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Topic: 6)are all German mathematicians like Peter Roquette,
Gunther Schmidt Karl-Otto Stöhr- as dumb as Franz, teachi
ng a Conic section is ellipse, when in truth it is an oval?

Replies: 11   Last Post: Oct 4, 2017 7:05 PM

 Messages: [ Previous | Next ]
 alouatta.coibensis@gmail.com Posts: 87 Registered: 9/12/15
Re: Princeton math dept agreeing with Franz for none has spotted the
error- as dumb as Franz, teaching a Conic section is ellipse, when in truth
it is an oval?

Posted: Oct 2, 2017 12:11 PM

On Monday, October 2, 2017 at 8:00:29 AM UTC-7, Archimedes Plutonium wrote:
> On Monday, October 2, 2017 at 1:37:38 AM UTC-5, Me wrote:
> > Let's consider the Sectioning of a Cylinder and a Cone.
> >
> > ^ x
> > E|
> > -+-
> > .' | `.
> > / | \
> > . | .
> > G | +c | H
> > . | .
> > \ | /
> > `. | ´
> > y <----------+ ´
> > F
> >

> > > The above is a view of a ellipse with center c and is produced by the
> > > Sectioning of a Cylinder as long as the cut is not perpendicular to the base,
> > > and as long as the cut involves two points not larger than the height of the
> > > cylinder walls. What we want to prove is that the cut is always a ellipse,
> > > which is a [certain] plane figure of two axes of symmetry with a Major Axis
> > > and Minor Axis and center at c.
> > >
> > > So, what is the proof that [cut] figure EGFH is always an ellipse in the
> > > cylinder section [as well as in the cone section]?

> >
> > Here's is an easy proof for it:
> >
> > Cylinder (side view):
> >
> > | | |
> > |-------+-------+ <= x = h
> > | | ´|
> > | | ´ |
> > | |´ |
> > | ´ | |
> > | ´ | |
> > x = 0 => ´-------|-------|
> > | r | |
> >
> > d(x) = r - (2r/h)x
> >
> > y^2 = r^2 - d(x)^2 = r^2 - r^2(2x/h - 1)^2 = r^2(1 - 4(x - h)^2/h^2
> >
> > => (1/r^2)y^2 + (4/h^2)(x - h)^2 = 1 ...equation of an ellipse
> >
> > Considerations:
> >
> > => y(h/2 + x')^2 = sqrt(r^2 - r^2(2(h/2 + x')/h - 1)^2) = r^2 - r^2(2x'/h)^2
> >
> > => y(h/2 + x') = r * (sqrt(1 - (2x'/h)^2) ...symmetric relative to h/2
> >
> > => y(h/2) = r (= Gc = cH)
> >
> > Cone (side view):
> >
> > .
> > /|\
> > / | \
> > /b | \
> > /---+---´ <= x = h
> > / |´ \
> > / ´ | \
> > / ´ | \
> > x = 0 => ´-------+-------\
> > / a | \
> >
> > 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/h^2
> >
> > => (1/ab)y(x)^2 + (4/h^2)(x - h)^2 = 1 ...equation of an ellipse
> >
> > Considerations:
> >
> > => y(h/2 + x')^2 = sqrt(ab - ab(2(h/2 + x')/h - 1)^2) = ab - ab(2x'/h)^2
> >
> > => y(h/2 + x') = sqrt(ab) * (sqrt(1 - (2x'/h)^2) ...symmetric relative to h/2
> >
> > => y(h/2) = sqrt(ab) (= Gc = cH)
> >
> > ======================================================
> >
> > It turns out that a cylinder can be considered as a special case of a cone here. Actually, the latter proof works for both cases, cone and cylinder.
> >
> > Cone/Cylinder (side view):
> >
> > / | \
> > /b | \
> > /---+---´ <= x = h
> > / |´ \
> > / ´ | \
> > / ´ | \
> > x = 0 => ´-------+-------\
> > / a | \
> >
> > (cone: b < a; cylinder: a = b = r)
> >
> > 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/h^2
> >
> > => (1/ab)y(x)^2 + (4/h^2)(x - h)^2 = 1 ...equation of an ellipse
> >
> > Considerations:
> >
> > => y(h/2 + x')^2 = sqrt(ab - ab(2(h/2 + x')/h - 1)^2) = ab - ab(2x'/h)^2
> >
> > => y(h/2 + x') = sqrt(ab) * (sqrt(1 - (2x'/h)^2) ...symmetric relative to h/2
> >
> > => y(h/2) = sqrt(ab) (= Gc = cH)
> >
> > ======================================================
> >
> > @Archie: Yes, this proves that (certain) cone sections "as depicted in my diagram" as well as (certain) cylinder sections (as described by you) are ellipses. qed
> >
> > 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 cone/cylinder section) above?

>
>
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The lists are getting longer. Is this part of your "proof"? Or are you finally getting completely unglued?

Date Subject Author
10/2/17 plutonium.archimedes@gmail.com
10/2/17 Me
10/2/17 plutonium.archimedes@gmail.com
10/2/17 alouatta.coibensis@gmail.com
10/2/17 Me
10/2/17 Dan Christensen
10/2/17 Peter Percival
10/2/17 Me
10/3/17 plutonium.archimedes@gmail.com
10/3/17 needspy@gmail.com
10/4/17 Jan Bielawski
10/4/17 Me