Search Loci: Convergence:
M. is execrable, but
Mr. C. is (x +
In N. Rose, Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.
Can You Really Derive Conic Formulae from a Cone?
Deriving the Symptom of the Acute-angled Cone
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.
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
Also in the axial plane are the pairs of similar triangles HDT and IEA, and BDT and BEA. From these we see that
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
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