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AP's thoughts as of 5JAN2019// three proofs that ellipse is never a conic

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  AP's thoughts as of 5JAN2019// three proofs that ellipse is never a conic Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-01-05 07:19 -0800
    Legitimate geometry question on cone-plane intersection moroney@world.std.spaamtrap.com (Michael Moroney) - 2019-01-05 17:48 +0000
      Re: Legitimate geometry question on cone-plane intersection Alan Mackenzie <acm@muc.de> - 2019-01-05 18:58 +0000
        Re: Legitimate geometry question on cone-plane intersection moroney@world.std.spaamtrap.com (Michael Moroney) - 2019-01-05 19:44 +0000
          Re: Legitimate geometry question on cone-plane intersection Peter Percival <peterxpercival@hotmail.com> - 2019-01-05 20:15 +0000
          Re: Legitimate geometry question on cone-plane intersection Alan Mackenzie <acm@muc.de> - 2019-01-07 03:15 +0000
            Re: Legitimate geometry question on cone-plane intersection moroney@world.std.spaamtrap.com (Michael Moroney) - 2019-01-07 04:44 +0000
        Re: Legitimate geometry question on cone-plane intersection Peter Percival <peterxpercival@hotmail.com> - 2019-01-05 20:13 +0000
          My third proof ellipse is never a conic is that the tip end of cone and bottom flared-out end can never yield a 1000to1 ellipse Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-01-05 13:44 -0800
            try wrapping a ellipse in a cone wrapper Re: My third proof ellipse is never a conic is that the tip end of cone and bottom flared-out end can never yield a 1000to1 ellipse Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-01-05 14:24 -0800
              Re: try wrapping a ellipse in a cone wrapper Re: My third proof ellipse is never a conic is that the tip end of cone and bottom flared-out end can never yield a 1000to1 ellipse Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-01-05 14:40 -0800
                researching if the ellipse has an internal universal constant of 1/137 Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-01-05 17:37 -0800
                  researching if the ellipse has an internal universal constant of 1/137 Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-01-05 21:13 -0800
                    Re: Archimedes "Village Idiot" Plutonium flunked the Math Test of a lifetime-generation test moroney@world.std.spaamtrap.com (Michael Moroney) - 2019-01-06 05:55 +0000
                  not even a constant percentage in the rectangles Re: researching if the ellipse has an internal universal constant of 1/137 Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-01-05 21:59 -0800
                    no percentage formula Re: researching if the ellipse has an internal universal constant of 1/137 Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-01-05 22:14 -0800
                      not even rectangles Re: no percentage formula Re: researching if the ellipse has an internal universal constant of 1/137 Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-01-05 23:02 -0800
                        directrix Re: not even rectangles Re: no percentage formula Re: researching if the ellipse has an internal universal constant of 1/137 Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-01-05 23:34 -0800
                          understanding the Ellipse well,is the same as understanding angular momentum of physics well Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-01-05 23:51 -0800
                            Re: understanding the Ellipse well,is the same as understanding angular momentum of physics well Me <franz.fritschee.ff@gmail.com> - 2019-01-06 03:10 -0800
            Re: My third proof ellipse is never a conic is that the tip end of cone and bottom flared-out end can never yield a 1000to1 ellipse Me <franz.fritschee.ff@gmail.com> - 2019-01-06 03:08 -0800
            Re: My third proof ellipse is never a conic is that the tip end of cone and bottom flared-out end can never yield a 1000to1 ellipse Alan Mackenzie <acm@muc.de> - 2019-01-06 11:17 +0000
              Re: My third proof ellipse is never a conic is that the tip end of cone and bottom flared-out end can never yield a 1000to1 ellipse moroney@world.std.spaamtrap.com (Michael Moroney) - 2019-01-06 18:42 +0000
    Re: AP's thoughts as of 5JAN2019// three proofs that ellipse is never a conic Jan <filmart@gmail.com> - 2019-01-06 21:47 -0800

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#430877 — AP's thoughts as of 5JAN2019// three proofs that ellipse is never a conic

Alright the first proof is my historical first proof, in late 2016, I was messing around with the sine function and the cone, ellipse came to mind in that playful messing around. And it struck me, my mind just could not see how a slant cut into the cone yields an ellipse-- but some odd shaped figure with only one axis of symmetry, not two axes that the ellipse requires. So my first proof was a hands on. I went for a stiff waxy paper (magazine cover) and rolled it into a cone shape, then inserted a Mason or Kerr jar lid and then tilted it to see if I could get a ellipse. No ellipse is ever possible for the lid has just one axis of symmetry.

The second proof is a standard High School type of geometry proof and is fully laid out here:


Proofs ellipse is never a conic, always a cylinder section by 
Archimedes Plutonium 
-------------------- 
AP's proof the ellipse is never a Conic Section, always a Cylinder section, and how the proof works 

Let us analyze AP's Proof 

On Friday, September 14, 2018 at 6:57:36 PM UTC-5, Archimedes Plutonium wrote: 

  
  Array:: Analytic Geometry proof that Cylinder section= Ellipse//Conic 
  section = Oval, never ellipse 
  
  Now I did 3 Experiments and 3 models of the problem, but it turns out 
  that one model is superior over all the other models. One model is the 
  best of all. 
  
  That model is where you construct a cone and a cylinder and then 
  implant a circle inside the cone and cylinder attached to a handle so 
  that you can rotate the circle inside. Mine uses a long nail that I 
  poked holes into the side of a cylinder and another one inside a cone 
  made from heavy wax paper of magazine covers. And I used a Mason or 
  Kerr used lid and I attached them to the nail by drilling two holes 
  into each lid and running a wire as fastener. All of this done so I 
  can rotate or pivot the circle inside the cylinder and cone. You need 
  a long nail, for if you make the models too small or too skinny, you 
  lose clarity. 
  
