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Topic:
Is MIT's Larry Guth, Sigurdur Helgason, Anette Hosoi as dumb as Dan Christensen in thinking ellipse is a conic section when it really is an OVAL (proofs at end of post)
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Last Post:
Oct 5, 2017 5:04 PM



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Re: Is MIT's Larry Guth, Sigurdur Helgason, Anette Hosoi as dumb as Dan Christensen in thinking ellipse is a conic section when it really is an OVAL (proofs at end of post)
Posted:
Oct 5, 2017 5:04 PM


On Thursday, October 5, 2017 at 9:36:06 AM UTC+2, Archimedes Plutonium is asking for proof.
We want to show that certain cone sections (as well as certain cylinder sections) are ellipses.
Some preliminaries:
Top view of the cone section and depiction of the coordinate system used in the proof below:
^ x  + <= x=h .'  `. /  \ .  .    .  . \  / `.  .´ y <+ <= x=0
Cone/Cylinder (side view): /  \ (with b <= a) /b  \ /+´ <= x = h / ´ \ / ´  \ / ´  \ x = 0 => ´+\ / a  \
2 cases: 1.) cone: b < a, 2.) cylinder: b = a = r.
Proof:
r(x) = a  ((ab)/h)x d(x) = a  ((a+b)/h)x
y(x)^2 = r(x)^2  d(x)^2 = ab  ab(2x/h  1)^2 = ab(1  4(x  h/2)^2/h^2
=> (1/ab)y(x)^2 + (4/h^2)(x  h/2)^2 = 1 ...equation of an ellipse
qed



