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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
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Me
Posts:
1,716
Registered:
1/23/16
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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 3:00 PM
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> Franz wrote:
>>
>> ...that certain cone 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?
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