  ARRAY, Analytic Geometry Proof, Cylinder Section is a Ellipse:: 
  
  
                E 
               __ 
        .-'              `-. 
      .'                    `. 
    /                         \ 
   ;                           ; 
  | G          c              | H 
   ;                           ; 
    \                         / 
     `.                     .' 
        `-.    _____  .-' 
                  F 
  


Alright, focus on the distance from c to F in the cone-cut compared to the distance from c to E 

In a Cylinder cut, those two distances are the same because a cylinder has two axes of symmetry. 

The side view of a cylinder is this 

|    | 
|    | 
|    | 

That allows cE to be the same distance as cF 


But the side view of the cone is 

     /\E 
    /c \ 
F /     \ 


The distance c to E is shorter because the slant of the side walls of the cone are in the direction of shortening cE, whereas the slant opposite c in cF make that distance larger than cE 

  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 plane figure of two axes of 
  symmetry with a Major Axis and Minor Axis and center at c. 
  
  Side view of Cylinder EGFH above with entry point cut at E and exit 
  point cut at F and where c denotes the central axis of the cylinder 
  and where x denotes a circle at c parallel with the base-circle of 
  cylinder 
  
  |                              | 
  |                              | E 
  |                              | 
  |                              | 
  |x            c              |x 
  |                              | 
  |                              | 
  |                              | 
  |F                            | 
  |                              | 
  |                              | 
  |                              | 
  
  

So we can see that the distance cE = cF in cylinder for the walls are Parallel to one another, giving distance symmetry 

But in the Cone, the walls are not parallel, shortening the distance cE compared to cF. Leaving only one axis of symmetry that of EF. The oval is the conic section of a cut at a slant, while the cylinder cut at a slant is a ellipse. The Oval has just one axis of symmetry. 

  So, what is the proof that figure EGFH is always an ellipse in the 
  cylinder section? The line segment GH is the diameter of the circle 
  base of cylinder and the cylinder axis cuts this diameter in half such 
  that Gc = cH. Now we only need to show that Fc = cE. This is done from 
  the right triangles cxF and cxE, for we note that by Angle-Side-Angle 
  these two right triangles are congruent and hence Fc = cE, our second 
  axis of symmetry and thus figure EGFH is always an ellipse. QED 
  
  
  
  Array proof:: Analytic Geometry proof that Conic section= Oval// never ellipse 
  
  ARRAY, Analytic Geometry Proof, Conic Section is a Oval, never an ellipse:: 
  
  
           A 
        ,'"   "`. 
     /            \ 
  C |     c       | D 
   \               / 
      ` . ___ .' 
           B 
  
  The above is a view of a figure formed from the cut of a conic with 
  center c as the axis of the cone and is produced by the Sectioning of 
  a Cone as long as the cut is not perpendicular to the base, and as 
  long as the cut is not a hyperbola, parabola or circle (nor line). 
  What we want to prove is that this cut is always a oval, never an 
  ellipse. An oval is defined as a plane figure of just one axis of 
  symmetry and possessing a center, c, with a Major Diameter as the axis 
  of symmetry and a Minor Diameter. In our diagram above, the major 
  diameter is AB and minor diameter is CD. 
  
  Alright, almost the same as with Cylinder section where we proved the 
  center was half way between Major Axis and Minor Axis of cylinder, 
  only in the case of the Conic, we find that the center is half way 
  between CD the Minor Diameter, but the center is not halfway in 
  between the Major Diameter, and all of that because of the reason the 
  slanted walls of the cone cause the distance cA to be far smaller than 
  the distance cB. In the diagram below we have the circle of x centered 
  at c and parallel to base. The angle at cx is not 90 degrees as in 
  cylinder. The angle of cAx is not the same as the angle cBx, as in the 
  case of the cylinder, because the walls of the cone-for line segments- 
  are slanted versus parallel in the cylinder. Triangles cAx and cBx are 
  not congruent, and thus, the distance of cA is not equal to cB, 
  leaving only one axis of symmetry AB, not CD. 
  
       /  \A 
   x/  c  \x 
  B/         \ 
  
  Hence, every cut in the Cone, not a hyperbola, not a parabola, not a 
  circle (not a line) is a Oval, never an ellipse. 
  
  QED 
  
  --Archimedes Plutonium 

The third proof was discovered just yesterday, in playing around with ellipses of long and skinny shapes.

So imagine a Sequence of Ellipses in terms of semimajor axis to semiminor axis:

10by1 ellipse, then 100by1 ellipse, then 1000by1 ellipse, etc etc

All of them legitimate ellipses only in a Sequence form.

So now, if an ellipse was ever a conic, there must be a cone which holds each and every one of these singular ellipses.

Now there sure enough is a Sequence of Cylinders that holds each and every one of those ellipses in terms of Radius of cylinders the Sequence is this:

10by1radius cylinder, then 100by1radius cylinder, then 1000by1radius cylinder, etc etc.

But, there never exists a single cone that can hold a 10by1 ellipse, nor a 100by1 ellipse, etc etc. Why is that?

Because no cone is uniform as we go from top to bottom of cone, and a ellipse needs uniformity.

Now this proof goes and impacts on another idea in mathematics-- the limit of calculus. Let me remind the reader the history of the Limit concept goes to Cauchy in around early mid 1800s. His students-- the bright students kept bothering Cauchy about the calculus for the calculus taught in the 1800s and up to modern day times is flawed. Severely flawed, and is the crippled calculus being taught today around the world in colleges. It is pathetically fake calculus for it has the students believe that you can have a rectangle with area yet have 0 width. And the reason you have such fakery tauted as true is because Old Math never defined infinity as a borderline which would give Calculus a small finite infinitesimal such as 1/10^604, so that the calculus integral can always be at least a summation of strips of area that are 1/10^604 wide. Always a finite positive width for the calculus integral. But in Cauchy's time, these bright students kept bothering him about 0 width, for that is what Cauchy was teaching and that is what college professors to this very day are teaching in colleges throughout the world-- teaching a huge lie. All because they are too lazy and too dumb to define infinity with a borderline.

