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Among answers of ConicsAssemblyQuery, I posted images that are not pretty. In particular, the parabola is elusive due to large parameter step. Please help.

The program attempts to translate and rotate a cutting plane ( by varying $ CC, \beta$ respectively ) acting on a cone of given semi-vertical angle $\alpha$. Change of sign in front of sqrt produces cone shell below $xz$ plane that can be Shown together.

al = .81; XYZ[x_, bt_, CC_] := {x, Tan[bt] x - CC,  Sqrt[x^2 (Tan[al]^2 - Tan[bt]^2) + 2 Tan[bt] x  CC - CC^2]};
 ParametricPlot3D[ XYZ[x, bt, 0.7], {x, 0, 3}, {bt, 0.85, 3}, Mesh -> {12, 30}]
 ParametricPlot3D[XYZ[x, 1.14, CC], {x, -1.3, 1.3}, {CC, -1.2, 1.2}, Mesh -> {12, 30}]
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  • $\begingroup$ Does this help PlotPoints -> 50? $\endgroup$ – Kuba Jan 28 '16 at 11:16
  • $\begingroup$ Disjunct parts at $ \epsilon =1$ for parabola to be attached properly and secondly when $z<0$ part of cone surface is together shown, the seam is quite visible. $\endgroup$ – Narasimham Jan 28 '16 at 11:41
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I post this based on the text of the hyperlink. This was done hurriedly and can be vastly improved (such as plotting conic). Others may wish to do. I am uncertain what the plots shown as graphics aim to do.

cone[a_, r_, t_] := {r, r Tan[a] Cos[t], r Tan[a] Sin[t]};
plane[b_, t_, u_, v_] := {0, t, 0} + {u, Tan[b] u, v};
Manipulate[
 Module[{p1, p2},
  p1 = ParametricPlot3D[{cone[\[Alpha], r, t], 
     cone[\[Alpha], r, t] {-1, 1, 1}}, {t, 0, 2 Pi}, {r, 0, 2}, 
    PlotStyle -> {{Yellow, Opacity[0.5]}}, Mesh -> None];
  p2 = ParametricPlot3D[
    plane[\[Beta], c, u, v], {u, -2, 2}, {v, -1, 1}, 
    PlotStyle -> {Blue, Opacity[0.5]}, Mesh -> None];
  Show[p1, p2, PlotRange -> Table[{-2, 2}, 3], 
   PlotLabel -> Row[{"\[Epsilon]=", Sin[\[Alpha]]/Sin[\[Beta]]}]]],
 {\[Alpha], Pi/6, Pi/3}, {\[Beta], Pi/6, Pi/3}, {c, -2, 2}]

enter image description here

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Have a look at PlotPoints and MaxRecursion the results are quite pleasing

ParametricPlot3D[XYZ[x, bt, 0.7], {x, 0, 3}, {bt, 0.85, 3}, 
Mesh -> {12, 30}, PlotPoints -> 50, MaxRecursion -> 5]

enter image description here

ParametricPlot3D[XYZ[x, 1.14, CC], {x, -1.3, 1.3}, {CC, -1.2, 1.2}, 
Mesh -> {12, 30}, PlotPoints -> 50, MaxRecursion -> 5]

enter image description here

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  • $\begingroup$ The first plot misses out parabola neighborhood when $ \beta \approx \alpha $. It divides cone into two parts. The second plot ( parallel planes cutting and producing ellipses of intersection) is fine. $\endgroup$ – Narasimham Jan 28 '16 at 16:25

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