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Topic: Synthetic Geometry proof and Analytical Geometry proof that Conic is
a Oval, never an ellipse// yes Apollonius was wrong

Replies: 5   Last Post: Oct 7, 2017 1:48 AM

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Me

Posts: 1,713
Registered: 1/23/16
Re: mounting evidence Re: ..proof that Conic is a Oval, never an
ellipse// yes Apollonius was wrong

Posted: Oct 6, 2017 2:52 PM
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On Friday, October 6, 2017 at 12:02:40 PM UTC+2, Archimedes Plutonium wrote:

> Now we have ...

...a simple proof that shows that certain conic sections are ellipses.

Some preliminaries:

Top view of the conic section and depiction of the coordinate system used in the proof below:

^ x
|
-+- <= x=h
.' | `.
/ | \
. | .
| | |
. | .
\ | /
`. | .´
y <----------+ <= x=0

Cone (side view):
.
/|\
/ | \
/b | \
/---+---´ <= x = h
/ |´ \
/ ´ | \
/ ´ | \
x = 0 => ´-------+-------\
/ a | \

Proof:

r(x) = a - ((a-b)/h)x and d(x) = a - ((a+b)/h)x, hence

y(x)^2 = r(x)^2 - d(x)^2 = ab - ab(2x/h - 1)^2 = ab(1 - 4(x - h/2)^2/h^2.

Hence (1/ab)y(x)^2 + (4/h^2)(x - h/2)^2 = 1 ...equation of an ellipse

qed

======================================================

@Archie: Yes, this proves that (certain) cone sections "as depicted in my diagram" are ellipses.

=> Appolonius was right.

Note, Archie, that there is no reference to Dandelin Spheres whatsoever.

Still not convinced? Can you point out an error in my simple calculation (of the shape of the coneic section) above?




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