So what Cauchy did to silence and muffle the bright smart young students of the world, is that Cauchy invented a liar's paradise that would sweep all smart students complaints under the rug of shame. Cauchy invented the Limit to silence smart students. So that if a bright smart student said to Cauchy, your integral calculus is the sum of rectangles of 0 width. Then Cauchy would react and reply-- well, you just do not understand the Limit. But the truth is, Cauchy and modern day math professors are liars and cheats of the truth of reality and the world. For the bright smart student understands that when you never define infinity properly-- you have contradictions of a 0 width rectangle.

And now, in geometry, we can display the hypocrisy and the vile vulgar contradiction of a Limit in math. For if you consider a cone and very elongated ellipses 10to1 , 100to1, 1000to1, in terms of semimajor and semiminor axes, a going to infinity sequence of long skinny ellipses, there are no Cones to handle these ellipses as conics. No, there are none. But Cauchy and today's dunce math professors would argue that in the Limit of cones-- you somehow pull out a 1000to1 ellipse or a 10000to1 ellipse. Ellipses are a geometrical Limit argument with Cones. Because no ellipse can ever be got from a cone, none. All ellipses come from a Cylinder, every single ellipse that exists has to come from a cylinder, never a cone.

So, I have three proofs that a ellipse is a cylinder cut, never a conic cut.

AP

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#430880 — Legitimate geometry question on cone-plane intersection

I have a geometry question for anyone who knows geometry regarding the
cone-plane intersection ellipse. Given:

  
       /  \A 
   Y/  C  \X 
  B/         \ 
  

Since the angle AXC must be less than 90 degrees and the angle BYC must be
greater than 90 degrees and the length XC = YC, segment AC must be less
than BC. Therefore the center of the ellipse cannot correspond to the axis
of the cone, but AB is the major axis of the ellipse with point C lying on
the major axis.

Does point C correspond to one of the foci of the ellipse? If so, does
anyone know the name of a mathematical proof of this? If not, does point C
have a name, and what is the formula for finding the distance from the
center of the ellipse to the point C?

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#430883 — Re: Legitimate geometry question on cone-plane intersection

Hello, Michael.

Michael Moroney <moroney@world.std.spaamtrap.com> wrote:
> I have a geometry question for anyone who knows geometry regarding the
> cone-plane intersection ellipse. Given:

> ? 
> ? ? ? ?/ ?\A 
> ? ?Y/ ?C ?\X 
> ? B/ ? ? ? ? \ 
> ? 

> Since the angle AXC must be less than 90 degrees and the angle BYC must be
> greater than 90 degrees and the length XC = YC, segment AC must be less
> than BC. Therefore the center of the ellipse cannot correspond to the axis
> of the cone, but AB is the major axis of the ellipse with point C lying on
> the major axis.

> Does point C correspond to one of the foci of the ellipse? If so, does
> anyone know the name of a mathematical proof of this? If not, does point C
> have a name, and what is the formula for finding the distance from the
> center of the ellipse to the point C?

I'm sorry that my newsreader can't display your diagram clearly enough
for me to make sense of it, but I know the answer to your question -
this is from a (British) Open University maths television program from,
I suppose, the mid to late 1970s.

Construct a sphere tangent to the cone (the line of contact being a
circle, of course) such that it is also tangent to the ellipse's plane.
The point where it touches the ellipse is a focus.  The other focus is
where the other such sphere touches the plane.

Extend the plane through the circle of contact of the sphere and cone
till it intersects with the ellipse's plane.  The line of intersection
is the directrix.

I can't remember the exact details of the proof, but it was fairly
simple, depending only on getting the appropriate projections to do the
reasoning.

The above also works for a hyperbola, parabola, and circle (sort of - it
has no directrix).

-- 
Alan Mackenzie (Nuremberg, Germany).

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#430885 — Re: Legitimate geometry question on cone-plane intersection

Alan Mackenzie <acm@muc.de> writes:

>Hello, Michael.

>Michael Moroney <moroney@world.std.spaamtrap.com> wrote:
>> I have a geometry question for anyone who knows geometry regarding the
>> cone-plane intersection ellipse. Given:

>> ? 
>> ? ? ? ?/ ?\A 
>> ? ?Y/ ?C ?\X 
>> ? B/ ? ? ? ? \ 
>> ? 

>> Since the angle AXC must be less than 90 degrees and the angle BYC must be
>> greater than 90 degrees and the length XC = YC, segment AC must be less
>> than BC. Therefore the center of the ellipse cannot correspond to the axis
>> of the cone, but AB is the major axis of the ellipse with point C lying on
>> the major axis.

>> Does point C correspond to one of the foci of the ellipse? If so, does
>> anyone know the name of a mathematical proof of this? If not, does point C
>> have a name, and what is the formula for finding the distance from the
>> center of the ellipse to the point C?

>I'm sorry that my newsreader can't display your diagram clearly enough
>for me to make sense of it, but I know the answer to your question -
>this is from a (British) Open University maths television program from,
>I suppose, the mid to late 1970s.

How about this:

       ^
      / \A 
    Y/ C \X 
   B/     \ 
   /       \   


>Construct a sphere tangent to the cone (the line of contact being a
>circle, of course) such that it is also tangent to the ellipse's plane.
>The point where it touches the ellipse is a focus.  The other focus is
>where the other such sphere touches the plane.

OK, this is the Dandelin Spheres proof that certain conic sections are an
ellipse.

https://en.wikipedia.org/wiki/Dandelin_spheres

My question is whether a focus of the ellipse is on the axis of the cone.
It appears from the first diagram that neither focus is on the cone's
axis. My question becomes whether this point (unlabeled in the diagram
but it is to the right of point P) has a name or any significance, perhaps
in orbital mechanics.

