When the ratio of the closest furthest point from 0 to the closest point to 0 gets large, the shape no longer looks much like an ellipse: it is always contained in a horizontal strip of length pi, but it can get arbitrarily long. One limiting example is when the circle goes through the origin: you get a curve that is asymptotic to two lines of the form imaginary(z) = constant, with distance pi between them. Here's a link to a plot, with the following Mma input. (I wish we could include pictures. Is anyone still inconvenienced by mail messages with images?) ParametricPlot[ {Re[#], Im[#]} &@Log[2 + 1.9999 Exp[2 Pi I t]], {t, -π, π}, PlotRange -> All, PlotPoints -> 50000] Bill On Dec 5, 2010, at 5:24 PM, Robert Munafo wrote:
I was working on a page about the Mandelbrot set as seen through an exponential (or logarithmic) coordinate transformation:
http://mrob.com/pub/muency/exponentialmap.html
and I ran across the need to describe the shape of an offset circle after its logarithm is taken. To be more precise:
If A is a circle (viewed as a set of points on the complex plane) whose distance from the origin is greater than its radius (i.e. the origin is outside the circle), and if B is the set of points you get by taking the (complex-valued) natural logarithm of each point in A, then what type of shape is B?
Is B an ellipse, some kind of superellipse (using a transcendental function perhaps)? If I need to give it a name, is there any name (like "quasi-ellipse") that doesn't already have some other meaning that would confuse my readers?
-- Robert Munafo -- mrob.com Follow me at: mrob27.wordpress.com - twitter.com/mrob_27 - youtube.com/user/mrob143 - rilybot.blogspot.com _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun