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Topic: How we get a Ellipse from a Conic, and how we get a Oval from
Cylinder Sections-- knifes that are V and asymmetrical V shaped

Replies: 27   Last Post: Oct 8, 2017 12:41 AM

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Re: How we get a Ellipse from a Conic, and how we get a Oval from
Cylinder Sections-- knifes that are V and asymmetrical V shaped

Posted: Oct 6, 2017 3:47 AM

On Friday, October 6, 2017 at 3:59:20 AM UTC+2, Archimedes Plutonium wrote:

> I already proved several times over that a cone is always a Oval section ...

No, you haven't.

In fact you will find a simple proof below 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:

^ 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