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.

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Consider an arbitrary ordinate (i.e. y value) TK constructed on the axis at T. We wish to determine the relationship between TK and AT, that is, the symptom of the conic. The ordinate TK is located in a horizontal plane that cuts the cone in the circle with diameter DG. In this horizontal plane construct the segments GK and DK, which results in a right triangle inscribed in a semicircle. (The triangle is right by Elements, Book III, Proposition 20). We also know that triangles GTK and KTD are similar (by Book VI, Prop. 8) and this implies

or
	(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

or
	(2)

Also in the axial plane are the pairs of similar triangles HDT and IEA, and BDT and BEA. From these we see that

or
and	(3)
.	(4)

Combining (3) and (4) we have

or
.	(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

.

(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

.

This does look like the equation of an ellipse.