Descritize or say mesh the conic section?

Questions
,
1

Rotating a non-vertical line such as the line y = -x about a vertical line such as the y axis at their point of intersection, a double napped right circular hollow cone is obtained. One of seven curves namely circle, ellipse, parabola, hyperbola, point, line and intersecting lines will be formed when a plane intersects the double napped right circular cone depending on the value of d ranging from d=-15 through to d=5 only existing to translate the plane so it may pass or not pass through the vertex and the value of a ranging from a=0 through to a=145 only existing to rotate the plane after it has been translated

How can I descritize or say mesh the conic section which results as the intersection?

mesh or say descritizing conic sections.nb (6.52 KB)

2

Considering yours in the notebook which is `

Manipulate[
 Show[RevolutionPlot3D[-x, {x, -10, 10}, RevolutionAxis -> "Z", 
   Boxed -> False, Axes -> False, Mesh -> None], 
  Graphics3D[{{Opacity[1], 
     White, {Rotate[
       Translate[
        Polygon[{{10, 10, 5}, {-10, 10, 5}, {-10, -10, 5}, {10, -10, 
           5}}], {0, 0, d}], a Degree, {0, 1, 0}]}}}, Boxed -> False],
   ImageSize -> 270], {{d, 0, "Translation"}, -15, 5, 
  Appearance -> "Labeled"}, {{a, 0, "Rotation"}, 0, 145, 
  Appearance -> "Labeled"}]

I tried using DiscretizeGraphics, DiscretizeRegion and RegionIntersection as follows

Manipulate[
 Show[RevolutionPlot3D[-x, {x, -10, 10}, RevolutionAxis -> "Z", 
   Boxed -> False, Axes -> False, Mesh -> None], 
  Graphics3D[{{Opacity[1], 
     White, {Rotate[
       Translate[
        Polygon[{{10, 10, 5}, {-10, 10, 5}, {-10, -10, 5}, {10, -10, 
           5}}], {0, 0, d}], a Degree, {0, 1, 0}]}}}, Boxed -> False],
   DiscretizeRegion@
   RegionIntersection[
    DiscretizeGraphics[
     RevolutionPlot3D[-x, {x, -10, 10}, RevolutionAxis -> "Z", 
      Boxed -> False, Axes -> False, Mesh -> None]], 
    DiscretizeGraphics[
     Graphics3D[{{Opacity[1], 
        White, {Rotate[
          Translate[
           Polygon[{{10, 10, 5}, {-10, 10, 5}, {-10, -10, 
              5}, {10, -10, 5}}], {0, 0, d}], a Degree, {0, 1, 0}]}}},
       Boxed -> False]]], 
  ImageSize -> 270], {{d, 0, "Translation"}, -15, 5, 
  Appearance -> "Labeled"}, {{a, 0, "Rotation"}, 0, 145, 
  Appearance -> "Labeled"}]

but it fails.

3

Thank you for the effort.

4
Open in Wolfram Cloud ↗
5

Thank you very much Roger but I want something like the blue curves in the pictures attached below

ellipse.png
ellipse.png200×200 9.03 KB

hyperbola.png
hyperbola.png200×200 9.61 KB

intersecting lines.png
intersecting lines.png200×200 9.33 KB

parabola.png
parabola.png200×200 8 KB

6

See this Wolfram Demonstration

7
Open in Wolfram Cloud ↗

Yes, having some trouble re posting and editing. Just having fun really with you graphic. Here is my update.

8

Thank you very much Hans Milton for linking me to that nice
demonstration, but I want to tinker with to see if
Mathematica can be manipulated so as to find and or solve for the
intersection of a RevolutionPlot3D and Graphics3D Polygon.