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Topic: 6) Is Wolfgang Rautenberg, Gerhard Ringel, Peter Roquette also trying
to teach-spread fake-math like Franz? For Conic is Oval, never ellipse
(proofs provided)

Replies: 3   Last Post: Oct 6, 2017 3:10 PM

 Messages: [ Previous | Next ]
 Me Posts: 1,716 Registered: 1/23/16
Re: 6) Is Wolfgang Rautenberg, Gerhard Ringel, Peter Roquette also
trying to teach-spread fake-math like Franz? For Conic is Oval, never ellipse
(proofs provided)

Posted: Oct 5, 2017 5:10 PM

On Thursday, October 5, 2017 at 9:38:21 AM UTC+2, Archimedes Plutonium want to see proofs...

...that certain cone sections as well as certain cylinder sections are ellipses.

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

Some preliminaries:

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

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

Cone/Cylinder (side view):

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

2 cases: 1.) cone: b < a, 2.) cylinder: b = a = r.

Proof:

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)^2/h^2

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

qed

Date Subject Author
10/5/17 plutonium.archimedes@gmail.com
10/5/17 Me
10/6/17 plutonium.archimedes@gmail.com
10/6/17 Me