Expression 37: left parenthesis, "Q" Subscript, "x" , Baseline , "Q" Subscript, "y" , Baseline , right parenthesisQx,Qy
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Expression 38: left parenthesis, "x" Subscript, 3 , Baseline , "y" Subscript, 3 , Baseline , right parenthesisx3,y3
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Expression 39: left parenthesis, "x" Subscript, 2 , Baseline , "y" Subscript, 2 , Baseline , right parenthesisx2,y2
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Expression 40: left parenthesis, "x" Subscript, 1 , Baseline , "y" Subscript, 1 , Baseline , right parenthesisx1,y1
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Expression 41: left parenthesis, "P" Subscript, "x" , Baseline , "P" Subscript, "y" , Baseline , right parenthesisPx,Py
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Expression 42: "X" Subscript, 1 , Baseline left parenthesis, "a" Subscript, "x" , Baseline , "b" Subscript, "y" , Baseline , right parenthesis equals StartFraction, left parenthesis, negative 4 "a" Subscript, "x" , Baseline minus StartRoot, left parenthesis, 16 "a" squared minus 4 left parenthesis, "a" squared plus "b" squared , right parenthesis left parenthesis, 4 minus "b" squared , right parenthesis , right parenthesis , EndRoot , right parenthesis Over 2 left parenthesis, "a" squared plus "b" squared , right parenthesis , EndFractionX1ax,by=−4ax−16a2x−4a2x+b2y4−b2y2a2x+b2y
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Expression 43: "X" Subscript, 2 , Baseline left parenthesis, "a" Subscript, "x" , Baseline , "b" Subscript, "y" , Baseline , right parenthesis equals StartFraction, left parenthesis, negative 4 "a" Subscript, "x" , Baseline plus StartRoot, left parenthesis, 16 "a" squared minus 4 left parenthesis, "a" squared plus "b" squared , right parenthesis left parenthesis, 4 minus "b" squared , right parenthesis , right parenthesis , EndRoot , right parenthesis Over 2 left parenthesis, "a" squared plus "b" squared , right parenthesis , EndFractionX2ax,by=−4ax+16a2x−4a2x+b2y4−b2y2a2x+b2y
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Expression 44: "P" Subscript, "x" , Baseline equals "X" Subscript, 1 , Baseline left parenthesis, "a" , "b" , right parenthesisPx=X1a,b
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Expression 45: "P" Subscript, "y" , Baseline equals "Y" left parenthesis, "a" , "b" , "P" Subscript, "x" , Baseline , right parenthesisPy=Ya,b,Px
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Expression 46: "Q" Subscript, "x" , Baseline equals "X" Subscript, 2 , Baseline left parenthesis, "a" , "b" , right parenthesisQx=X2a,b
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Expression 47: "Q" Subscript, "y" , Baseline equals "Y" left parenthesis, "a" , "b" , "Q" Subscript, "x" , Baseline , right parenthesisQy=Ya,b,Qx
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Expression 48: "Y" left parenthesis, "a" Subscript, "x" , Baseline , "b" Subscript, "y" , Baseline , "x" Subscript, "n" , Baseline , right parenthesis equals StartFraction, left parenthesis, negative 2 minus "a" Subscript, "x" , Baseline "x" Subscript, "n" , Baseline , right parenthesis Over "b" Subscript, "y" , Baseline , EndFractionYax,by,xn=−2−axxnby
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Distance
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Expression 50: "N" left parenthesis, "u" Subscript, 1 , Baseline , "u" Subscript, 2 , Baseline , "v" Subscript, 1 , Baseline , "v" Subscript, 2 , Baseline , right parenthesis equals StartRoot, left parenthesis, "u" Subscript, 1 , Baseline minus "v" Subscript, 1 , Baseline , right parenthesis squared plus left parenthesis, "u" Subscript, 2 , Baseline minus "v" Subscript, 2 , Baseline , right parenthesis squared , EndRootNu1,u2,v1,v2=u1−v12+u2−v22
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Expression 51: "d" Subscript, "a" "p" , Baseline equals "N" left parenthesis, "x" Subscript, 1 , Baseline , "y" Subscript, 1 , Baseline , "P" Subscript, "x" , Baseline , "P" Subscript, "y" , Baseline , right parenthesisdap=Nx1,y1,Px,Py
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Expression 52: "d" Subscript, "b" "q" , Baseline equals "N" left parenthesis, "x" Subscript, 2 , Baseline , "y" Subscript, 2 , Baseline , "Q" Subscript, "x" , Baseline , "Q" Subscript, "y" , Baseline , right parenthesisdbq=Nx2,y2,Qx,Qy
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Expression 53: "d" Subscript, "b" "p" , Baseline equals "N" left parenthesis, "x" Subscript, 2 , Baseline , "y" Subscript, 2 , Baseline , "P" Subscript, "x" , Baseline , "P" Subscript, "y" , Baseline , right parenthesisdbp=Nx2,y2,Px,Py
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Expression 54: "d" Subscript, "a" "q" , Baseline equals "N" left parenthesis, "x" Subscript, 1 , Baseline , "y" Subscript, 1 , Baseline , "Q" Subscript, "x" , Baseline , "Q" Subscript, "y" , Baseline , right parenthesisdaq=Nx1,y1,Qx,Qy
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the Poincare distance
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Expression 56: "D" Subscript, "p" , Baseline equals absolute value left parenthesis, log left parenthesis, StartFraction, left parenthesis, "d" Subscript, "a" "p" , Baseline times "d" Subscript, "b" "q" , Baseline , right parenthesis Over left parenthesis, "d" Subscript, "b" "p" , Baseline times "d" Subscript, "a" "q" , Baseline , right parenthesis , EndFraction , right parenthesis , right parenthesisDp=abslogdap·dbqdbp·daq
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Expression 57: "f" left parenthesis, "x" , right parenthesis equals absolute value left parenthesis, log left parenthesis, StartFraction, left parenthesis, "d" Subscript, "a" "p" , Baseline times "x" , right parenthesis Over left parenthesis, "d" Subscript, "b" "p" , Baseline times "d" Subscript, "a" "q" , Baseline , right parenthesis , EndFraction , right parenthesis , right parenthesisfx=abslogdap·xdbp·daq
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graph of the P-distance as function of Euclidean distance of red point to orange
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Expression 59: left parenthesis, "d" Subscript, "b" "q" , Baseline , "f" left parenthesis, "d" Subscript, "b" "q" , Baseline , right parenthesis , right parenthesisdbq,fdbq
