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Extra info for AIAA 2002-5664 Analysis Design and Optimization of Non-Cylindrical Fuselage for Blended-Wi

Example text

As a preliminary,we shall have to make a few definitions. Specifically,let A and B be two bounded fuzzy sets and let M be a hypersurface in E" defined by an equation h (J-)= 0, with all points for which h ( x 1 2 0 being an one jar which h ( s ) 5 O being on s i c k of H and all points .. the other side. Ilet K!! be a number dependent on H such rhat f ~ , ( x ) S K o, n~ one side of ( x 3 S K H on t h e other side. I,et . The n u m b ~ rD,, Vote that the sets in : r ~ ~ e s t i uhave n k! in common, 21 ' 1-MH will be called the degrpe of seporurion rcf A and k?

Now ,since A and B are convex Q This way of exxpwwing convexity was suggested to the writer by his mlkague* I?. krlekamp. and hence + L[h-[ ( 1 -41x2] 2 ~ i n [ ~ i n [ j ; d , . A fuzzy set +4is l ? x) r,,= in I, then exists a for all J- on the side of H which does not contain the origin. For,consider the set L = (2:/

Inturvuctitnr 7'hrlt D = 1 - M . In plain words ,the theorem the highest degree of separation of t w o convex fuzzy sets A and B that can be Grnmunt. This is illus~rntedin Fig. 5 for Prwf: rr = 1. MR). Note It is convenient tu that the latter case rules out A C U or B C A . re 1. For concreteness, assume that M,,