Re: how to determine the center or foci of an ellipse from a slope
- From: "Roger Stafford" <ellieandrogerxyzzy@xxxxxxxxxxxxxxxxxxxxxx>
- Date: Wed, 21 May 2008 00:03:04 +0000 (UTC)
John Kellerham <johnkellerham@xxxxxxxxx> wrote in message
<6153535.1211321967244.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxx>...
Dear Mr. Stafford,not an easy problem to describe.
My apology, I admit the question was vague. I'll give it another try, but it's
(x,y) I know the tangent (or slope) of some function intersecting the point.
I have a 2d rectangular grid of point in the xy-plane and at each point P
The data can be plotted as a slope field when dealing with differential
equations. From the physics behind the problem I know that the function
which intersects lets say point P1(x,y) is an ellipse, which can have its half
axes at an angle with respect to the xy-coordinate system. At a different grid
point which I have given the tangent belongs to a different ellipse. But the
ellipse which is tangent to the slope given in P2(x,y) has the same center, the
same half axis orientation and even the same focal points as the ellipse being
tangent to the slope given in point P1(x,y). So, essentially I have an infinite
amount of ellipses which all have the same focal points.
belongs to all of the ellipses based on the tangent slopes in each grid point.
What I'm trying to do is to find the center and the focal points which
The problem which I have is that the grid points don't necessarily belong to
the same ellipse, which makes it somewhat hard to determine the center and
focal points of the ellipse.
--------------
Best Regards,
Thomas
It looks as though "Helper" may have come very close to understanding your
problem, Thomas. I'm sorry that I couldn't. Some questions remain in my
mind however. You have a certain function, let us call it f(x,y), for which you
possess the gradient - the first partial derivatives of f with respect to x and to
y. At each point in your field, you know that an ellipse of the family you
describe runs through the point with a tangent line at that point in a certain
relation to the gradient direction of f(x,y). What is that relationship? Is the
gradient parallel to the tangent direction, or is it orthogonal? Do you regard
the magnitude of the gradient as significant to your problem?
In any event, one way or another, you know the direction of tangent lines to
a family of ellipses at all points in a field, or at least at all points in a grid, and
you want to know the common foci of the family, presuming that you are
right that they do have common foci. And you know nothing more than that.
We can simply assume the tangent direction is a field of unit vectors. Or we
can assume the outward orthogonal vectors to the ellipses are a known field
of unit vectors.
Yes, that does sound like a difficult problem. What you are seeking are
envelopes of these tangent directions, and you are already aware that such
envelopes will turn out to be a family of ellipses with common foci. It would
be interesting to find out how it is that you know this. That information
might conceivably be useful in seeking an effective solution.
Well, here is a possible approach making use of some one of the routines in
Mathworks' optimization toolbox. Suppose you make an initial guess as to
the location of the two foci, call them points f1 and f2. Then select a set of at
least four, probably more, of strategically located points surrounding f1 and
f2. At each of these points it is possible to compute the tangent direction
that an ellipse with foci at f1 and f2 ought to have. This is an elementary
calculation. Then compare the results with the known tangent directions that
should prevail at these points and compute some measure of how far off the
fit is. The idea is to get the routine to continually adjust the positions of f1
and f2 so as to minimize this measure. I suppose it might be some function
like 'fminsearch' or 'fminunc'. I am not very knowledgeable about this
toolbox since I do not have it on my primitive system. Of course it would
remain to be seen whether the fit obtained in this manner would still be a
good one for the entire field. If your hypothesis is correct, it should
presumably also be a good fit if the selected test points were properly
chosen.
It occurs to me that to make the above workable, you would need to lock in
the selected points to fixed grid points while allowing the f1 and f2 foci to be
continuously variable. That is all right except that improperly selected grid
points may end up poorly positioned with respect to the foci. It suggests
more than one try to come reasonably close to a situation where they are well
located.
I promise to think about the problem some more. I haven't bothered to
work through the "elementary" tangent direction computation above for
ellipses in case this method doesn't appeal to you. If you need them, they
can be readily supplied.
Roger Stafford
.
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- Re: how to determine the center or foci of an ellipse from a slope field
- From: Roger Stafford
- Re: how to determine the center or foci of an ellipse from a slope field
- From: John Kellerham
- Re: how to determine the center or foci of an ellipse from a slope field
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