>Extend the plane through the circle of contact of the sphere and cone
>till it intersects with the ellipse's plane.  The line of intersection
>is the directrix.

>I can't remember the exact details of the proof, but it was fairly
>simple, depending only on getting the appropriate projections to do the
>reasoning.

https://en.wikipedia.org/wiki/Dandelin_spheres explains it quite well,
other than glossing over why the distance P-F1 is equal to the distance
P-P1.

>The above also works for a hyperbola, parabola, and circle (sort of - it
>has no directrix).

Interesting that this proof works for other conic sections as well.

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#430888 — Re: Legitimate geometry question on cone-plane intersection

Michael Moroney wrote:
> Alan Mackenzie <acm@muc.de> writes:
> 
>> Hello, Michael.
> 
>> Michael Moroney <moroney@world.std.spaamtrap.com> wrote:
>>> I have a geometry question for anyone who knows geometry regarding the
>>> cone-plane intersection ellipse. Given:
> 
>>> ?
>>> ? ? ? ?/ ?\A
>>> ? ?Y/ ?C ?\X
>>> ? B/ ? ? ? ? \
>>> ?
> 
>>> Since the angle AXC must be less than 90 degrees and the angle BYC must be
>>> greater than 90 degrees and the length XC = YC, segment AC must be less
>>> than BC. Therefore the center of the ellipse cannot correspond to the axis
>>> of the cone, but AB is the major axis of the ellipse with point C lying on
>>> the major axis.
> 
>>> Does point C correspond to one of the foci of the ellipse? If so, does
>>> anyone know the name of a mathematical proof of this? If not, does point C
>>> have a name, and what is the formula for finding the distance from the
>>> center of the ellipse to the point C?
> 
>> I'm sorry that my newsreader can't display your diagram clearly enough
>> for me to make sense of it, but I know the answer to your question -
>> this is from a (British) Open University maths television program from,
>> I suppose, the mid to late 1970s.
> 
> How about this:
> 
>         ^
>        / \A
>      Y/ C \X
>     B/     \
>     /       \
> 
> 
>> Construct a sphere tangent to the cone (the line of contact being a
>> circle, of course) such that it is also tangent to the ellipse's plane.
>> The point where it touches the ellipse is a focus.  The other focus is
>> where the other such sphere touches the plane.
> 
> OK, this is the Dandelin Spheres proof that certain conic sections are an
> ellipse.
> 
> https://en.wikipedia.org/wiki/Dandelin_spheres
> 
> My question is whether a focus of the ellipse is on the axis of the cone.

No!  Except in the degenerate case that the ellipse is a circle.

> It appears from the first diagram that neither focus is on the cone's
> axis. My question becomes whether this point (unlabeled in the diagram
> but it is to the right of point P) has a name or any significance, perhaps
> in orbital mechanics.
> 
>> Extend the plane through the circle of contact of the sphere and cone
>> till it intersects with the ellipse's plane.  The line of intersection
>> is the directrix.
> 
>> I can't remember the exact details of the proof, but it was fairly
>> simple, depending only on getting the appropriate projections to do the
>> reasoning.
> 
> https://en.wikipedia.org/wiki/Dandelin_spheres explains it quite well,
> other than glossing over why the distance P-F1 is equal to the distance
> P-P1.
> 
>> The above also works for a hyperbola, parabola, and circle (sort of - it
>> has no directrix).
> 
> Interesting that this proof works for other conic sections as well.
> 


-- 
"He who will not reason is a bigot;
he who cannot is a fool;
he who dares not is a slave."
  - Sir William Drummond

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#430979 — Re: Legitimate geometry question on cone-plane intersection

Hello, Michael.

Michael Moroney <moroney@world.std.spaamtrap.com> wrote:
> Alan Mackenzie <acm@muc.de> writes:

[ .... ]

>>Construct a sphere tangent to the cone (the line of contact being a
>>circle, of course) such that it is also tangent to the ellipse's plane.
>>The point where it touches the ellipse is a focus.  The other focus is
>>where the other such sphere touches the plane.

> OK, this is the Dandelin Spheres proof that certain conic sections are an
> ellipse.

> https://en.wikipedia.org/wiki/Dandelin_spheres

Thanks.  I didn't know this elegant construction had a name and a well
know discoverer.

> My question is whether a focus of the ellipse is on the axis of the cone.

I think it's clear, from the above, that it's not.  Intuitively, if one
focus was on the cone's axis, why wouldn't the other one be, too?  As
one tilts the section plane from an ellipse, through a circle, back to
an ellipse, there would be an abrupt jump from "left" to "right" in
"which" focus was on the axis.  I agree, this would all need to be
backed up by some maths.  :-)

> It appears from the first diagram that neither focus is on the cone's
> axis. My question becomes whether this point (unlabeled in the diagram
> but it is to the right of point P) has a name or any significance, perhaps
> in orbital mechanics.

Intriguing question.  I don't know the answer.

[ .... ]

-- 
Alan Mackenzie (Nuremberg, Germany).

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#430987 — Re: Legitimate geometry question on cone-plane intersection

Alan Mackenzie <acm@muc.de> writes:

>Hello, Michael.

>Michael Moroney <moroney@world.std.spaamtrap.com> wrote:
>> Alan Mackenzie <acm@muc.de> writes:

>[ .... ]

>>>Construct a sphere tangent to the cone (the line of contact being a
>>>circle, of course) such that it is also tangent to the ellipse's plane.
>>>The point where it touches the ellipse is a focus.  The other focus is
>>>where the other such sphere touches the plane.

>> OK, this is the Dandelin Spheres proof that certain conic sections are an
>> ellipse.

>> https://en.wikipedia.org/wiki/Dandelin_spheres

>Thanks.  I didn't know this elegant construction had a name and a well
>know discoverer.

It is an elegant proof.

