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Subject: Re: curvature and radius of curvature of a plane curve
Date: Mon, 10 Aug 2009 17:41:02 +0000 (UTC)
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"Michael Snyder" <> wrote in message <h5pcvf$903$>...
> Hi Roger,
> Good points.  I have worked with the images for so long that I had forgotten that there was an implicit reference point for cw and ccw.  I feel silly; an arc segment would have to have beginning point for it to have a cw or ccw direction.  
> Assume there is a reference point directly in the center of the matrix and I want to evaluate each arc&#8217;s cw or ccw direction in regards to that center reference.  In other words, the closest endpoint of an arc segment to the center, is defined as the beginning point for a segment.
> Think of the physics cloud chamber photographs of electron and positron paths.  It is easy just to look at the decaying spiral paths and know if it was an electron or positron. I had completely forgotten that an EM field was the implicit reference frame for such detectors.
> I am not dealing with complete spirals, but with just some spiral arcs (less than 2pi of curvature) decaying into a center point.  My problem being that have both possible types, cw and ccw arcs crossing each other in the same image and I want write a program to separate them into two images.   
> So I have to find the lines, and find out which end point of the line is closest to the center, and then characterize each line as cw or ccw, and then separate a complex image into two simpler ones.  No Problem.   <smile>
> I think I can do most of that with a radial search, but not sure about the characterizing the cw or ccw directions.  On a side note, wonder if there is already code out there from particle physics, evaluating overlaying particle spirals would be a much harder task.
> Thanks,
> Michael

Hello again Michael,

  Your criterion, "the closest endpoint of an arc segment to the center, is defined as the beginning point for a segment", would do the job all right, but it seems very unnatural to me.  How is such a "reference" center related to "spiral arcs ... decaying into a center point"?  A spiral arc decaying toward a central point, as opposed to a strictly circular arc, does indeed have an inherent clockwise versus counterclockwise orientation, assuming the decay process is sufficiently evident in the arc's varying curvature.  Why not use this direction of "decay" as your criterion rather than some artificial reference point in the "center of the matrix"?

  Of course all this begs the most fundamental question.  Such determinations of clockwise versus counterclockwise orientation are vastly simpler than the truly formidable task of detecting intersecting arcs in the first place and then separating them from one another.
Roger Stafford