From: mert0407 AT sable DOT ox DOT ac DOT uk (George Foot)
Newsgroups: comp.os.msdos.djgpp
Subject: Re: fwd: Re: ellipses at an angle
Date: Thu, 06 Feb 1997 22:00:18 GMT
Organization: Oxford University
Lines: 56
Message-ID: <32fa5063.108734727@news.ox.ac.uk>
References: <32FA01D0 DOT 50EA AT dtechs DOT com>
NNTP-Posting-Host: mc31.merton.ox.ac.uk
To: djgpp AT delorie DOT com
DJ-Gateway: from newsgroup comp.os.msdos.djgpp

On Thu, 6 Feb 1997 16:07:45 GMT, Mark Teel <mteel AT dtechs DOT com> wrote:

>Our friend Mr. Chambers stated (and I paraphrase):
>"an ellipse can be defined as the sum of the distances from any point on
>the ellipse to the two foci is equal to 1".  Is this correct?

Yes. An ellipse can be defined in this way; it's just not convenient
in this case, IMHO.

>>we get the equation above. x^2/a^2 + y^2/b^2 = 1 is as general as this
>>can get. It is also completely irrelevant for "ellipses at an angle"
>>as the subject says.
>
>This is not what I'm saying at all, and if you had followed the thread
>from the beginning, you would not have made such an assumption about
>what I was saying.  But to shoot you down anyway, "x^2/a^2 + y^2/b^2 = 1
>is as general as this can get" is FALSE.  This only describes ellipses
>centered at the origin.

By 'this' I meant 'this equation'. But I'm sorry, you're quite right,
I didn't follow the thread from the beginning. I was getting fed up of
seeing it and decided to give a useful, accurate answer (i.e. the
polar equation in my posting) in the hope that this would kill it.

My argument above was simply that setting the LHS equal to 1 or a
constant doesn't make any difference. I had assumed that the poster
knew that this equation only holds if you take your axes to be the
long and short axes of the ellipse. Evidently I was mistaken, but as I
wrote above, this thread shouldn't be this long. Let's stop arguing
and drop it. I think we both agree, anyway...

>You may be right, I don't program shapes or mathematical graphs very
>often.  I just wanted to be clear as to the general definition of an
>ellipse.  The parametric treatment is perfectly acceptable to me.

You can define the ellipse in several ways, I have been taught to
define it as a conic section with 0<e<1, and I have been taught to
define conic sections as sets of ponits whose distances from a point
(focus) are a fixed multiple of their distances from a line
(directrix). From this definition it is simple to prove that the sum
of the distances from the focii is constant for an ellipse.

>  I am
>amazed (and sometimes disturbed) at how righteous some software folks
>become when rigorous definition is presented.  I am a mathematician who
>makes his living designing software - I find both to be interesting (and
>sometimes intersecting).

Well, a rigorous definition should be just that - rigorous. I don't
know... maybe when you leave education the emphasis changes. But I'm
certainly finding it necessary to give concise, watertight proofs to
all the problems.

Sorry to extend this further,

George