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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
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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 5, 2017 5:18 PM


On Thursday, October 5, 2017 at 6:38:55 PM UTC+2, Archimedes Plutonium wrote:
> the Conic section is always a Oval
No, it isn't.
> never an Ellipse.
Nope, *some* cone sections *are* ellipses (see proof below).
> So, that is true and I proved it.
No, it's NOT true, and you didn't prove it.
You will find a simple proof below that shows that certain cone sections as well as certain cylinder sections are ellipses.
It turns out that a cylinder can be considered as a special case of a cone in this context. Actually, the presented proof works for both cases, cone and cylinder.
Some preliminaries:
Top view of the cone/cylinder section and depiction of the coordinate system used in the proof:
^ x  + <= x=h .'  `. /  \ .  .    .  . \  / `.  .´ y <+ <= x=0 Cone/Cylinder (side view): /  \ /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



