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2-Imp-parade in sci.math of imps with fake-arguments that ellipse is a conic

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  2-Imp-parade in sci.math of imps with fake-arguments that ellipse is a conic Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-09-29 16:42 -0700
    Re: Archimedes "Total Failure" Plutonium flunked the Math Test of a lifetime-generation test moroney@world.std.spaamtrap.com (Michael Moroney) - 2018-09-30 01:59 +0000

#424541 — 2-Imp-parade in sci.math of imps with fake-arguments that ellipse is a conic

2Newsgroups: sci.math
Date: Sat, 29 Sep 2018 09:02:43 -0700 (PDT)

Subject: Imp-parade in sci.math of imps with fake-arguments that ellipse is a conic
From: Archimedes Plutonium <plutonium....@gmail.com>
Injection-Date: Sat, 29 Sep 2018 16:02:43 +0000

Imp-parade in sci.math of imps with fake-arguments that ellipse is a conic

First we start out with a dumb German named Franz hiding behind "Me" 

His argument amounts to drawing pictures and writing equations that have no linkage to one another, and ending his argument with "there, I told you so" 

Franz imp doing ellipse 


On Tuesday, September 19, 2017 at 7:27:03 PM UTC-5, Me (Franz) wrote: 
> On Tuesday, September 19, 2017 at 9:00:30 PM UTC+2, Archimedes Plutonium wrote: 
> 
> > So, you have an entry cut, say here 
> >        entry cut up high 
> >     /\/ 
> >   /   \ 
> > //   
> > exit 
> > 
> > So, you have entry cut up high where the curvature is small ... 
> > and you have exit cut below where curvature is large ... 
> 
> Your error is the following, you do not taking into account that the entry cut as well as the exit cut are not _horizontal_! 
> 
> If they WERE your argument would be correct, i.e. if we had to consider: 
> 
>      /\ 
>      ----- 
>    /   \ 
> ---- 
>  //   
> 
> But actually, the "cutting plain" is tilted. The slightly strange (and certainly counterintuitive) effect of this tilt, is that the resulting cone section actually is an ellipse. 
> 

On Saturday, October 7, 2017 at 5:59:31 PM UTC-5, Me wrote: 
> On Sunday, October 8, 2017 at 12:14:45 AM UTC+2, Archimedes Plutonium wrote: 
> > On Saturday, October 7, 2017 at 4:56:40 PM UTC-5, Me (Franz) wrote: 
> > > 
> > > You will find a simple proof below that shows that certain conic sections 
> > > as well as cylinder sections are ellipses. 
> 
> Some preliminaries: 
> 
> Top view of the conic and cylinder section and depiction of the coordinate system used in the proof: 
> 
>               ^ x 
>               | 
>              -+- <= x=h 
>          .'   |   `. 
>         .     |     .   
>         |     |     | 
>         '     |     '       
>          `.   |   .´   
>  y <----------+ <= x=0     
>               
> Cone/Cylinder (side view): 
>   
>                / | \ 
>               /b |  \ 
>              /---+---´ <= x = h 
>             /    |´   \ 
>            /   ´ |     \ 
>           / ´    |      \ 
> x = 0 => ´-------+-------\ 
>         /    a   |        \ 
> 
> 2 cases: cone: b < a, cylinder: b = a = r. 
> 
> 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 



2nd IMP 
What can be learned from this misfit QBWR, is that a mind can be so soaked in hatred, almost every sentence is a ad hominem, that the poor fellow is drowning in hatred, and assembles an argument, never to the question of whether ellipse is a conic, but assembles an argument to tease out every possible attack to place on AP. 


