, and I would like to know how this function behaves on this surface. For example, take

f[x_,y_,z_] := {x^3 - 2 x y + z^2 x - 5 z^2 y^2 - 4 x y z, 
                Sin[10x] + Cos[5y] + Cos[20z]}

I was thinking of making a list with some of the point that lie on this surface. Then it is easy to evaluate the function on these points. The way I would like to generate the list is to click with the mouse on a few points that I think are interesting. Is that possible?

What are the alternatives to using the mouse? There most be some list available with all points that were used to draw this graph. How can I take some point of that list and add them as point to the 3D plot to visualize them?

Update

The methods of Szabolcs and Heike work fine on the example $x^3 + y^2 - z^2 =0$. Now I try to apply the same to

$$ 2316 a^{12} c^6+500 a^{11} b c^5+10296 a^{10} b^2 c^4+1624 a^{10} c^5+656 a^9 b^3 c^3- \\ - 3856 a^9 b c^4+41 a^8 b^4 c^2+808 a^8 b^2 c^3+784 a^8 c^4+a^7 b^5 c+24 a^7 b^3 c^2- \\ - 176 a^7 b c^3+2 a^6 b^4 c+16 a^6 b^2 c^2+32 a^6 c^3 = 0 $$

f[a_, b_, c_] := 2 a^6 b^4 c + a^7 b^5 c + 16 a^6 b^2 c^2 + 24 a^7 b^3 c^2 
+ 41 a^8 b^4 c^2 + 32 a^6 c^3 - 176 a^7 b c^3 + 808 a^8 b^2 c^3 
+ 656 a^9 b^3 c^3 + 784 a^8 c^4 - 3856 a^9 b c^4 
+ 10296 a^10 b^2 c^4 + 1624 a^10 c^5 + 500 a^11 b c^5 + 2316 a^12 c^6
pts = {};

Substituting this function into Heike's solution does not work. Clicking does not result in points on the surface. Also Szabolcs's FindInstance does not work. What goes wrong here?

.

Answer 1

The location that the mouse is pointing to on a 3D surface can be found by starting with MousePosition["Graphics3DBoxIntercepts"]. This will give you the two points where the line perpendicular to the screen at the mouse pointer intersects the three-dimensional bounding box.

We can calculate the intersection of this line with the surface to find a point. Here is a simple implementation to track that point dynamically:

fun defines the surface:

fun[{x_, y_, z_}] := x^3 + y^2 - z^2

Let's plot it:

plot = ContourPlot3D[
   fun[{x, y, z}] == 0, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}]

Now let's try to find the intersection of the mouse pointer with the surface using FindRoot. There might be several intersections, so I specified the box intersection point closer to the viewer as a starting point for FindRoot (t == 0 in the code). This does not guarantee that the closest (i.e. visible) point will be found, but it makes it more likely.

Show[plot,
 Graphics3D[{
   Red,
   Dynamic@Quiet@Check[
      Sphere[#, Scaled[0.01]]& @ Module[{p1, p2, t},
        {p1, p2} = MousePosition[{"Graphics3DBoxIntercepts", Graphics3D}];
        (p2 - p1) t + p1 /. FindRoot[fun[(p2 - p1) t + p1], {t, 0, 0, 1}]
        ],
      {}]}]
 ]

Now that we have the point on the surface, you can do with it whatever you want (calculate another functions, etc.) You can use EventHandler to just record clicks instead of tracking values dynamically.

---

To address your other question about how to get a number of points on the surface. One way is to use FindInstance.

FindInstance[
 fun[{x, y, z}] == 0 && Thread[-2 < And[x, y, z] < 2, And], {x, y, 
  z}, Reals, 10]

This will give you 10 points that are precisely on the surface (this uses exact calculations). Let's show them:

Show[plot, Graphics3D[{Red, Sphere[{x, y, z}, Scaled[0.01]] /. %}]]

To get the points generated by ContourPlot3D, extract them from the GraphicsComplex object is creates. These coordinates will not be quite as precise as they are meant for visualization only.

First@Cases[plot, GraphicsComplex[points_, ___] :> points, Infinity]

Let's show those points:

Graphics3D@Point[%]

Answer 2

This solution is pretty similar to Szabolcs's solution. I've added an EventHandler to make it easier to select points in the plot. Here f[x,y,z]==0 is the equation of the surface, and pts contains a list of selected points on this surface. You can add points to the list by right-clicking somewhere on the plot.

f[x_, y_, z_] := x^3 + y^2 - z^2;
pts = {};

plot = ContourPlot3D[
   f[x, y, z] == 0, {x, -2, 2}, {y, -2, 2}, {z, -2, 2},
   Mesh -> False, ColorFunctionScaling -> False, 
   BoundaryStyle -> None];
Dynamic@EventHandler[
  Show[plot, Graphics3D[{Red, Sphere[#, .05] & /@ pts}]],
  {{"MouseClicked", 2} :> Module[{p, sol, pt},
     p = CurrentValue[{"MousePosition", "Graphics3DBoxIntercepts"}];
     If[p =!= None, 
      sol = Quiet@
        Check[FindRoot[
          f[x, y, z] /. Thread[{x, y, z} -> l p[[1]] + (1 - l) p[[2]]],
          {l, .5, 0, 1}], None];
      If[sol =!= None, 
       AppendTo[pts, l p[[1]] + (1 - l) p[[2]] /. sol]]]]}]