From: mailbox@dmitry-kazakov.de   
      
   On Mon, 07 Apr 2008 10:50:11 -0400, Kirk Zurell wrote:   
      
   > bint wrote:   
      
   >>    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?   
   >   
   > I can give one geometry example that is illustrative. I tried to   
   > create a fuzzy circle drawing program, a fuzzy equivalent to the   
   > x^2 + y^2 = r^2 algebra. I created 'width' and 'height'   
   > linguistic variables, and created rules like 'IF width IS very   
   > wide THEN height IS very tall' and so forth.   
      
   1. Implication is not same as equivalence. You wanted to say forall Circles   
   x in R <=> y in S. But IF A THEN B is modus ponens A, A=>B |= B, which is a   
   very different thing. Firstly it is abut an implication, and secondly,   
      
   2. you are mixing inference rules with predicates. Implication => is not   
   same as deduction |=.   
      
   3. Logic is not geometry. Though it is true that the equation x^2 + y^2 =   
   r^2 implies under certain conditions certain statements about axes of a   
   circle defined in a certain way. But it would be silly to expect fuzzy   
   logic to capture that. Same is true for crisp logic too. Neither is   
   geometry.   
      
   > It didn't work. Any actual curves that I got I couldn't expand   
   > upon, or could be attributed to errors.   
   >   
   > I'm fully willing to admit it may be my own lack of vision, but   
   > in the end I concluded there was a good reason this approach   
   > mightn't work.   
      
   Yes it cannot. Because in first place you need a clear approach to   
   geometry. Once you had it, you could generalize it onto a fuzzy case. In   
   this order.   
      
   > Consider a perfect circle, not the formula but the figure itself.   
   > That indescribable twist that is continuous yet always changing.   
   > People have tried for years to describe it outside of pure   
   > mathematics. It's not just a relationship between width and   
   > height; there's some other magic in there. If we can't describe   
   > its qualities accurately in human linguistic terms, we might not   
   > be able to encode it in fuzzy logic.   
      
   Fuzzy offers a model of uncertainty suitable for this cases. At least it is   
   believed that human reasoning is fuzzy.   
      
   You do not need to describe a circle accurately. You can do it   
   inaccurately, that's the whole idea.   
      
   Your problem was not in accuracy of a description, but rather in a lack of   
   any description.   
      
   Consider your example, with x^2 + y^2 = r^2. That fixes a description of   
   some set of pairs (x,y). A fuzzy approach to extend this would be to take   
   this model (but there can be countless number of other models) and then   
   look where uncertainty can come from. For instance, when r is not all   
   known. So you replace a crisp number r with a fuzzy number. That will   
   immediately give you a fuzzy set of pairs (x,y) in R^2, which can be called   
   "fuzzy circle."   
      
   Note that this will yet tell nothing about axes and their relations. You   
   will need to define them first. It could turn a non-trivial problem.   
   Neither it is in geometry, BTW.   
      
   --   
   Regards,   
   Dmitry A. Kazakov   
   http://www.dmitry-kazakov.de   
      
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