QBWR imp doing ellipse 

On Friday, September 28, 2018 at 1:13:56 PM UTC-5, qbwr...@gmail.com wrote: 
> On Thursday, September 27, 2018 at 5:36:05 PM UTC-7, Archimedes Plutonium wrote: 
> > On Thursday, September 27, 2018 at 3:54:29 PM UTC-5, qbwr...@gmail.com wrote: 
> > > On Thursday, September 27, 2018 at 1:15:17 PM UTC-7, Archimedes Plutonium wrote: 
> > > ... 
> > > > So how do we corral this mechanism to apply it and prove FLT? In the case of Generalized, at the end it was used to say, 7^3 + 7^4 = 14^3 with prime divisor 7 is true then in exp 3, after dividing by 7 then 1^3 + 1^3 = 2^3 but that is not the case, and so that mechanism destroys all chances of a proof in exp3 and higher. 
> > > 
> > > You have made a mistake that you need to correct or realize you can't fix. It appears that you do not know how to correctly divide numbers with exponents. 
> > 
> > For these years I have known you-- under various fake names-- all your posts and replies were "attack ads". Attack ads, and I just plain will not accept attack ads. 
> 
> No "various fake names", no "attack ads", just simply and sometimes even 
> politely clearly pointing out his worst mistakes. Almost always doing that 
> quietly via email so he wouldn't have to be publicly humiliated, but it 
> appears that he finally couldn't accept that and blocked email from me. 
> 
> > So, can you admit the Ellipse is never a conic, but that the oval was the conic section never the ellipse. And that the ellipse is always a cylinder section, never a conic. I have spent years posting that proof and fact. 
> > 
> > So, if you want to continue to reply in my threads, start out with saying -- You accept the fact the ellipse is never a conic. 
> > 
> > Because if you cannot come to that truth-fact, you do not belong in mathematics at all, for you are then just a stupid attack dog of mathematics, and nothing further that you write or post to sci.math is worth my time. 
> > 
> > If you cannot admit, then I will make efforts to bounce you out of sci.math as just another one of those Stalker Attack Dogs. 
> > 
> > I QBWR: admits that a ellipse is __________ never a conic section, always a _________ cylinder section. 
> > 
> > 
> > AP 
> 
> By coincidence I was most of the way through writing a very long very simple 
> very detailed description of the intersection of the cone and plane, 
> simple enough that a bright fifteen year old student would be able to 
> understand and verify it. Even Archie might be able to do this, if he 
> really, really tried. That isn't ready yet, but if that is what he wants to 
> challenge me with then let's see what the mathematics says. 
> 
> If the math says the ellipse is ever not a conic section and I haven't made 
> a mistake then I will have to admit Archie is correct. 
> 
> Note that none of what follows is ASSUMING that the intersection is an 
> ellipse. He has dismissed arguments because he claimed they assumed the 
> conclusion. This is simply going to do some calculations, find some points 
> and see what properties those points have. 
> 
> Here is the challenge that Archie wrote recently: 
> > If Jones thinks AP is wrong, why cannot Jones point to a mistake in the reasoning 
> 
> That seems reasonable. A different proof isn't going to convince Archie. 
> Let's look at Archie's exact argument and see if we can find any mistake in 
> reasoning. 
> 
> Archie wrote: 
> > Proof:: Take a cone in general, and build a circle that rotates on a 
> > axis. Rotate the circle just a tiny bit for it is bound to get stuck 
> > or impeded by the upward slanted walls of the cone. Rotate as far as 
> > you possibly can. Now filling in the area upwards is far smaller than 
> > filling in the area downwards. Hence, only 1 axis of symmetry, not 2 
> > axes of symmetry. Define Oval as having 1 axis of symmetry. Thus a 
> > oval, never an ellipse. QED 
> 
> So his exact argument seems to be: For any cone and plane, If area1 < area2 
> then the intersection is not an ellipse. 
> 
> Suppose we have a cone with vertex at the origin and symmetric about the z 
> axis 
> and a plane sloping in the x direction and not sloping in the y direction. 
> 
>    z^2 = x^2 + y^2 
>    z = 7/8 x + 7/8 
> 
> Step 1: Is area1 < area2? 
> 
> Having precise high resolution multi-color graphics seems essential to 
> convince a fifteen year old mind. Unfortunately Archie is terrified of 
