Cone image refinement

Question

Trying to make a nice three-dimensional graphics of cone intersected by a plane I choose a slight rearrangement of an existing approach in Mathematica (i.e. books by S.Mangano and S.Wagon). The code beneath is assumed to show so-called Dandelin construction : the inner and outer spheres tangent internally to a cone and also to a plane intersecting the cone. Tangency points of spheres to the plane at the same time are foci of the ellipse.

 Block[{r1, r2, m, h1, h2, C1, C2, M, MC1, MC2, T1, T2, cone, slope, plane},
   {r1, r2} = {1.4, 3.4};
    m = Tan[70.*Degree];
    h1 := r1*Sqrt[1 + m^2];
    h2 := r2*Sqrt[1 + m^2];
    C1 := {0, 0, h1};
    C2 := {0, 0, h2};
    M = {0, MC1 + h1};
    MC2 = MC1*(r2/r1);
    MC1 = (r1*(h2 - h1))/(r1 + r2);
    T1 = C1 + r1*{-Sqrt[1 - r1^2/MC1^2], 0, r1/MC1};
    T2 = C2 + r2*{Sqrt[1 - r2^2/MC2^2], 0, -(r2/MC2)};

    cone[m_, h_] := RevolutionPlot3D[{t, m*t}, {t, 0, h/m}, Mesh -> False][[1]];
    slope = (T2[[3]] - T1[[3]])/(T2[[1]] - T1[[1]]);
    plane = ParametricPlot3D[{t, u, slope*t + M[[2]]}, {t, -2*m, 12/m}, {u, -3, 3},
                              Boxed -> False, Axes -> False][[1]];
    Graphics3D[{{Gray, Opacity[0.39], cone[m, 1.2*(h2 + r2)]},
                {Opacity[0.5], Sphere[C1, r1], Sphere[C2, r2]},
                {LightBlue, Opacity[0.6], plane},
                 PointSize[0.0175], Point[T1], Point[T2]},
                 Boxed -> False, Lighting -> "Neutral", 
                 ViewPoint -> {-1.8, -2.5, 1.5}, ImageSize -> 950]]

Here is the graphics :

The problem is with the white spots around the both spheres near tangency points. Putting the above code to Manipulate[...GrayLevel[z]...{z,0,1} ] we can easliy "remove" the spots as z tends to 1.

1. Can anyone see a different approach to removing the white spots ? I prefer GrayLevel[z] with z < 0.5.

2. I have been intrigued with a slightly different pattern of the spots on the lower and upper spheres on the graphics . Have you got any ideas how this could be explained ?

Answer 1

You might want to make the spheres a tiny bit smaller:

Sphere[C1, .98 r1], Sphere[C2, .98 r2]

It's a hack, but it avoids the intersection problem.

Alternatively, you can up the PlotPoints on the cone:

PlotPoints -> 100

but that will make the rendering slower.

Edit: Or a combination of these to help with speed and quality.

Answer 2

Why has no one suggested to just use the built-in Cone[] primitive?

cone[m_, h_] := {EdgeForm[], Cone[{{0, 0, h}, {0, 0, 0}}, h/m]};

This works fine here (no white spots). Also, it's not a hack or workaround. The purpose of the empty EdgeForm[] is to remove the black outline of the cone base.

_{I just realized that Cone[] has a solid base, also very visible on the included image. So this is not exactly the same as the original RevolutionPlot version.}

Answer 3

You could construct the cone using a Tube with varying radii:

cone[m_, h_] := {CapForm[None], Tube[{{0, 0, 0}, {0, 0, h}}, {0, h/m}]};