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Hy everyone,

I have a set of 2D cartesian points (x,y coordinates) lying inside an arbitrary closed contour , something like this: arbitrary_closed_contour

by 'arbitrary' I mean that the closed curve does not resemble any circle or ellipse.

What I want to do know is to map each pair of x,y coordinates to unit circle, thus obtaining something like this: points mapped to unit circle

How to accomplish this? I must confess up to this point I'm clueless!

Thanks for any suggestion.

Update: (based on @bubba comments)

Some clarifications:

  1. I want to map each pair of (x,y) coordinates to the unit disk, not the unit circle.
  2. The points I want to map emanate all from the same origin of coordinates. They're also distributed along a set of rays radially distributed
  3. Each of these rays have two extrema points ((xA,yA) and (xB,yB)) each of them located at a different distance from the origin. Constrain: when mapped each of these points should be located on r=1
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    You have tagged your question (conformal-geometry), but the text of your question doesn't require the map to be conformal. Please clarify this discrepancy.2012-10-18
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    Given that a ray starts at the origin and extends to infinity in some definite direction $\phi\in{\mathbb R}/(2\pi)$, what do you mean by the two "extrema points" $(xA,yA)$, $(xB,yB)$ on such a ray?2012-10-18
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    I suppose he means that these two points are the ones (among those on this particular given ray) that have minimum and maximum distance from the origin, respectively.2012-10-22
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    @msotaquira: Maybe we don't share the same concept of "ray". To me (and most other people, I think), a "ray" is a semi-infinite line. In other words, it starts at some fixed point (the origin, typically) and it shoots off to infinity in some direction. So, a ray isn't just an infinite line; it's a "half-line". Are your points $(xA,yA)$ and $(xB,yB)$ on the same ray, or are they merely on the same line? Saying it another way -- is the origin allowed to be in between $(xA,yA)$ and $(xB,yB)$?2012-10-22

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