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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)

Replies: 5   Last Post: Oct 5, 2017 5:04 PM

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Registered: 1/23/16
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
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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 - ((a-b)/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



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