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Topic: (5)are all Univ.Western Ontario mathematicians like
Ajneet Dhillon, Matthias Franz, John Jardine as dumb as Dan
Christensen, teaching a Conic section is ellipse, when in t
ruth it is an oval?

Replies: 4   Last Post: Oct 3, 2017 12:10 PM

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Posts: 1,713
Registered: 1/23/16
Re: (5)are all Univ.Western Ontario mathematicians l
ike Ajneet Dhillon, Matthias Franz, John Jardine as dumb as
Dan Christensen, teaching a Conic section is ellipse, when
in truth it is an oval?

Posted: Oct 3, 2017 6:04 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

On Tuesday, October 3, 2017 at 11:16:41 AM UTC+2, Archimedes Plutonium wrote:

> Why teach students fakery, eh, eh??

Right. Better teach them facts.

> Analytic Geometry Proof, Cylinder Section [as well as a certain type of cone
> section] is a[n] Ellipse:


^ x
E|
-+- <= x=h
.' | `.
/ | \
. | .
G | +c | H
. | .
\ | /
`. | .´
y <----------+ <= x=0
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]?


It turns out that a cylinder can be considered as a special case of a cone here. Actually, there's a simple proof which works for both cases, cone and cylinder.

(@Archie: This shows that there is no essential difference between these two cases.)

Cone/Cylinder (side view):

/ | \ (with b <= a)
/b | \
/---+---´ <= x = h
/ |´ \
/ ´ | \
/ ´ | \
x = 0 => ´-------+-------\
/ a | \

(cone: b < a, cylinder: b = a = 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

Some 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 (hence Ec = cF)

=> 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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