(u + v) / (1 - u * v) (complex) works for v = (x, 0)
39
i believe the sum formula is something of the form (u + v) / (1 - u*v), as an extension of the 1d/2d case, but that introduces the problems:
40
- what is u*v? how do we define the product of two R^2s?
41
- if u*v is R^2, how do we define the 1 from which it is subtracted?
42
- if u*v is R^2, how do we define the R^2/R^2 division necessary in the overall formula?
43
some differential calculus on the 1d/2d case gives these insights:
44
- u*v should be defined s.t. Λ[u→0] u*v = 0 and Λ[v→0] u*v = 0
45
- 1 in the denominator should be a multiplicative/divisive identity for however such operations are defined here (this is kinda just like a duh math fact)
46
- multiplication should be distributive
47
half of the number of lines around each dimension of the sphere surface
48
Expression 49: "K" equals 6K=6
11
100100
49
bound magnitude of projective computation
50
Expression 51: "B" equals 99B=99
negative 10−10
9999
51
rotated sphere lines
52
Expression 53: left bracket, "P" left parenthesis, "O" plus "R" Subscript, "P" , Baseline left parenthesis, "S" left parenthesis, "q" , left parenthesis, "t" times StartFraction, 2 "k" Over "k" squared plus 1 , EndFraction , "t" times left parenthesis, 1 minus StartFraction, 2 Over "k" squared plus 1 , EndFraction , right parenthesis , right parenthesis , right parenthesis , right parenthesis , right parenthesis for "k" equals left bracket, negative 1 , negative 1 plus StartFraction, 1 Over "K" , EndFraction , ...1 , right bracket plus StartFraction, 0 Over 3 "K" , EndFraction , right bracketPO+RPSq,t·2kk2+1,t·1−2k2+1fork=−1,−1+1K,...1+03K
domain t Minimum: negative "B"−B
less than or equal to "t" less than or equal to≤t≤
domain t Maximum: "B"B
53
Expression 54: left bracket, "P" left parenthesis, "O" plus "R" Subscript, "P" , Baseline left parenthesis, "S" left parenthesis, "q" , left parenthesis, "k" times left parenthesis, 1 minus StartFraction, 2 Over "t" squared plus 1 , EndFraction , right parenthesis , "k" times StartFraction, 2 "t" Over "t" squared plus 1 , EndFraction , right parenthesis , right parenthesis , right parenthesis , right parenthesis for "k" equals left bracket, StartFraction, StartRoot, 1 minus "j" squared , EndRoot Over "j" minus 1 , EndFraction for "j" equals left bracket, negative 1 , negative 1 plus StartFraction, 1 Over "K" , EndFraction , ...1 minus StartFraction, 1 Over "K" , EndFraction , right bracket plus StartFraction, 0 Over 3 "K" , EndFraction , right bracket , right bracketPO+RPSq,k·1−2t2+1,k·2tt2+1fork=1−j2j−1forj=−1,−1+1K,...1−1K+03K
domain t Minimum: negative "B"−B
less than or equal to "t" less than or equal to≤t≤
Hidden Label: "q" equals left parenthesis, "q" Subscript, "x" , Baseline , "q" Subscript, "y" , Baseline , right parenthesisq=qx,qy
Label
equals=
left parenthesis, negative 0.0 1 , 0.0 1 , right parenthesis−0.01,0.01
58
Hidden Label: "S" left parenthesis, left parenthesis, 1 , 0 , right parenthesis , "q" , right parenthesisS1,0,q
Label
equals=
left parenthesis, 0.9 8 0 0 0 3 9 2 , 0.0 1 9 6 0 4 , right parenthesis0.98000392,0.019604
59
transformed planar coordinates
60
Hidden Label: left bracket, "S" left parenthesis, left parenthesis, "j" , "k" , right parenthesis , "q" , right parenthesis for "j" equals left bracket, negative 3 , negative 2.5 , ...3 , right bracket , "k" equals left bracket, negative 3 , negative 2.5...3 , right bracket , right bracketSj,k,qforj=−3,−2.5,...3,k=−3,−2.5...3
Label
equals=
left parenthesis, negative 3.2 0 2 , negative 3.1 8 1 , right parenthesis−3.202,−3.181
left parenthesis, negative 2.6 3 9 , negative 3.1 7 8 , right parenthesis−2.639,−3.178
left parenthesis, negative 2.0 8 2 , negative 3.1 6 9 , right parenthesis−2.082,−3.169
left parenthesis, negative 1.5 3 2 , negative 3.1 5 5 , right parenthesis−1.532,−3.155
left parenthesis, negative 0.9 8 6 8 , negative 3.1 3 5 , right parenthesis−0.9868,−3.135
left parenthesis, negative 0.4 4 7 9 , negative 3.1 1 , right parenthesis−0.4479,−3.11
left parenthesis, 0.0 8 4 9 4 , negative 3.0 8 , right parenthesis0.08494,−3.08
left parenthesis, 0.6 1 1 9 , negative 3.0 4 5 , right parenthesis0.6119,−3.045
left parenthesis, 1.1 3 3 , negative 3.0 0 5 , right parenthesis1.133,−3.005
left parenthesis, 1.6 4 8 , negative 2.9 6 , right parenthesis1.648,−2.96
left parenthesis, 2.1 5 7 , negative 2.9 1 1 , right parenthesis2.157,−2.911
left parenthesis, 2.6 6 , negative 2.8 5 8 , right parenthesis2.66,−2.858
left parenthesis, 3.1 5 8 , negative 2.8 0 1 , right parenthesis3.158,−2.801
left parenthesis, negative 3.1 9 9 , negative 2.6 1 8 , right parenthesis−3.199,−2.618
left parenthesis, negative 2.6 4 2 , negative 2.6 2 1 , right parenthesis−2.642,−2.621