>> My question is whether a focus of the ellipse is on the axis of the cone.

>I think it's clear, from the above, that it's not.  Intuitively, if one
>focus was on the cone's axis, why wouldn't the other one be, too?

Yes, it is now obvious from the diagram that the cone's axis does not pass
through a focus, and cannot do so, except for the special case when the
ellipse is a circle and the cutting plane is parallel to the base.

Since the ellipse is off-center to the cone's axis, I thought that the
axis might pass through a focus instead.

>As
>one tilts the section plane from an ellipse, through a circle, back to
>an ellipse, there would be an abrupt jump from "left" to "right" in
>"which" focus was on the axis.  I agree, this would all need to be
>backed up by some maths.  :-)

A circle is centered and the center and foci are the same.

As the plane tilts the other way, the ellipse is off-center the other
side. There would be no 'jump' since the switch is at the circle when
the center and foci are the same.

>> It appears from the first diagram that neither focus is on the cone's
>> axis. My question becomes whether this point (unlabeled in the diagram
>> but it is to the right of point P) has a name or any significance, perhaps
>> in orbital mechanics.

>Intriguing question.  I don't know the answer.

So far I haven't found an answer.

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#430887 — Re: Legitimate geometry question on cone-plane intersection

Alan Mackenzie wrote:
> Hello, Michael.
> 
> Michael Moroney <moroney@world.std.spaamtrap.com> wrote:
>> I have a geometry question for anyone who knows geometry regarding the
>> cone-plane intersection ellipse. Given:
> 
>> ?
>> ? ? ? ?/ ?\A
>> ? ?Y/ ?C ?\X
>> ? B/ ? ? ? ? \
>> ?
> 
>> Since the angle AXC must be less than 90 degrees and the angle BYC must be
>> greater than 90 degrees and the length XC = YC, segment AC must be less
>> than BC. Therefore the center of the ellipse cannot correspond to the axis
>> of the cone, but AB is the major axis of the ellipse with point C lying on
>> the major axis.
> 
>> Does point C correspond to one of the foci of the ellipse? If so, does
>> anyone know the name of a mathematical proof of this? If not, does point C
>> have a name, and what is the formula for finding the distance from the
>> center of the ellipse to the point C?
> 
> I'm sorry that my newsreader can't display your diagram clearly enough
> for me to make sense of it, but I know the answer to your question -
> this is from a (British) Open University maths television program from,
> I suppose, the mid to late 1970s.
> 
> Construct a sphere tangent to the cone (the line of contact being a
> circle, of course) such that it is also tangent to the ellipse's plane.
> The point where it touches the ellipse is a focus.  The other focus is
> where the other such sphere touches the plane.
> 
> Extend the plane through the circle of contact of the sphere and cone
> till it intersects with the ellipse's plane.  The line of intersection
> is the directrix.
> 
> I can't remember the exact details of the proof

The OP may like to look up "spheres of Dandelin".  This 
https://en.wikipedia.org/wiki/Dandelin_spheres, for example, has a nice 
picture.

> , but it was fairly
> simple, depending only on getting the appropriate projections to do the
> reasoning.
> 
> The above also works for a hyperbola, parabola, and circle (sort of - it
> has no directrix).
> 


-- 
"He who will not reason is a bigot;
he who cannot is a fool;
he who dares not is a slave."
  - Sir William Drummond

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#430890 — My third proof ellipse is never a conic is that the tip end of cone and bottom flared-out end can never yield a 1000to1 ellipse

On Saturday, January 5, 2019 at 2:13:53 PM UTC-6, Peter Percival wrote:

> 
> The OP may like to look up "spheres of Dandelin".  This 
> https://en.wikipedia.org/wiki/Dandelin_spheres, for example, has a nice 
> picture.
> 

Hi, Peter, I am hoping OP does not mean me. 

For my third proof that the ellipse is never a conic, is simple.

Get three figures and elongate.

So you have a ellipse that elongates from 10to1, to 100to1 to 1000to1 etc etc
A cylinder that elongates from 10to1radius, to 100to1radius to 1000to1 radius etc etc

Now, finally you elongate a cone in 10to1radius to 100to1radius, etc etc.

Notice that as you elongate a ellipse and a cylinder, they are symmetrical from side to side and from top to bottom. And you can fit the ellipse inside the cylinder and make a cut to get the ellipse

But notice as you elongate a cone, you always have a sharp tip top and a flaring-out at the bottom. In other words, no ellipse comes out of a cone.

Simple proof.

AP

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#430896 — try wrapping a ellipse in a cone wrapper Re: My third proof ellipse is never a conic is that the tip end of cone and bottom flared-out end can never yield a 1000to1 ellipse

On Saturday, January 5, 2019 at 3:44:55 PM UTC-6, Archimedes Plutonium wrote:
> On Saturday, January 5, 2019 at 2:13:53 PM UTC-6, Peter Percival wrote:
> 
> > 
> > The OP may like to look up "spheres of Dandelin".  This 
> > https://en.wikipedia.org/wiki/Dandelin_spheres, for example, has a nice 
> > picture.
> > 

Alright, the Dandelin spheres is an excellent example of where you use a "false assumption" in a proof and end up -- you can prove anything is possible-- the false assumption Dandelin used was he assumed the conic cut was an ellipse when in fact it never is an ellipse, but rather an oval. So the Dandelin proof is a pure slab of worthless garbage. But in modern times we have many fake proofs-- just to mention a few-- the Appel & Haken garbage of 4 Color Mapping with their assumptions of reductio ad absurdum and countries devoid of borders. Then the Andrew Wiles nonsense of Fermat's Last Theorem with his reductio ad absurdum assumption and no well defined-- what is infinity. Then the Green-Tao nonsense in number theory where they never understood-- first you need a definition of what it means to be infinite. Then the utter nonsense of Hales on Kepler Packing when the imp never defines infinity in the first place.

