From: mailbox@dmitry-kazakov.de   
      
   On Thu, 10 Apr 2008 09:54:41 -0400, Kirk Zurell wrote:   
      
   > Bill Silvert wrote:   
   >> On Apr 7, 3:50 pm, Kirk Zurell  wrote:   
   >>> I can give one geometry example that is illustrative. I tried to   
   >>> create a fuzzy circle drawing program, a fuzzy equivalent to the   
   >>   
   >> I think that drawing a circle misses the point. Drawing a round shape   
   >> is much closer to what could be considered fuzzy geometry.   
   >   
   > I think that's what the original poster was looking for. I was   
   > sharing my efforts at specifying a line figure "linguistically"   
   > rather than through algebra.   
   >   
   >> But given that a circle has a crisp definition, you cannot draw a   
   >> fuzzy circle unless you define what you mean.   
   >   
   > I was looking for a circle-like figure, specified linguistically,   
   > which I thought satisfied the OP's query:   
      
   I think this probably could be done as well, provided you first defined   
   crisp circles linguistically. You would need a geometric framework that   
   describes shapes linguistically. Surely it will limit the set of possible   
   shapes to some algebra of canonical shapes and their transformations,   
   described linguistically. Then you would extend that on the fuzzy case.   
   That could work.   
      
   >> Do you know of any examples where fuzzy logic has been used to control   
   >> the shape of an object, or the path of something, in a way that would have   
   >> been difficult otherwise?   
   >   
   > Dmitry gave us a more rigorous perspective on this, focusing on   
   > 'smudged' circles rather than 'circle-like' figures.   
      
   Wait, but Bill"s approach is no less rigorous. It really depends on what is   
   understood under shapes. If you knew that all shapes are crisp curves in   
   your application domain, why shouldn't you use this knowledge? Compare this   
   with classical statistical approach. You never consider deterministic   
   things random. Why should you? Same here.   
      
   Further, I think it is a mistake to believe that there is single fuzzy   
   geometry. Just technically, you should stop somewhere. Truth values are   
   themselves crisp numbers. If you fuzzify them (fuzzy sets-2), then the   
   second order truth values will be crisp. There is always a firm crisp basis   
   below any such generalization. For this reason we can generalize certain   
   parts of geometric properties, but we cannot do them all. Therefore there   
   always be many fuzzy geometries.   
      
   --   
   Regards,   
   Dmitry A. Kazakov   
   http://www.dmitry-kazakov.de   
      
   --- SoupGate-Win32 v1.05   
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