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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

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plutonium.archimedes@gmail.com

Posts: 18,231
Registered: 3/31/08
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 11:00 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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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Date Subject Author
10/2/17
Read 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?
plutonium.archimedes@gmail.com
10/2/17
Read Re: 6)are all German mathematicians like Peter Roque
tte, Gunther Schmidt Karl-Otto Stöhr- as dumb as Franz, te
aching a Conic section is ellipse, when in truth it is an ov
al?
Me
10/2/17
Read 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?
plutonium.archimedes@gmail.com
10/2/17
Read 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?
alouatta.coibensis@gmail.com
10/2/17
Read 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?
Me
10/2/17
Read Re: 6)are all German mathematicians like Peter Roque
tte, Gunther Schmidt Karl-Otto Stöhr- as dumb as Franz, te
aching a Conic section is ellipse, when in truth it is an ov
al?
Dan Christensen
10/2/17
Read Re: 6)are all German mathematicians like Peter Roquette
, Gunther Schmidt Karl-Otto Stöhr- as dumb as Franz, tea
ching a Conic section is ellipse, when in truth it is an oval
?
Peter Percival
10/2/17
Read Re: 6)are all German mathematicians like Peter Roque
tte, Gunther Schmidt Karl-Otto Stöhr- as dumb as Franz, tea
ching a Conic section is ellipse, when in truth it is an ova
l?
Me
10/3/17
Read Re: 6)are all German mathematicians like Peter Roque
tte, Gunther Schmidt Karl-Otto Stöhr- as dumb as Franz, te
aching a Conic section is ellipse, when in truth it is an ov
al?
plutonium.archimedes@gmail.com
10/3/17
Read Re: 6)are all German mathematicians like Peter Roque
tte, Gunther Schmidt Karl-Otto Stöhr- as dumb as Franz, te
aching a Conic section is ellipse, when in truth it is an ov
al?
needspy@gmail.com
10/4/17
Read Re: 6)are all German mathematicians like Peter Roque
tte, Gunther Schmidt Karl-Otto Stöhr- as dumb as Franz, te
aching a Conic section is ellipse, when in truth it is an ov
al?
Jan Bielawski
10/4/17
Read Re: 6)are all German mathematicians like Peter Roque
tte, Gunther Schmidt Karl-Otto Stöhr- as dumb as Franz, te
aching a Conic section is ellipse, when in truth it is an ov
al?
Me

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