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Consider the acute-angled cone with vertex O and a plane intersecting a generating line OG at a right angle at point A. The plane intersects the cone in the oxytome with diameter AB.
A dynamic view of this construction will be helpful in what follows.
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(1) |
Now consider the similar triangles TAG and TDH in the plane through O, G, and D, the axial plane. The triangles are similar because they each have a right angle and opposite vertical angles. This in turn implies
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or |
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(2) |
Also in the axial plane are the pairs of similar triangles HDT and IEA, and BDT and BEA. From these we see that
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(3) |
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(4) |
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(5) |
Notice also that in triangle IEA the line OL bisects AE so it must also bisect AI, making IA = 2AL. Putting this together with (1) and (2) we have
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(5) |
This might not look like an equation that we recognize, but if we let KT = y, the distance from the center of the ellipse to T be x, AB = 2a, and 2AL = p we have


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This does look like the equation of an ellipse.
Gary S. Stoudt, "Can You Really Derive Conic Formulae from a Cone? - Deriving the Symptom of the Acute-angled Cone," Loci (August 2010)