Re: how to determine the center or foci of an ellipse from a slope

I believe I have an effective 3-step method for determining
your foci that will minimize the number of points that need
be sampled. I am in a rush right now, so excuse me if I
make any specific mistake, but I am just intending to give
the general idea.

The first step involves determining the center of the
ellipses. Select an arbitrary point "p" and determine its
slope. Then, perform an angular search around that point
looking for an angle "tht" at which the slope is the same
as that at "p". The center of the ellipse will lie along
the line defined by this angle through "p".

Do the same at another arbitrary point and the intersection
of these two lines will give you the center "c" of the
ellipse.

The second step involves finding one of the major or minor
axes. Do another angular search around "c" looking for an
angle "tht" at which the slope is perpendicular to "tht".
This will give you one of the major or minor axes.

Using "c" and "tht" you can easily transform your ellipses
to ones centered at the origin with the major/minor axis
oriented along the x-axis.

The third step involves determining the distance of the
focus from the center. Evalute the transformed slope "s"
at some transformed point "p" where the angle of the line
from "c" to "p" is "tht" radians clockwise from the x-axis.

The angle of this slope should be

s = -b/a/tan(tht)

where a is the length of the semi-axis along the x-axis and
b is the one along the y-axis. Therefore:

b/a = -s*tan(tht)

I need to run right now, but I believe once you have (b/a),
it will be elementary to determine the distance from the
center to a focus.

The number of points that need be evaluated during these 3
angular searches can be minimized using some appropriate
optimization function like FMINSEARCH or FMINUNC.

I hope this helps.
.

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