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October 1

http://en.wikipedia.org/wiki/Ellipse#General_polar_form

In the section of this article "General polar form", the equaitons are given for expressing the polar form of the equation for an ellipse if we know a few parameters like the semi major and semi minor axes etc. To this end, the equation is given defining Q(theta). This equation contains the square root of the difference of two terms. Those terms are R(theta) and another term that varies independently of R(theta).

R(theta) is itself the sum of two terms, one of which includes (b**2 - a**2) and the other is (a**2 + b**2).

Since a and b can be arbitrarily close to one another, the first term involving (b**2 - a**2) can be arbitrarily small. also, a and b can be arbitrarily small, so that (a**2 + b**2) can be arbitrarily small. Consequently, R(theta) can be arbitrarily small.

Returning our attention to the square root term above, it seems that the argument under the square root can become negative since R(theta) can be arbitrarily small and the term subtracted from it can be positive and larger than R(theta). But then the square root is not defined, at least in terms of real numbers.

This leads me to wonder if there is some mistake in the given equations, since in physical reality the ellipse exists in the real plane, nd we should not expect to encounter negative square roots.

I am not expert in this area, so I might misunderstand and perhaps there is no problem, but I cannot seem to resolve this. I cannot seem to find other articles the cite the same equations (although I will keep looking). I would appreciate any clarification of any confusion on my part, or correction of a typo etc in the article. — Preceding unsigned comment added by 68.98.184.146 (talk) 00:17, 1 October 2012 (UTC)[reply]

I haven't verified the equations in detail, but, in all cases where the origin does not lie inside the ellipse, it is inevitable that r will have no real values for certain values of theta. 81.159.110.168 (talk) 01:42, 1 October 2012 (UTC)[reply]

That's right. You'll notice that if we put a plus minus sign before the radical, the number of real solutions for r(theta) equals the number of real solutions for the square root in the expression for Q(theta) -- namely either zero or two, or one in a razor's edge case. If the origin is outside the ellipse and you draw a ray from the origin, it either does not intersect the ellipse (no real solution; expression under the radical is negative), or it intersects the ellipse twice (expression under the radical is positive, giving two real solutions), or it has a single tangency point with the ellipse (expression under the radical is zero, so the number of distinct real roots is one). On the other hand, if the origin is inside the ellipse, and you allow the ray to have positive or negative length, then there are two intersections of the ray (actually line in both directions) with the ellipse; presumably the formula always gives real values for the square root when the origin is inside, and presumably the formula for r(theta) then gives one positive and one negative solution.

That section could use a drawing for clarity. The best I could find was in angular diameter

The angles the two tangent lines make with the horizontal are the minimum and maximum value for θ, outside of that, the value under the square root is negative. Ssscienccce (talk) 21:51, 1 October 2012 (UTC)[reply]

Like the previous respondent, I haven't verified the equations, but they appear to be consistent with the above analysis. I'll tag that sub-section as citation-needed. Duoduoduo (talk) 15:06, 1 October 2012 (UTC)[reply]

Complex functions

Can someone give an explicit 1:1 conformal map from the square to the half disk? Money is tight (talk) 08:06, 1 October 2012 (UTC)[reply]

I'd guess "no". A conformal map is supposed to preserve angles between crossing lines in the plane. The corners of the square can be thought of as the intersection point of two orthogonal straight lines (restricted to the square) crossing each other at an angle of 90 degrees. The images of these lines under the presumed conformal map (wich must be parts of the circle with a common point on the circle) can't possibly cross at 90 degrees. (It would be 180 degrees because its a circle.)

But, there is a homeomorphism (continuous and continuously invertible map) from the filled square in the real 2-dimensional plane to the filled circle. Divide each vector in the filled square with its length. YohanN7 (talk) 09:25, 1 October 2012 (UTC)[reply]

There is a map, it's guaranteed by the Riemann mapping theorem. But the proof is non constructive and I need an explicit function. Money is tight (talk) 11:20, 1 October 2012 (UTC)[reply]

I guess I was wrong then. But you must probably replace "square" by "interior of the square", and likewise for the circle. Better: use "open disk". Edit: "Open" is in the hypothesis of the Riemann mapping theorem.

I now recall something vaguely similar from a complex analysis text. [Churchill & Brown?]. I'll have a look later this evening. YohanN7 (talk) 11:46, 1 October 2012 (UTC)[reply]

The maps you need are discussed at http://www.encyclopediaofmath.org/index.php/Conformal_mapping (examples 5 and 9). Both are of the Schwarz–Christoffel mapping type (a digon and a tetragon), and the upper half-plane to square is given in our article here "explicitly". A source is Zeev Nehari's book "Conformal mapping". —Kusma (t·c) 12:07, 1 October 2012 (UTC)[reply]

You can get an idea of why something simple doesn't work from a little puzzle you might like at The Problem of the Right Wiggly Triangles. Dmcq (talk) 12:28, 1 October 2012 (UTC)[reply]

I don't agree with the solution for nr 2, that's not a wiggly triangle, that's a wiggly pentagon! Ssscienccce (talk) 22:00, 1 October 2012 (UTC)[reply]

I believe Paul Daniels catchphrase at the Bunko Booth about covers that "You'll like this...not a lot, but you'll like it." ;-) Dmcq (talk) 23:40, 1 October 2012 (UTC)[reply]

YohanN7, yes I do mean open square and open half disk, sorry for not pointing that out. Kusma, in Schwarz–Christoffel mapping which specific square does the half plane get mapped to (i.e. what are the coordinates of the vertices)? The integral is very difficult to evaluate. Money is tight (talk) 00:46, 2 October 2012 (UTC)[reply]

You might be able figure this out using the Jacobi elliptic functions, which map a rectangle into the upper half plane. But I have no experience doing this, I can just point you to the literature :( —Kusma (t·c) 10:12, 2 October 2012 (UTC)[reply]