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
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How can I descritize or say mesh the conic section which results as the intersection?
Extract the two conic sections from the manipulation by copying and pasting the RevolutionPlot
RevolutionPlot3D[-x,{x,-10,10},RevolutionAxis"Z",BoxedFalse,AxesFalse,MeshNone]
The Mesh, MeshStyle, RGBColor are suggestion bar options.
RevolutionPlot3D[-x,{x,-10,10},PlotStyleRGBColor[0.,0.,0.],MeshStyleRGBColor[0.97,1.,0.72],MeshAutomatic,RevolutionAxis"Z",BoxedFalse,AxesFalse]
Return the opposing conics into the manipulate command and set the Translation controls to -15 Translation and zero Rotation. Observe the plane Polygon[{{10,10,5},{-10,10,5},{-10,-10,5},{10,-10,5}}],{0,0,d}] and then set the Translation to minus 5 keeping the Rotation at zero. Make a second observation and then finish moving the plane to the final extrema of the two shapes then read {{d,0,”Translation”},-15,5 in the line of code. Observe at the upper bound of translation 5 a circle is inscribed inside of the plane. The circle is blue and is not any of the command line “choices” or graphics. This is the elusive calculated point needed. At both of the cone bases it appears as an inscribed circle and touches the edges of the plane at exactly four points a Degree,{0,1,0}]}}} Set the controls to Translation -7.6922 and Rotation zero and consider the area of the four almost triangles at the edges. Push play. Stop the Translation about 3.14 and zero Rotation.
Manipulate[Show[RevolutionPlot3D[-x,{x,-10,10},MeshAutomatic,MeshStyleRGBColor[0.,0.,0.],RevolutionAxis"Z",BoxedFalse,AxesFalse],Graphics3D[{{Opacity[1],White,{Rotate[Translate[Polygon[{{10,10,5},{-10,10,5},{-10,-10,5},{10,-10,5}}],{0,0,d}],aDegree,{0,1,0}]}}},BoxedFalse],ImageSize270],{{d,0,"Translation"},-15,5,Appearance"Labeled"},{{a,0,"Rotation"},0,145,Appearance"Labeled"}]
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Translation
3.14
Rotation
0
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Translation
3.14
Rotation
0
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Kwame Acheampong https://community.wolfram.com/groups/-/m/t/1903037
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"}]