So when you assume something bad, while making a proof, you can end up proving the World is Flat and has green monsters.

> 
> Hi, Peter, I am hoping OP does not mean me. 
> 
> For my third proof that the ellipse is never a conic, is simple.
> 
> Get three figures and elongate.
> 
> So you have a ellipse that elongates from 10to1, to 100to1 to 1000to1 etc etc
> A cylinder that elongates from 10to1radius, to 100to1radius to 1000to1 radius etc etc
> 
> Now, finally you elongate a cone in 10to1radius to 100to1radius, etc etc.
> 
> Notice that as you elongate a ellipse and a cylinder, they are symmetrical from side to side and from top to bottom. And you can fit the ellipse inside the cylinder and make a cut to get the ellipse
> 
> But notice as you elongate a cone, you always have a sharp tip top and a flaring-out at the bottom. In other words, no ellipse comes out of a cone.
> 
> Simple proof.
> 

Very simple proof, for we imagine a candy that is ellipse in shape and elongated. Now we ask, what wrapper can we use to wrap the candy-- in place of a slant cut.

Well, you place the ellipse in a cylinder and it wraps real real nicely

But you try to place the ellipse-- a long pencil shaped ellipse inside a cone, well, it is stopped near the sharp tip end and has too much flaring out at the bottom. So a candy company would never use a cone shape wrapper because near the top is wasted space and near the bottom is junk flared out that has to be dealt with.

With the Cylinder wrapper, we just have the cylinder be longer than the ellipse and then tuck in the ends.

And we can see, in our mind's eye that the ellipse comes from the cylinder, because it is uniform all the way through, just as the ellipse is uniform because of its two axes of symmetry.

Now the ellipse inside a snugly fit cylinder, say the 100by1ellipse, 100 semimajor axis and 1 semiminor, fit inside a cylinder that is 200long and radius 1, diameter2. Now that ellipse can rotate from the center axis and where its endpoints can then rest on the cylinder's endpoints and thus be a slant cut.

But, if you tried doing that inside a cone, you have all that area to fill in, especially near the bottom, because the cone keeps flaring near the bottom, hence no symmetry, only 1 axis of symmetry and not a conic cut.

AP

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#430897 — Re: try wrapping a ellipse in a cone wrapper Re: My third proof ellipse is never a conic is that the tip end of cone and bottom flared-out end can never yield a 1000to1 ellipse

Sad the day has come that a candy wrapper company of candy that is elongated ellipse shaped has to tell that bonehead mathematician at Stanford, CalTech, Harvard, MIT, that a ellipse is never a conic but always a cylinder section. For you just cannot fit that ellipse into a cone, you need a cylinder for a fit.

AP

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#430907 — researching if the ellipse has an internal universal constant of 1/137

I feel almost like Tycho Brahe and Kepler-- first you need the hard core data. Then you can find out the theory behind the data.

analysis Re: problem with interior squares

Now that I have the squares for these two ellipses, try to see if 1/137 is a constant 

Semimajor axis = 5 
Semiminor axis = 2 
Foci distance from center = 4.58 using formula F^2 = semimajor axis^2 - semiminor axis^2 
Circumference = 23.013 
Area = 31.416 

Looking for the square
4*(1.85^2) + 25y^2 = 100 
4*(3.42) + 25y^2 = 100 
13.69 + 25y^2 = 100 
25y^2 = 86.31 
sqrt3.45 
y = 1.846 

Yes, but where did the .15 subtracted from 2 come from?
.0072u = .15 
20.83 

Now the maximum interior rectangle
4*(4.53^2) +25y^2 = 100 
4*20.52 + 25y^2 = 100 
82.08 + 25y^2 = 100 
25y^2 = 12.40 
y = .846 

That gives me a rectangle of 9.06by1.692 which is area of 15.32 

Yes, but where did the .05 that was subtracted from 4.58 come from?
.0072u = .05
u = 6.94

ELLIPSE 10by8 
Semimajor axis = 10 
Semiminor axis = 8 
Foci distance of 6 from center, reckoned by F^2 = semimajor^2 - semiminor^2 
Circumference = 56.723 
Area = 251.327 

interior square

64(6.25^2) + 100y^2 = 6400 
64(39.06) + 100y^2 = 6400 
2500 + 100y^2 = 6400 
100y^2 = 3900 
y = 6.24 

That is a square of 12.5by12.5 = 156.25
Yes, but where did a .25 that was added to 6 come from ? 

.0072u = .25
u = 34.72

Maximum interior rectangle
64*(7.1^2) + 100y^2 = 6400 
64*50.41 + 100y^2 = 6400 
100y^2 = 6400-3226 
y^2 = 31.74 
y = 5.63 

14.2*11.26 = 160.00 

But where did the 1.1 that was added to 6 come from?
.0072u = 1.1
u = 152.7

AP

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#430917 — researching if the ellipse has an internal universal constant of 1/137

No progress, none at all

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#430918 — Re: Archimedes "Village Idiot" Plutonium flunked the Math Test of a lifetime-generation test

Math Failure Archimedes Plutonium <plutonium.archimedes@gmail.com> fails:

>Subject: Re: researching if the ellipse has an internal universal constant of 1/137

>No progress, none at all

Of course not. You started off with a conclusion and are trying to find
any proof of the conclusion. A real mathematician or scientist would
start off with a hypothesis and look for evidence that the hypothesis
is either false or true.

That is why the governor of South Dakota has declared that Archimedes 
Plutonium is the official State Math and Physics Failure of South Dakota.

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#430919 — not even a constant percentage in the rectangles Re: researching if the ellipse has an internal universal constant of 1/137

No success yet with 1/137, but even more troubling is no success yet with percentages.


