Expression 45: "R" Subscript, "y" , Baseline equals "Y" left parenthesis, "c" , "d" , "R" Subscript, "x" , Baseline , right parenthesisRy=Yc,d,Rx
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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: "S" Subscript, "x" , Baseline equals "X" Subscript, 2 , Baseline left parenthesis, "c" , "d" , right parenthesisSx=X2c,d
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Expression 48: "Q" Subscript, "y" , Baseline equals "Y" left parenthesis, "a" , "b" , "Q" Subscript, "x" , Baseline , right parenthesisQy=Ya,b,Qx
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Expression 49: "S" Subscript, "y" , Baseline equals "Y" left parenthesis, "c" , "d" , "S" Subscript, "x" , Baseline , right parenthesisSy=Yc,d,Sx
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Expression 50: "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 52: "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 53: "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 54: "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 55: "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 56: "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 58: "D" Subscript, "p" , Baseline left parenthesis, "x" Subscript, 11 , Baseline , "y" Subscript, 11 , Baseline , "x" Subscript, 21 , Baseline , "y" Subscript, 21 , Baseline , "P" Subscript, "x" 1 , Baseline , "P" Subscript, "y" 1 , Baseline , "Q" Subscript, "x" 1 , Baseline , "Q" Subscript, "y" 1 , Baseline , right parenthesis equals absolute value left parenthesis, log left parenthesis, StartFraction, left parenthesis, "N" left parenthesis, "x" Subscript, 11 , Baseline , "y" Subscript, 11 , Baseline , "P" Subscript, "x" 1 , Baseline , "P" Subscript, "y" 1 , Baseline , right parenthesis times "N" left parenthesis, "x" Subscript, 21 , Baseline , "y" Subscript, 21 , Baseline , "Q" Subscript, "x" 1 , Baseline , "Q" Subscript, "y" 1 , Baseline , right parenthesis , right parenthesis Over left parenthesis, "N" left parenthesis, "x" Subscript, 21 , Baseline , "y" Subscript, 21 , Baseline , "P" Subscript, "x" 1 , Baseline , "P" Subscript, "y" 1 , Baseline , right parenthesis times "N" left parenthesis, "x" Subscript, 11 , Baseline , "y" Subscript, 11 , Baseline , "Q" Subscript, "x" 1 , Baseline , "Q" Subscript, "y" 1 , Baseline , right parenthesis , right parenthesis , EndFraction , right parenthesis , right parenthesisDpx11,y11,x21,y21,Px1,Py1,Qx1,Qy1=abslogNx11,y11,Px1,Py1·Nx21,y21,Qx1,Qy1Nx21,y21,Px1,Py1·Nx11,y11,Qx1,Qy1
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Expression 59: "D" Subscript, "p" , Baseline left parenthesis, "x" Subscript, 1 , Baseline , "y" Subscript, 1 , Baseline , "x" Subscript, 2 , Baseline , "y" Subscript, 2 , Baseline , "P" Subscript, "x" , Baseline , "P" Subscript, "y" , Baseline , "Q" Subscript, "x" , Baseline , "Q" Subscript, "y" , Baseline , right parenthesisDpx1,y1,x2,y2,Px,Py,Qx,Qy
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Expression 60: "D" Subscript, "p" , Baseline left parenthesis, "x" Subscript, 3 , Baseline , "y" Subscript, 3 , Baseline , "x" Subscript, "i" , Baseline , "y" Subscript, "i" , Baseline , "R" Subscript, "x" , Baseline , "R" Subscript, "y" , Baseline , "S" Subscript, "x" , Baseline , "S" Subscript, "y" , Baseline , right parenthesisDpx3,y3,xi,yi,Rx,Ry,Sx,Sy
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Expression 61: "D" Subscript, "p" , Baseline left parenthesis, "x" Subscript, 1 , Baseline , "y" Subscript, 1 , Baseline , "x" Subscript, "i" , Baseline , "y" Subscript, "i" , Baseline , "P" Subscript, "x" , Baseline , "P" Subscript, "y" , Baseline , "Q" Subscript, "x" , Baseline , "Q" Subscript, "y" , Baseline , right parenthesisDpx1,y1,xi,yi,Px,Py,Qx,Qy
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Expression 62:
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Expression 63: "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 65: left parenthesis, "d" Subscript, "b" "q" , Baseline , "f" left parenthesis, "d" Subscript, "b" "q" , Baseline , right parenthesis , right parenthesisdbq,fdbq