> getting another nasty filthy infection and thus he won't look at any web 
> site hosting my graphics. 
> 
> Thus I can't show him those graphics do actually clearly show that his claim 
> that the area inside the intersection and outside the circle and below the 
> diameter of the circle is less than the area inside the intersection and 
> outside the circle and above the diameter of the circle is correct. at least 
> when the intersection is not a circle, parabola or hyperbola. 
> 
> But perhaps nothing in this is clear enough. I could use Archie's "trapezoid 
> calculus" and approximate those two areas sufficiently well to make a 
> stronger case, but the number of calculations needed would exceed the 
> attention span of the reader. 
> 
> So skipping that and accepting that the condition in his 'If' is true, that 
> is +1 point for Archie. 
> 
> Now how can we very, very, very simply test whether the intersection is an 
> ellipse? Perhaps by testing whether it has one or two planes of symmetry? 
> 
> The first step seems to determine the maximum and minimum values of x and z. 
> The graphic shows this will happen when y=0. Archie has claimed similar 
> things. So we accept that without further evidence. That simplifies Archie's 
> system to 
> 
>    z^2 = x^2 
>    z = 7/8 x + 7/8 
> 
> Eliminate z by substituting 7/8 x + 7/8 for z in the first equation gives 
>    (7/8 x + 7/8)^2 = x^2 
> so 
>    49/64 x^2 + 49/32 x + 49/64 == x^2 
> so 
>    x^2 - (49/64 x^2 + 49/32 x + 49/64) = 0 
> so 
>    15/64 x^2 - 49/32 x - 49/64 == 0 
> 
> That is a quadratic equation and the bright fifteen year old can find the 
> two solutions. 
> 
> From that and the original two equations the bright student can find the 
> minimum and maximum x and z that lie on the plane and on the cone when y=0. 
> Thus those two extreme points are 
> 
>    {-0.466667, 0, 0.466667} and {7, 0, 7} 
> 
> Now we prompt the student to find more points that lie on the plane and on 
> the cone by selecting values of x which lie evenly spaced between 
> -0.466667 and 7 and perform similar calculations to find the y and z values. 
> 
> Note +/- is used to indicate that either a positive or negative value for y 
> is in the intersection due to symmetry in y. 
> 
>    {-0.466667, 0,          0.466667} 
>    { 0.466667, +/-1.19548, 1.28333} 
>    { 1.4,      +/-1.56525, 2.1} 
>    { 2.33333,  +/-1.75,    2.91667} 
>    { 3.26667,  +/-1.80739, 3.73333} 
>    { 4.2,      +/-1.75,    4.55} 
>    { 5.13333,  +/-1.56525, 5.36667} 
>    { 6.06667,  +/-1.19548, 6.18333} 
>    { 7,        0,          7} 
> 
> The specific values chosen for x are not special, they only need to include 
> the minimum, the maximum and be evenly spaced between those. 
> 
> You should get out your calculator and carefully check that every one of 
> those points do both lie on that plane and on that cone. 
> 
> I am fairly confident that all these are correct, but you should no more 
> take my word for anything than anyone in the world would ever take your 
> word for anything. 
> 
> Because the intersection is offset from the origin that does not obviously 
> show whether there is or is not a second plane of symmetry. 
> 
> To check this we use Archie's "scooting" method. So we scoot the points by 
> subtracting (7-0.466667)/2 from all the x values and subtracting 
> (7+0.466667)/2 from all the z values. That will center all points about 
> {0,0,0} and clearly show any symmetry or lack of symmetry. 
> 
> The scooted result is 
> 
>    {-3.73333,    0,         -3.26667} 
>    {-2.8,       +/-1.19548, -2.45} 
>    {-1.86667,   +/-1.56525, -1.63333} 
>    {-0.933333,  +/-1.75,    -0.816667} 
>    { 0,         +/-1.80739,  0} 
>    { 0.933333,  +/-1.75,     0.816667} 
>    { 1.86667,   +/-1.56525,  1.63333} 
>    { 2.8,       +/-1.19548,  2.45} 
>    { 3.73333,    0,          3.26667} 
> 
> As you can see by inspecting that table of points, there is a plane of 
> symmetry above and below the middle of that table and there is a plane of 
> symmetry with + and - y. 
> 
> The skeptical student can repeat these calculations for any values of x and 
> for values equally distant from that midpoint, all to verify the 
> intersection has two planes of symmetry. 
> 
> There is nothing difficult in these calculations. Even Archie could have 
> done this half a decade ago to check his astonishing claim. But demolishing 
