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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 ]
 Me Posts: 1,716 Registered: 1/23/16
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?

Posted: Oct 2, 2017 2:37 AM

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?

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