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Topic: 2)Gottingen's Max Wardetzky, Ingo Witt, are you as
dumb and messy about Conics as Franz? Oval is the conic sect
ion, never ellipse

Replies: 3   Last Post: Oct 6, 2017 3:00 PM

 Messages: [ Previous | Next ]
 Me Posts: 1,716 Registered: 1/23/16
Re: 2)Gottingen's Max Wardetzky, Ingo Witt, are you
as dumb and messy about Conics as Franz? Oval is the conic
section, never ellipse

Posted: Oct 6, 2017 2:57 PM

On Friday, October 6, 2017 at 10:54:16 AM UTC+2, Archimedes Plutonium wrote:
> Franz wrote:
> >
> > ...that certain cone sections as well as certain cylinder sections are
> > ellipses.
> >

Right. You will find a simple proof for that claim below.

Some preliminaries:

Top view of the conic section and depiction of the coordinate system used in the proof:

^ x
|
-+- <= x=h
.' | `.
/ | \
. | .
| | |
. | .
\ | /
`. | .´
y <----------+ <= x=0

Cone (side view):
.
/|\
/ | \
/b | \
/---+---´ <= x = h
/ |´ \
/ ´ | \
/ ´ | \
x = 0 => ´-------+-------\
/ a | \

Proof:

r(x) = a - ((a-b)/h)x and d(x) = a - ((a+b)/h)x, hence

y(x)^2 = r(x)^2 - d(x)^2 = ab - ab(2x/h - 1)^2 = ab(1 - 4(x - h/2)^2/h^2.

Hence (1/ab)y(x)^2 + (4/h^2)(x - h/2)^2 = 1 ...equation of an ellipse

qed

======================================================

@Archie: Yes, this proves that (certain) cone sections "as depicted in my diagram" are ellipses.

Still not convinced? Can you point out an error in my simple calculation (of the shape of the coneic section) above?

Date Subject Author
10/6/17 plutonium.archimedes@gmail.com
10/6/17 bursejan@gmail.com
10/6/17 Me
10/6/17 Me