> his claims wouldn't serve Archie's goals. 
> 
> There is a book titled "Principles" by Ray Dalio which explains how to 
> organize your life so that you expose your mistakes and thus avoid making 
> the same mistakes again and again. Carefully applying the contents of that 
> book might help Archie and others. 
> 
> Note that all these calculations really should be done using exact fractions 
> and square roots. I have all those numbers, but I hoped that by simplifying 
> them to be decimals that Archie might take the time to his calculator to 
> verify every detail of what I have shown. 
> 
> Thus there are two planes of symmetry in the intersection and based on the 
> results of the mathematics I am forced to write, quoting Archie's demand: 
> 
> I QBWR: admits that a ellipse is ALWAYS a conic section, always a CYLINDER section 
> 
> I wrote a much longer post showing how the circular disk is dropped into the 
> cone, how that wedges against the cone, what the center of that disk is, what the points of intersection of the cone and disk and plane are, how 
> those intersection points are a distance of 1 from the center of the disk, 
> how points f1 and f2 can be calculated to determine that the sum of the 
> distance from calculated points f1 and f2 to any point on the intersection 
> are always the same no matter what point is chosen in the intersection. 
> 
> Archie wouldn't read that or bother to work through it step by step to 
> verify that there are no errors in it. 
> 
> But his conclusion that the inequality of the two areas thus implies the 
> intersection is not an ellipse is simply false. 
> 
> Archie did ask one other question recently: 
> > AP' s proof has been in sci.math within 5 years time, and (why) only now is Jones raising issues. 
> 
> I believe the answer to that question is almost certainly that people have 
> simply given up on trying to carefully point out the simplest mistakes to 
> Archie, that Archie makes mistakes that a bright 15 year old would never 
> make. If anyone does point out a specific mistake that Archie made then he 
> completely misses the point and responds with ranting that he is being hated 
> or he responds with insults and more posts pasting the same mistakes that he 
> has repeated for years or decades without ever having carefully fixed his 
> mistakes. 
> 
> He has never used mathematics to verify his oval claim, he doesn't even know 
> how to do that. He fondled his paper cone, made a conjecture and screeched 
> it for half a decade while being told hundreds of times that he was wrong 
> without ever having checked his work. No one was able to explain his error 
> in terms that a nine year old could understand. 
> 
> Archie did make a post a year or two ago. In that he said that he realized 
> that once his dog brain really got its teeth into something that he just 
> couldn't give it up, even when he knew it was wrong, even when he knew that 
> it made him look stupid to everyone, he realized that he just couldn't give 
> it up. That does show a surprising bit of self awareness. Within a day 
> Archie realized what he had admitted to the world and he deleted that post. 
> 
> I believe this explains much of Archie's behavior. He desperately needs to 
> see himself as a "super genius logic brain." So he must make astonishing 
> claims. And if he ever realizes that they are wrong he simply cannot give 
> them up or admit that he is wrong. 
> 
> Thus he has screeched "oval" and "semicircle trigonometry" and "cell numbers" 
> and "infinity is pi*10^604" long enough that he will never be able 
> to see his mistakes in these. 
> 
> Archie either has a memory problem or he only pastes copies of what he has 
> previously posted. This is demonstrated by the handful of cases where Archie 
> actually recognized that he had made a mistake and then a few weeks later is 
> posting the same mistakes claiming that it is correct again. One recent 
> example of this was where he was shown that that repeatedly rounding numbers 
> to a couple of digits when doing his polynomial interpolation resulted in 
> errors. He stated that he realized he did have to teach fractions in his 
> textbook. But this directly contradicted his prior and soon following posts 
> in which claimed that the only numbers which exist are his cell numbers. 
> 
> Bean Counter 


