锥形图像细化 [英] Cone image refinement

问题描述

试图制作一个与平面相交的圆锥体的三维立体图形，我选择Mathematica中已有方法（即S.Mangano和S.Wagon的书籍）的轻微重排。下面的代码假定显示所谓的丹德林结构：内部和外部球体在内部与锥体相切，也与锥体相交的平面相切。

 区块[{r1，r2，m， h1，h2，C1，C2，M，MC1，MC2，T1，T2，锥体，斜面，平面}，
 {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  - >假] [[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}，
盒装 - >假，轴 - >假] [[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  - >假，照明 - > 中性，
 ViewPoint  - > {-1.8，-2.5,1.5}，ImageSize-> 950]] 
  

以下是图形：

问题在于切点附近两个球体周围的白色斑点。将上面的代码放到 Manipulate [... GrayLevel [z] ... {z，0,1}] 我们可以很容易地移除这些点，因为z倾向于1.

1. 任何人都可以看到不同的方法去除白点吗？我更喜欢 GrayLevel [z] ，其中z < 0.5。




2. 我一直对图形上较低和较高球体上的斑点略有不同的模式感兴趣。你有什么想法可以解释这个问题吗？

为什么有没有人建议只使用内置的 Cone [] 原始语言？

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

这里工作正常（没有白点）。此外，这不是一个黑客或解决方法。空的 EdgeForm [] 的目的是去除锥形底座的黑色轮廓。







_{我刚刚意识到， code> Cone [] 有一个坚实的基础，在包含的图片上也非常明显。所以这与原来的 RevolutionPlot 版本并非完全相同。}




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 ?

解决方案

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.}

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