title: Cone image refinement
body: <p>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.</p>
  
  <pre><code> 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 -&gt; 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 -&gt; False, Axes -&gt; 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 -&gt; False, Lighting -&gt; "Neutral", 
                   ViewPoint -&gt; {-1.8, -2.5, 1.5}, ImageSize -&gt; 950]]
  </code></pre>
  
  <p>Here is the graphics : 
   <img src="https://i.stack.imgur.com/UIpLC.jpg" alt="Dandelin construction"></p>
  
  <p>The problem is with the white spots around the both spheres near tangency points. Putting the above code to <code>Manipulate[...GrayLevel[z]...{z,0,1} ]</code> we can easliy "remove" the spots as z tends to 1.  </p>
  
  <ol>
  <li><p>Can anyone see a different approach to removing the white spots ?  I prefer  <code>GrayLevel[z]</code> with z &lt; 0.5.</p></li>
  <li><p>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 ?</p></li>
  </ol>