In computer vision/image processing, a logarithmic map was suggested ~30 years ago as a way to focus attention/computational cycles on areas of greatest interest. For example, the fovea of the human eye has the highest density of pixels (as well as all of the color pixels), and the density of pixels falls off pretty quickly the further from the fovea that one goes. By utilizing a logarithmic map, one can theoretically achieve any desired resolution, at the cost of moving the center point -- either mechanically or digitally. I believe that a number of common geometric objects (circles, lines, etc.) were analyzed by these researchers to understand how they mapped under this logarithmic transformation. At 02:24 PM 12/5/2010, 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