On Saturday, January 5, 2019 at 7:37:49 PM UTC-6, Archimedes Plutonium wrote:
> I feel almost like Tycho Brahe and Kepler-- first you need the hard core data. Then you can find out the theory behind the data.
> 
> analysis Re: problem with interior squares
> 
> Now that I have the squares for these two ellipses, try to see if 1/137 is a constant 
> 
> Semimajor axis = 5 
> Semiminor axis = 2 
> Foci distance from center = 4.58 using formula F^2 = semimajor axis^2 - semiminor axis^2 
> Circumference = 23.013 
> Area = 31.416 
> 
> Looking for the square
> 4*(1.85^2) + 25y^2 = 100 
> 4*(3.42) + 25y^2 = 100 
> 13.69 + 25y^2 = 100 
> 25y^2 = 86.31 
> sqrt3.45 
> y = 1.846 
> 
> Yes, but where did the .15 subtracted from 2 come from?
> .0072u = .15 
> 20.83 
> 
> Now the maximum interior rectangle
> 4*(4.53^2) +25y^2 = 100 
> 4*20.52 + 25y^2 = 100 
> 82.08 + 25y^2 = 100 
> 25y^2 = 12.40 
> y = .846 
> 
> That gives me a rectangle of 9.06by1.692 which is area of 15.32 
> 

Now one would suspect that the exterior Minimum Rectangle versus interior maximum rectangle would have a universal constant percentage in all ellipses.

In the above we have 15.32/40 = 38% and looking what we have for the next ellipse below.


> Yes, but where did the .05 that was subtracted from 4.58 come from?
> .0072u = .05
> u = 6.94
> 
> ELLIPSE 10by8 
> Semimajor axis = 10 
> Semiminor axis = 8 
> Foci distance of 6 from center, reckoned by F^2 = semimajor^2 - semiminor^2 
> Circumference = 56.723 
> Area = 251.327 
> 
> interior square
> 
> 64(6.25^2) + 100y^2 = 6400 
> 64(39.06) + 100y^2 = 6400 
> 2500 + 100y^2 = 6400 
> 100y^2 = 3900 
> y = 6.24 
> 
> That is a square of 12.5by12.5 = 156.25
> Yes, but where did a .25 that was added to 6 come from ? 
> 
> .0072u = .25
> u = 34.72
> 
> Maximum interior rectangle
> 64*(7.1^2) + 100y^2 = 6400 
> 64*50.41 + 100y^2 = 6400 
> 100y^2 = 6400-3226 
> y^2 = 31.74 
> y = 5.63 
> 
> 14.2*11.26 = 160.00 
> 

Exterior Minimum Rectangle is area 20*16 = 320 and so 160/320 = 50%

I would have thought the two rectangles would be a constant percentage in all ellipses.


> But where did the 1.1 that was added to 6 come from?
> .0072u = 1.1
> u = 152.7
> 
> AP

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#430920 — no percentage formula Re: researching if the ellipse has an internal universal constant of 1/137

On Saturday, January 5, 2019 at 11:59:36 PM UTC-6, Archimedes Plutonium wrote:
> No success yet with 1/137, but even more troubling is no success yet with percentages.
> 
> 
> On Saturday, January 5, 2019 at 7:37:49 PM UTC-6, Archimedes Plutonium wrote:
> > I feel almost like Tycho Brahe and Kepler-- first you need the hard core data. Then you can find out the theory behind the data.
> > 
> > analysis Re: problem with interior squares
> > 
> > Now that I have the squares for these two ellipses, try to see if 1/137 is a constant 
> > 
> > Semimajor axis = 5 
> > Semiminor axis = 2 
> > Foci distance from center = 4.58 using formula F^2 = semimajor axis^2 - semiminor axis^2 
> > Circumference = 23.013 
> > Area = 31.416 
> > 
> > Looking for the square
> > 4*(1.85^2) + 25y^2 = 100 
> > 4*(3.42) + 25y^2 = 100 
> > 13.69 + 25y^2 = 100 
> > 25y^2 = 86.31 
> > sqrt3.45 
> > y = 1.846 
> > 
> > Yes, but where did the .15 subtracted from 2 come from?
> > .0072u = .15 
> > 20.83 

Now the square area is almost the same as the interior rectangle area 3.7^2 = 13.68 versus 15.32



> > 
> > Now the maximum interior rectangle
> > 4*(4.53^2) +25y^2 = 100 
> > 4*20.52 + 25y^2 = 100 
> > 82.08 + 25y^2 = 100 
> > 25y^2 = 12.40 
> > y = .846 
> > 
> > That gives me a rectangle of 9.06by1.692 which is area of 15.32 
> > 
> 
> Now one would suspect that the exterior Minimum Rectangle versus interior maximum rectangle would have a universal constant percentage in all ellipses.
> 
> In the above we have 15.32/40 = 38% and looking what we have for the next ellipse below.
> 
> 
> > Yes, but where did the .05 that was subtracted from 4.58 come from?
> > .0072u = .05
> > u = 6.94
> > 
> > ELLIPSE 10by8 
> > Semimajor axis = 10 
> > Semiminor axis = 8 
> > Foci distance of 6 from center, reckoned by F^2 = semimajor^2 - semiminor^2 
> > Circumference = 56.723 
> > Area = 251.327 
> > 
> > interior square
> > 
> > 64(6.25^2) + 100y^2 = 6400 
> > 64(39.06) + 100y^2 = 6400 
> > 2500 + 100y^2 = 6400 
> > 100y^2 = 3900 
> > y = 6.24 
> > 
> > That is a square of 12.5by12.5 = 156.25
> > Yes, but where did a .25 that was added to 6 come from ? 
> > 
> > .0072u = .25
> > u = 34.72
> > 
> > Maximum interior rectangle
> > 64*(7.1^2) + 100y^2 = 6400 
> > 64*50.41 + 100y^2 = 6400 
> > 100y^2 = 6400-3226 
> > y^2 = 31.74 
> > y = 5.63 
> > 
> > 14.2*11.26 = 160.00 
> > 
> 
> Exterior Minimum Rectangle is area 20*16 = 320 and so 160/320 = 50%
> 
> I would have thought the two rectangles would be a constant percentage in all ellipses.
> 

And the square area here is almost the same as the interior rectangle area of 156.25 versus 160

Is there a defect of 1/137 as to why they cannot be equal?

