Re: [math-fun] Q re ellipses in the complex plane
Thanks, Rich! Perhaps I should generalize my question: are there any interesting uses of _ellipses_ (not just circles) in the _complex_ plane ? At 01:28 PM 2/12/2013, rcs@xmission.com wrote:
No. Elliptic functions have two periods, with a non-real ratio. The periods define a "period parallelogram". The input is usally scaled so that one period is real, typically either 1 or 2pi. Interesting special cases arise when one period is 1 and the other is i or w = 1/2 + i sqrt3 /2. If I remember correctly, each pargrm must have either two poles or a double pole (at least) and a matching number of zeros. Salamin's the expert at this. Abramowitz & Stegun is an OK beginning introduction. I don't know about DLMF.
Rich
--- Quoting Henry Baker <hbaker1@pipeline.com>:
Are elliptical regions (those conic sections with 2 foci) in the complex plane particularly interesting in the study of "elliptic functions" ??
I don't know enough about elliptic functions to have any insights here.
Actually, the _only_ connection between ellipses and the complex plane I can think of is Marden's Theorem: http://en.wikipedia.org/wiki/Marden%27s_theorem There must be more... At 02:19 PM 2/12/2013, Henry Baker wrote:
Thanks, Rich!
Perhaps I should generalize my question: are there any interesting uses of _ellipses_ (not just circles) in the _complex_ plane ?
At 01:28 PM 2/12/2013, rcs@xmission.com wrote:
No. Elliptic functions have two periods, with a non-real ratio. The periods define a "period parallelogram". The input is usally scaled so that one period is real, typically either 1 or 2pi. Interesting special cases arise when one period is 1 and the other is i or w = 1/2 + i sqrt3 /2. If I remember correctly, each pargrm must have either two poles or a double pole (at least) and a matching number of zeros. Salamin's the expert at this. Abramowitz & Stegun is an OK beginning introduction. I don't know about DLMF.
Rich
--- Quoting Henry Baker <hbaker1@pipeline.com>:
Are elliptical regions (those conic sections with 2 foci) in the complex plane particularly interesting in the study of "elliptic functions" ??
I don't know enough about elliptic functions to have any insights here.
"HB" == Henry Baker <hbaker1@pipeline.com> writes:
HB> Actually, the _only_ connection between ellipses and the complex plane I can think of is Marden's Theorem: HB> http://en.wikipedia.org/wiki/Marden%27s_theorem Silly question based on that: The article doesn't mention what occurs when the three zeros of p(z) are collinear. Presumably if one allows for degenerate triangles, ellipses then the theorem holds? But what happens to the inellipse's foci? Does it perhaps hold that if the zeros of p(z) are collinear, then they are also the union of the zeros of p'(z) and p''(z)? -JimC -- James Cloos <cloos@jhcloos.com> OpenPGP: 1024D/ED7DAEA6
Where do the Marden ellipse foci go when the 3 vertices are collinear? Suppose A,B,C are the 3 triangle vertices, with their opposite line segments a,b,c, respectively, as usual. Suppose we go to the limit where a=b+c. As the triangle collapses, the foci move further and further into the corners B,C. In the limiting case, the foci are _at_ B,C. It should be possible to use Maxima (or equivalent) to make an animated .gif image that shows this process in action. Start with an equilateral triangle standing on its base, in which case the inscribed ellipse is actually a circle <=> the foci are at the same place in the center. Allow the top vertex to fall vertically until it hits the base and goes through to become an upside down equilateral triangle. Draw the inscribed ellipse and the 2 foci during this process. Q: What is the curve of the locus of points of the foci during this animation? At 05:03 PM 2/12/2013, James Cloos wrote:
"HB" == Henry Baker <hbaker1@pipeline.com> writes: HB> Actually, the _only_ connection between ellipses and the complex plane I can think of is Marden's Theorem: HB> http://en.wikipedia.org/wiki/Marden%27s_theorem
Silly question based on that:
The article doesn't mention what occurs when the three zeros of p(z) are collinear. Presumably if one allows for degenerate triangles, ellipses then the theorem holds? But what happens to the inellipse's foci?
Does it perhaps hold that if the zeros of p(z) are collinear, then they are also the union of the zeros of p'(z) and p''(z)?
