FractionalGeometry-Chap2

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Unformatted text preview: ubsets of the plane, and the distance between these sets is measured by the Hausdorff metric h. From the parameter space P , take points p = (rp , sp , θp , ϕp , ep , fp ) and q = (rq , sq , θq , ϕq , eq , fq ) Because we are viewing θ and ϕ as lying on circles, the angular distance between θp and θq , for example, is a(θp , θq ) = |θp − θq |(mod π ) Why (mod π ) instead of (mod 2π )? Because the greatest angular distance between two points on a circle os π , not 2π . Then the distance dP (p, q ) is defined by dP (p, q ) = max |rp − rq |, |sp − sq |, |a(θp , θq )|, |a(ϕp , ϕq )|, |ep − eq |, |fp − fq | 88 CHAPTER 2. ITERATED FUNCTION SYSTEMS Then between IFSs {T1 , . . . , Tn } with parameters {p1 , . . . , pn } and {Q1 , . . . , Qn } with parameters {q1 , . . . , qn } the distance is D({T1 , . . . , Tn }, {Q1 , . . . , Qn }) = max dP (p1 , q1 ), · · · , dP (pn , qn ) Why do we measure the distance between points (x1 , y1 ) and (x2 , y2 ) by max{|x1 − x2 |, |y1 − y2 |} instead of (x1 −...
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