I am surprised and disappointed that no percentage formula for all ellipses

AP

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#430925 — not even rectangles Re: no percentage formula Re: researching if the ellipse has an internal universal constant of 1/137

Alright, not even a percentage formula success for ellipse perimeters. One would think the external rectangle is intimately connected to the size of the internal rectangle.


> > > Semimajor axis = 5 
> > > Semiminor axis = 2 
> > > Foci distance from center = 4.58 using formula F^2 = semimajor axis^2 - semiminor axis^2 
> > > Circumference = 23.013 
> > > Area = 31.416 
> > > 
> > > Looking for the square
> > > 4*(1.85^2) + 25y^2 = 100 
> > > 4*(3.42) + 25y^2 = 100 
> > > 13.69 + 25y^2 = 100 
> > > 25y^2 = 86.31 
> > > sqrt3.45 
> > > y = 1.846 
> > > 
> > > Yes, but where did the .15 subtracted from 2 come from?
> > > .0072u = .15 
> > > 20.83 
> 
> Now the square area is almost the same as the interior rectangle area 3.7^2 = 13.68 versus 15.32
> 
> 
> 
> > > 
> > > Now the maximum interior rectangle
> > > 4*(4.53^2) +25y^2 = 100 
> > > 4*20.52 + 25y^2 = 100 
> > > 82.08 + 25y^2 = 100 
> > > 25y^2 = 12.40 
> > > y = .846 
> > > 
> > > That gives me a rectangle of 9.06by1.692 which is area of 15.32 
> > > 

So the internal rectangle perimeter is 2(9.06 +1.692) = 21.5 while the external rectangle perimeter is 28 for a percentage of 21.5/28 = 77%



> > 
> > Now one would suspect that the exterior Minimum Rectangle versus interior maximum rectangle would have a universal constant percentage in all ellipses.
> > 
> > In the above we have 15.32/40 = 38% and looking what we have for the next ellipse below.
> > 
> > 
> > > Yes, but where did the .05 that was subtracted from 4.58 come from?
> > > .0072u = .05
> > > u = 6.94
> > > 
> > > ELLIPSE 10by8 
> > > Semimajor axis = 10 
> > > Semiminor axis = 8 
> > > Foci distance of 6 from center, reckoned by F^2 = semimajor^2 - semiminor^2 
> > > Circumference = 56.723 
> > > Area = 251.327 
> > > 
> > > interior square
> > > 
> > > 64(6.25^2) + 100y^2 = 6400 
> > > 64(39.06) + 100y^2 = 6400 
> > > 2500 + 100y^2 = 6400 
> > > 100y^2 = 3900 
> > > y = 6.24 
> > > 
> > > That is a square of 12.5by12.5 = 156.25
> > > Yes, but where did a .25 that was added to 6 come from ? 
> > > 
> > > .0072u = .25
> > > u = 34.72
> > > 
> > > Maximum interior rectangle
> > > 64*(7.1^2) + 100y^2 = 6400 
> > > 64*50.41 + 100y^2 = 6400 
> > > 100y^2 = 6400-3226 
> > > y^2 = 31.74 
> > > y = 5.63 
> > > 
> > > 14.2*11.26 = 160.00 
> > > 

Here we have a perimeter of internal rectangle as 2(14.2+11.26) = 50.92 with external rectangle perimeter of 72 thus 50.92/72 = 70%


> > 
> > Exterior Minimum Rectangle is area 20*16 = 320 and so 160/320 = 50%
> > 
> > I would have thought the two rectangles would be a constant percentage in all ellipses.
> > 
> 
> And the square area here is almost the same as the interior rectangle area of 156.25 versus 160
> 
> Is there a defect of 1/137 as to why they cannot be equal?
> 
> I am surprised and disappointed that no percentage formula for all ellipses
> 

Very puzzling. 

AP

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#430926 — directrix Re: not even rectangles Re: no percentage formula Re: researching if the ellipse has an internal universal constant of 1/137

I have always shunned the directrix. Perhaps it will lift me out of this ellipse stalmate.

The directrix is a distance of semimajor axis^2 / distance from center to focus and parallel to minor axis.

Can one think of the directrix as some parameter of physics? Is it the sweeping out of equal area in equal time intervals?

I have always neglected the directrix, perhaps I need to look more closely.

AP

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#430927 — understanding the Ellipse well,is the same as understanding angular momentum of physics well

On Sunday, January 6, 2019 at 1:35:02 AM UTC-6, Archimedes Plutonium wrote:
> I have always shunned the directrix. Perhaps it will lift me out of this ellipse stalmate.
> 
> The directrix is a distance of semimajor axis^2 / distance from center to focus and parallel to minor axis.
> 
> Can one think of the directrix as some parameter of physics? Is it the sweeping out of equal area in equal time intervals?

What I meant is in physics history you can go back to a time where the planets move in orbit by Angels pushing the planet and so the directrix made me think of angels pushing a planet in its track while the other part of the directrix is a pull inward to the focal point. A push along the track by directrix and a pull inward by the focal point.

> 
> I have always neglected the directrix, perhaps I need to look more closely.
> 

Understanding the ellipse well, is the same as understanding Angular Momentum of Physics well.

AP

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#430934 — Re: understanding the Ellipse well,is the same as understanding angular momentum of physics well

On Sunday, January 6, 2019 at 8:51:17 AM UTC+1, Archimedes Plutonium wrote:

> Understanding the ellipse well, is the same as understanding Angular Momentum
> of Physics well.

If you say so.

But then, not understanding ellipses at all means...

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