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[GIF] Move Along (Envelopes of conic sections)


Envelopes of conic sections

Move Along

Consider a circle of latitude at height $z=-3/4$ on the unit sphere. The orthogonal complements of unit vectors on the circle are planes through the origin in $\mathbb{R}^3$, which we can also interpret as lines in $\mathbb{RP}^2$, the real projective plane. In order to visualize these lines, take the intersection of the planes with the plane $z=1$. These lines will form the envelope of a circle in the $z=1$ plane.

Now, as we rotate the unit sphere around the $y$-axis, the lines in the $z=1$ plane will form the envelopes of a family of conics. What the animation actually shows are the points of intersections of the lines corresponding to 50 equally-spaced points on this rotating circle on the unit sphere.

Here's the code:

DynamicModule[{m, p, a, θ = 0., t = π/1., r = -0.75, lines,
   linepairs, dots, cols = RGBColor /@ {"#E84A5F", "#2A363B"}},
  m = RotationMatrix[
    t, {Cos[θ] Sin[ϕ], Sin[θ] Sin[ϕ], 
  p[s_] = m.{Sqrt[1 - r^2] Cos[s], Sqrt[1 - r^2] Sin[s], r};
  a = m.{0, 0, 1};
  lines = 
       0}, {0, -p[s][[3]]/p[s][[2]]}}], {s, 0, 2 π - 2 π/50, 
     2 π/50}];
  linepairs = Subsets[lines, {2}];
  dots = DeleteCases[RegionIntersection @@ # & /@ linepairs, 
  Graphics[{White, PointSize[.005], cols[[1]], dots}, 
   ImageSize -> 540, Frame -> False, Background -> cols[[-1]], 
   PlotRange -> 5], {ϕ, 0, π/2}]
POSTED BY: Clayton Shonkwiler
6 months ago

enter image description here - another post of yours has been selected for the Staff Picks group, congratulations! We are happy to see you at the top of the "Featured Contributor" board. Thank you for your wonderful contributions, and please keep them coming!

POSTED BY: Moderation Team
6 months ago

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