From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Subject: Re: Reconstructing an ellipse from some of its tangents
Date: 14 Sep 1999 17:50:05 GMT
Newsgroups: sci.math
I'm coming in late here; I hope I'm not misunderstanding the question.
It looks like we're asking for the ellipses which are tangent to n given
lines, with now the extra proviso from the original poster,
In article <37DA6736.447DCD86@psu.edu>, Alan Horwitz wrote:
>I've assumed, for now, that all ellipses have major and minor axes parallel to
>the coordinate axes.
In that case, we may reformulate the question as follows: given a set of
n lines, find all vertical scalings which make the lines tangent to a
common circle, and find the possible circles.
> For n=3, the set of admissible centers lie on a certain curve inside the
>triangle(assuming no two tangents are parallel).
For n=3 we observe that given three (pairwise non-parallel) lines there
are precisely four circles tangent to all three -- one inside the triangle
they form, the other three outside. As we apply all scalings, the four centers
will each follow along a curve.
>I haven't looked at n=4 yet. All of the
>computations have been done directly, without using any general methods for
>fiting a conic to five given tangents.
If the lines are L1 L2 L3 L4 then by the preceding analysis there are
four circles tangent to all of L1 L2 L3 and four tangent to L1 L2 L4.
You would not expect any of the eight to coincide, in general.
Indeed, we don't expect there to be ellipses of the sort you describe
tangent to all four lines at once. Think of the four centers of circles
tangent to L1 L2 L3 moving as the picture is scaled vertically, and likewise
the four centers corresponding to L1 L2 L4. Now you have two sets of
four points moving across the plane as time progresses. It's reasonable
to hope that the paths of any two of them cross; it's not reasonable
to expect the points to reach those points of intersection at the same
moment. You have an over-constrained system (generically speaking of course).
For example I just ran through the numbers when the lines are
y=x, y=-2x, y=5-3x, and y=7-4x. The first three lines are all tangent to
each of four circles. Scale vertically, and the four centers move. If you
scale by about 0.589, for example, one of the four centers moves to
around (1.01, .170). Likewise lines 1, 3, and 4 are all tangent to four
circles. Scale, and the centers move. In fact, one of the centers does
happen to pass through this same (1.01, .170) -- but only when we
scale by about 0.789 . Equivalently, there are two ellipses with the same
x-coordinate 1.01.., each a little taller than wide, each tangent to L1 and
L2, one tangent to L3 and the other tangent to L4. But no single ellipse
with this orientation will be tangent to all four lines.
What you should be able to concoct is a condition on four lines which
is equivalent to the existence of a common tangent ellipse parallel to
the axes. Geometrically, it's the condition that there exist a
vertical scaling of the four lines which makes them all tangent to a
circle; I don't know an easier way to say it. If you intend to do it
algebraically (ugh!) then this should again apply:
>Grobner bases have also come in handy.
dave