IMP kibo M.Moroney 


imp MM 
Finally we get an imp who thinks he is a comedian in Usenet but sadly is a psychotic clown dumped into the sewer gutter of Boston, and spends the day stalking Usenet. 
But to this imp's credit, he makes the most lucid criticism, and thus, beats out the dumb German Franz and the morose downbeat moron QBWR, for MM thinks a mistake is that the cone center on the section offsets the sectioning. But MM is never a mathematics mind, for he cannot even understand that 938 is not short of 945 by 12%, although even today he never recognizes that it is short by less than 1%. So, we cannot expect this loser comedian turned Usenet stalker to ever get much correct when it comes to mathematics. For it never entered his nonmath mind that you can simply nest the entire cone inside a cylinder and every section cut that the plane going through the cylinder is going to carve out a ellipse, and yet that very same plane is going to carve out not another ellipse in the conic but rather a oval in the conic, and all two figures-- cone and cylinder have the same center-axis( central axis is the same in cone nested inside cylinder). 

And so, what does the loser comedian kibo-MM do, instead of admitting his errors, he stalks even harder, and now the MM nonmath loser is posting the dumb German Franz's failed argument, without giving Franz his rightful due-- source-- a dumb German named Franz. 

And so, what does the loser comedian kibo-MM do, instead of admitting his errors, he stalks even harder, and now the MM nonmath loser is posting the dumb German Franz's failed argument, without giving Franz his rightful due-- source-- a dumb German named Franz. 


On Saturday, September 29, 2018 at 1:02:16 AM UTC-5, Michael Moroney wrote: 
> qbwrfmlx@gmail.com writes: 
> 
> >On Thursday, September 27, 2018 at 5:36:05 PM UTC-7, Archimedes Plutonium wrote: 
> 
> >> For these years I have known you-- under various fake names-- all your 
> >> posts and replies were "attack ads". Attack ads, and I just plain will 
> >> not accept attack ads. 
> 
> >No "various fake names", no "attack ads", just simply and sometimes even 
> >politely clearly pointing out his worst mistakes. Almost always doing that 
> >quietly via email so he wouldn't have to be publicly humiliated, but it 
> >appears that he finally couldn't accept that and blocked email from me. 
> 
> Archie must have some sort of megalomania or narcissistic personality 
> disorder, he is more interested in seeing his name out there than the 
> actual truth of science and math. Which is why he gets so upset over being 
> shown he is wrong - it ruins his supposed positive image as some sort of 
> great mathematician or scientist (his view of himself). What he doesn't 
> seem to realize is that his bizarre reactions make him appear much worse 
> than simply being shown to be wrong. 
> 
> >By coincidence I was most of the way through writing a very long very simple 
> >very detailed description of the intersection of the cone and plane, 
> >simple enough that a bright fifteen year old student would be able to 
> >understand and verify it. Even Archie might be able to do this, if he 
> >really, really tried. That isn't ready yet, but if that is what he wants to 
> >challenge me with then let's see what the mathematics says. 
> 
> I'd like to see it when you're done. However no matter how good it is, 
> Archie will never accept anything that doesn't make him out as some sort 
> of super mathematician. I'm not sure he even understands what a math proof 
> even is. 
> 
> >That seems reasonable. A different proof isn't going to convince Archie. 
> 
> As I just said. 
> 
> >Let's look at Archie's exact argument and see if we can find any mistake in 
> >reasoning. 
> ... 
> >Because the intersection is offset from the origin that does not obviously 
> >show whether there is or is not a second plane of symmetry. 
> 
> One of his so-called "proofs" claims there is no second axis of symmetry 
> because the part of the intersection curve near the tip of the cone is 
> smaller than the end away from the tip. But that assumes the axis of 
> symmetry of the curve is the axis of the cone. It is not; the center of 
> the ellipse is closer to the intersection away from the tip of the cone, 
> not the axis of the cone. (does the axis of the cone always go through a 
> focus of the ellipse?) 
> 
> >I believe the answer to that question is almost certainly that people have 
> >simply given up on trying to carefully point out the simplest mistakes to 
> >Archie, that Archie makes mistakes that a bright 15 year old would never 
> >make. If anyone does point out a specific mistake that Archie made then he 
> >completely misses the point and responds with ranting that he is being hated 
> >or he responds with insults and more posts pasting the same mistakes that he 
> >has repeated for years or decades without ever having carefully fixed his 
> >mistakes. 
> 
> Yes he attacks anyone who proves him wrong, and most people don't want to 
> bother with that type of illogical reaction. (Look what he does to me!) 
> 
> >He has never used mathematics to verify his oval claim, he doesn't even know 
> >how to do that. He fondled his paper cone, made a conjecture and screeched 
> >it for half a decade while being told hundreds of times that he was wrong 
> >without ever having checked his work. No one was able to explain his error 
> >in terms that a nine year old could understand. 
> 
> I have noticed something about him. He'll make a conjecture and babble 
> about it for a few days. Then at some point the conjecture locks itself 
> in his mind as an absolute proof, a proof he "discovered". He mentally 
> changes the history of his conjecture, it seems to me. His ellipse claim 
> is just one example; it started out as a conjecture. 
> 
> >Archie did make a post a year or two ago. In that he said that he realized 
> >that once his dog brain really got its teeth into something that he just 
> >couldn't give it up, even when he knew it was wrong, even when he knew that 
> >it made him look stupid to everyone, he realized that he just couldn't give 
> >it up. That does show a surprising bit of self awareness. 
> 
> He must have remembered to take his meds that day... :-) 
> 
> > Within a day 
> >Archie realized what he had admitted to the world and he deleted that post. 
> 
> >I believe this explains much of Archie's behavior. He desperately needs to 
> >see himself as a "super genius logic brain." So he must make astonishing 
> >claims. And if he ever realizes that they are wrong he simply cannot give 
> >them up or admit that he is wrong. 
> 
> ... 
> 
> I see that Archie has had a nuclear meltdown over your post and a couple 
> of others. (That must be the real reason he changed his name to Plutonium) 
> I find it truly bizarre for a white guy to accuse anyone who doesn't 
> blindly accept his claims to be a "racist" (or "rascist") when there is no 
> hint of that. So far he has started or replied to at least 10 topics since 
> you posted your reply as part of his meltdown. 

