Oh, is that what you think the original question was? On 2013-02-19, at 5:24 PM, Henry Baker wrote:
But you missed the original question:
As the triangle collapses, _why_ does the ellipse collapse to [-1,1]/sqrt(3) instead of to [-1,1] ?
After all, the ellipse gets thinner & thinner; why not get thin enough to move the foci all the way into the triangle's base vertices ?
--- Interestingly, as an equilateral triangle collapses to smaller & smaller isosceles triangles, the ellipse foci start together at the center of the triangle, and then move downwards on either side in a _circular_ locus to hit the base of the triangle at +-1/sqrt(3), where the radius of the circle is r=1/sqrt(3).
This would be an interesting quiz problem, as I would never have guessed the circularity of the locus.
At 04:40 PM 2/19/2013, Dan Asimov wrote:
My point is that the ellipses for c > 0 do not approach the line segment [-1,1].
They approach the line segment [-sqrt(â…“), sqrt(â…“)].
On 2013-02-19, at 4:33 PM, Henry Baker wrote:
Re: "This is not what happens":
Tell that to Maxima. P(z) has roots at +-1, +ci.
P'(z) has roots at +ci/3+-sqrt(3-c^2)/3 = +-1/sqrt(3) when c=0.
(%i1) declare(z,complex); (%o1) done (%i2) P(z):=(z-1)*(z+1)*(z-%i*c); (%o2) P(z) := (z - 1) (z + 1) (z - %i c) (%i3) solve(P(z),z); (%o3) [z = %i c, z = - 1, z = 1] (%i4) solve(diff(P(z),z),z); 2 2 sqrt(3 - c ) - %i c sqrt(3 - c ) + %i c (%o4) [z = - -------------------, z = -------------------] 3 3 (%i5) %,c=0; 1 1 (%o5) [z = - -------, z = -------] sqrt(3) sqrt(3) (%i6) %,numer; (%o6) [z = - 0.57735026918963, z = 0.57735026918963]
I.e., these numbers are not +-1.
At 03:33 PM 2/19/2013, Dan Asimov wrote:
This is not what happens.
The limit of the ellipse as c -> 0 is indeed a degenerate ellipse that is precisely the line segment between its foci, +-sqrt(Ⅶ“).