"HB" == Henry Baker <hbaker1@pipeline.com> writes:
HB> Where do the Marden ellipse foci go when the 3 vertices are collinear? HB> Suppose A,B,C are the 3 triangle vertices, with their opposite line segments a,b,c, respectively, as usual. HB> Suppose we go to the limit where a=b+c. As the triangle collapses, HB> the foci move further and further into the corners B,C. In the HB> limiting case, the foci are _at_ B,C. That was my intuition (or memory? I haven't really thought about such stuff much in the last 25 or so). But a simple poly with real (and therefore collinear) zeros such as 0=(x-1)*(x-2)*(x-3) seems to show otherwise. The roots of the 1st diff are bounded by 2+-sqrt(3)/3, and the root of the 2nd diff is 2, (as calculated and plotted by maxima). Or does this imply the that limit of an ellipse inscribed in a triangle is not the same as the limit of said triangle when the triangle is collapsed to a line segment? -JimC -- James Cloos <cloos@jhcloos.com> OpenPGP: 1024D/ED7DAEA6
Re Marden's Theorem: Interesting! Consider the polynomial P(z)=(z-1)*(z+1)*(z-c*%i), which describes an isosceles triangle sitting on the real number line from -1 to +1 with altitude c. If I compute P'(z) and find its rightmost zero, I get 2 sqrt(3 - c ) + %i c (%o18) z = ------------------- 3 whose real part is 2 sqrt(3 - c ) (%o21) ------------ 3 so long as |c|<=sqrt(3). As c->0, this real part ->1/sqrt(3). So the foci of the ellipse _don't_ approach the corners after all. I hadn't thought it possible that an ellipse could approach a line segment while still keeping its foci away from the line segment ends, but I was wrong! At 03:17 PM 2/18/2013, James Cloos wrote:
"HB" == Henry Baker <hbaker1@pipeline.com> writes:
HB> Where do the Marden ellipse foci go when the 3 vertices are collinear? HB> Suppose A,B,C are the 3 triangle vertices, with their opposite line segments a,b,c, respectively, as usual.
HB> Suppose we go to the limit where a=b+c. As the triangle collapses, HB> the foci move further and further into the corners B,C. In the HB> limiting case, the foci are _at_ B,C.
That was my intuition (or memory? I haven't really thought about such stuff much in the last 25 or so). But a simple poly with real (and therefore collinear) zeros such as 0=(x-1)*(x-2)*(x-3) seems to show otherwise.
The roots of the 1st diff are bounded by 2+-sqrt(3)/3, and the root of the 2nd diff is 2, (as calculated and plotted by maxima).
Or does this imply the that limit of an ellipse inscribed in a triangle is not the same as the limit of said triangle when the triangle is collapsed to a line segment?
-JimC -- James Cloos <cloos@jhcloos.com> OpenPGP: 1024D/ED7DAEA6
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(⅓). --Dan P.S. Could we *please* given credit where credit is due, and call the theorem in question "Siebeck's Theorem", since it was discovered 149 years ago by someone named Siebeck? For unknown reasons, Dan Kalman, in his 2008 article in the Monthly, decided to call it Marden's Theorem. (There were indeed articles by one Morris Marden about the theorem: one in 1945, and another in 1966.) But the theorem was discovered by Jörg Siebeck in 1864. Kalman acknowledges this in his article. See the unfortunately named Wikipedia article: < http://en.wikipedia.org/wiki/Marden's_theorem >. On 2013-02-18, at 5:19 PM, Henry Baker wrote:
Re Marden's Theorem:
Interesting!
Consider the polynomial P(z)=(z-1)*(z+1)*(z-c*%i), which describes an isosceles triangle sitting on the real number line from -1 to +1 with altitude c.
If I compute P'(z) and find its rightmost zero, I get
2 sqrt(3 - c ) + %i c (%o18) z = ------------------- 3
whose real part is 2 sqrt(3 - c ) (%o21) ------------ 3
so long as |c|<=sqrt(3).
As c->0, this real part ->1/sqrt(3).
So the foci of the ellipse _don't_ approach the corners after all.
I hadn't thought it possible that an ellipse could approach a line segment while still keeping its foci away from the line segment ends, but I was wrong!
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(â ).
participants (3)
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Dan Asimov -
Henry Baker -
James Cloos