4th IMP, Jan

On Friday, September 28, 2018 at 10:52:31 PM UTC-5, Jan wrote:
> On Friday, September 28, 2018 at 7:45:42 PM UTC-7, Archimedes Plutonium wrote:
> > Eastside complains no-one at UCLA math dept can do High School Geometry, thus failing to see that ellipses are never conic sections
> 
> Ellipses are conic sections, here is the proof: 
> https://en.wikipedia.org/wiki/Dandelin_spheres 
> 
> 
> BTW, do you really think, in all seriousness, that the UCLA math dept. has nothing better
> to do than "researching" _trigonometry_??? Muahahahaha!
> 
> --
> Jan

Should I include that worthless airhead Jan Bielawski with his argument of a Dandelin proof. A worthless stalker in sci.math from California for about 23 or 24 years stalking AP? Talking to Jan is like talking to a fence post. Should I include Jan, a vivid hate monger, full of hatred. For AP believes anyone that commits stalking is an insane person, at the get-go, no more questions needed. Why bother, for you are dealing with insane persons, and the best wisdom is walk away and avoid the insane. 

Of course, any fool that thinks Dandelin argument is proof that the ellipse is a conic, is a logical misfit. Because in the Dandelin argument, Dandelin is never using a ellipse but an oval, for Dandelin assumed he was using an ellipse, but he never proved he was using an ellipse. And so like a Liaring lie made to one person, becomes mushroomed into more liaring lies in order to keep the first lie in place. 

It is not worth listing Jan as the 4th imp in any detail. The above says more than any Jan post on the sci.math. A worthless stalking hatemongering imp for 23 years now, is Jan. 

AP

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#424549 — Re: Archimedes "Total Failure" Plutonium flunked the Math Test of a lifetime-generation test

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

>Imp-parade in sci.math of imps with fake-arguments that ellipse is a conic

>First we start out with a dumb German named Franz hiding behind "Me"

So you *still* don't have any proof that Franz's proof is wrong.
FAIL.

SInce you asked, here it is again!


Below you will find a simple *proof* that shows that certain conic 
sections are ellipses.

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

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