Expression 30: "v" Subscript, "n" "o" "r" "m" "a" "l" "i" "z" "e" "M" "a" "n" "y" , Baseline left parenthesis, "l" , right parenthesis equals Start sum from "n" equals left bracket, 1 , ... , length left parenthesis, "l" , right parenthesis , right bracket to left bracket, 1 , ... , length left parenthesis, "l" , right parenthesis , right bracket , end sum, left brace, mod left parenthesis, "n" , 4 , right parenthesis equals 1 : StartFraction, "l" left bracket, "n" , right bracket Over "l" left bracket, "n" plus 3 , right bracket , EndFraction , left brace, mod left parenthesis, "n" , 4 , right parenthesis equals 2 : StartFraction, "l" left bracket, "n" , right bracket Over "l" left bracket, "n" plus 2 , right bracket , EndFraction , left brace, mod left parenthesis, "n" , 4 , right parenthesis equals 3 : StartFraction, "l" left bracket, "n" , right bracket Over "l" left bracket, "n" plus 1 , right bracket , EndFraction , 1 , right brace , right brace , right bracevnormalizeManyl=1,...,lengthl∑n=1,...,lengthlmodn,4=1:lnln+3,modn,4=2:lnln+2,modn,4=3:lnln+1,1
30
Expression 31: "S" equals StartFraction, 1 Over tan left parenthesis, StartNestedFraction, "p" Subscript, "c" "a" "m" "e" "r" "a" "F" "O" "V" , Baseline NestedOver 2 , EndNestedFraction times StartNestedFraction, pi NestedOver 180 , EndNestedFraction , right parenthesis , EndFractionS=1tanpcameraFOV2·π180
equals=
3.0 1 4 8 9 2 6 1 8 53.0148926185
31
Expression 32: "v" Subscript, "p" "e" "r" "s" "p" "e" "c" "t" "i" "v" "e" "M" "a" "n" "y" , Baseline left parenthesis, "l" , "f" , "r" , right parenthesis equals Start sum from "n" equals left bracket, 1 , ... , length left parenthesis, "l" , right parenthesis , right bracket to left bracket, 1 , ... , length left parenthesis, "l" , right parenthesis , right bracket , end sum, left brace, mod left parenthesis, "n" , 4 , right parenthesis equals 1 : "S" times "l" left bracket, "n" , right bracket , left brace, mod left parenthesis, "n" , 4 , right parenthesis equals 2 : "S" times "l" left bracket, "n" , right bracket , left brace, mod left parenthesis, "n" , 4 , right parenthesis equals 3 : "l" left bracket, "n" , right bracket , negative StartFraction, "f" times "r" Over "f" minus "r" , EndFraction times "l" left bracket, "n" minus 1 , right bracket , right brace , right brace , right bracevperspectiveManyl,f,r=1,...,lengthl∑n=1,...,lengthlmodn,4=1:S·ln,modn,4=2:S·ln,modn,4=3:ln,−f·rf−r·ln−1
32
Expression 33: "v" Subscript, "a" "l" "l" "T" "f" "o" "r" "m" "s" , Baseline left parenthesis, "l" , "c" Subscript, "f" "a" "r" , Baseline , "c" Subscript, "n" "e" "a" "r" , Baseline , "y" , "p" , "t" Subscript, "x" , Baseline , "t" Subscript, "y" , Baseline , "t" Subscript, "z" , Baseline , right parenthesis equals Start sum from "n" equals left bracket, 1 , ... , length left parenthesis, "l" , right parenthesis , right bracket to left bracket, 1 , ... , length left parenthesis, "l" , right parenthesis , right bracket , end sum, left brace, mod left parenthesis, "n" , 4 , right parenthesis equals 1 : "S" cos left parenthesis, "y" , right parenthesis "l" left bracket, "n" , right bracket plus "S" sin left parenthesis, "y" , right parenthesis "l" left bracket, "n" plus 2 , right bracket plus "S" left parenthesis, "t" Subscript, "x" , Baseline cos left parenthesis, "y" , right parenthesis plus "t" Subscript, "z" , Baseline sin left parenthesis, "y" , right parenthesis , right parenthesis "l" left bracket, "n" plus 3 , right bracket , left brace, mod left parenthesis, "n" , 4 , right parenthesis equals 2 : "S" sin left parenthesis, "p" , right parenthesis sin left parenthesis, "y" , right parenthesis "l" left bracket, "n" minus 1 , right bracket plus "S" cos left parenthesis, "p" , right parenthesis "l" left bracket, "n" , right bracket minus "S" sin left parenthesis, "p" , right parenthesis cos left parenthesis, "y" , right parenthesis "l" left bracket, "n" plus 1 , right bracket plus "S" "t" Subscript, "y" , Baseline cos left parenthesis, "p" , right parenthesis "l" left bracket, "n" plus 2 , right bracket , left brace, mod left parenthesis, "n" , 4 , right parenthesis equals 3 : negative cos left parenthesis, "p" , right parenthesis sin left parenthesis, "y" , right parenthesis "l" left bracket, "n" minus 2 , right bracket plus sin left parenthesis, "p" , right parenthesis "l" left bracket, "n" minus 1 , right bracket plus cos left parenthesis, "p" , right parenthesis cos left parenthesis, "y" , right parenthesis "l" left bracket, "n" , right bracket plus left parenthesis, "t" Subscript, "y" , Baseline sin left parenthesis, "p" , right parenthesis plus cos left parenthesis, "p" , right parenthesis left parenthesis, negative "t" Subscript, "x" , Baseline sin left parenthesis, "y" , right parenthesis plus "t" Subscript, "z" , Baseline cos left parenthesis, "y" , right parenthesis , right parenthesis , right parenthesis "l" left bracket, "n" plus 1 , right bracket , StartFraction, "c" Subscript, "f" "a" "r" , Baseline "c" Subscript, "n" "e" "a" "r" , Baseline Over "c" Subscript, "f" "a" "r" , Baseline minus "c" Subscript, "n" "e" "a" "r" , Baseline , EndFraction cos left parenthesis, "p" , right parenthesis sin left parenthesis, "y" , right parenthesis "l" left bracket, "n" minus 3 , right bracket minus StartFraction, "c" Subscript, "f" "a" "r" , Baseline "c" Subscript, "n" "e" "a" "r" , Baseline Over "c" Subscript, "f" "a" "r" , Baseline minus "c" Subscript, "n" "e" "a" "r" , Baseline , EndFraction sin left parenthesis, "p" , right parenthesis "l" left bracket, "n" minus 2 , right bracket minus StartFraction, "c" Subscript, "f" "a" "r" , Baseline "c" Subscript, "n" "e" "a" "r" , Baseline Over "c" Subscript, "f" "a" "r" , Baseline minus "c" Subscript, "n" "e" "a" "r" , Baseline , EndFraction cos left parenthesis, "p" , right parenthesis cos left parenthesis, "y" , right parenthesis "l" left bracket, "n" minus 1 , right bracket minus StartFraction, "c" Subscript, "f" "a" "r" , Baseline "c" Subscript, "n" "e" "a" "r" , Baseline Over "c" Subscript, "f" "a" "r" , Baseline minus "c" Subscript, "n" "e" "a" "r" , Baseline , EndFraction left parenthesis, "t" Subscript, "y" , Baseline sin left parenthesis, "p" , right parenthesis plus cos left parenthesis, "p" , right parenthesis left parenthesis, negative "t" Subscript, "x" , Baseline sin left parenthesis, "y" , right parenthesis plus "t" Subscript, "z" , Baseline cos left parenthesis, "y" , right parenthesis , right parenthesis , right parenthesis "l" left bracket, "n" , right bracket , right brace , right brace , right bracevallTformsl,cfar,cnear,y,p,tx,ty,tz=1,...,lengthl∑n=1,...,lengthlmodn,4=1:Scosyln+Ssinyln+2+Stxcosy+tzsinyln+3,modn,4=2:Ssinpsinyln−1+Scospln−Ssinpcosyln+1+Stycospln+2,modn,4=3:−cospsinyln−2+sinpln−1+cospcosyln+tysinp+cosp−txsiny+tzcosyln+1,cfarcnearcfar−cnearcospsinyln−3−cfarcnearcfar−cnearsinpln−2−cfarcnearcfar−cnearcospcosyln−1−cfarcnearcfar−cneartysinp+cosp−txsiny+tzcosyln
33
Expression 34: "p" Subscript, "w" "o" "r" "l" "d" "S" "p" "a" "c" "e" , Baseline equals "v" Subscript, "r" "o" "t" "a" "t" "e" "Y" "M" "a" "n" "y" , Baseline left parenthesis, "p" Subscript, "m" "d" "l" "Y" "a" "w" , Baseline , "v" Subscript, "r" "o" "t" "a" "t" "e" "X" "M" "a" "n" "y" , Baseline left parenthesis, "p" Subscript, "m" "d" "l" "P" "i" "t" "c" "h" , Baseline , "p" Subscript, "m" "d" "l" , Baseline , right parenthesis , right parenthesispworldSpace=vrotateYManypmdlYaw,vrotateXManypmdlPitch,pmdl
equals=
11
1.0 3 9 1 8 9 4 4 0 8 51.03918944085
0.9 5 9 2 1 0 7 7 2 4 7 40.959210772474
11
negative 1−1
1.0 3 9 1 8 9 4 4 0 8 51.03918944085
0.9 5 9 2 1 0 7 7 2 4 7 40.959210772474
11
negative 1−1
negative 0.9 5 9 2 1 0 7 7 2 4 7 4−0.959210772474
1.0 3 9 1 8 9 4 4 0 8 51.03918944085
11
11
negative 0.9 5 9 2 1 0 7 7 2 4 7 4−0.959210772474
1.0 3 9 1 8 9 4 4 0 8 51.03918944085
64 element list
34
Expression 35: "p" Subscript, "v" "i" "e" "w" "S" "p" "a" "c" "e" , Baseline equals "v" Subscript, "r" "o" "t" "a" "t" "e" "X" "M" "a" "n" "y" , Baseline left parenthesis, negative "p" Subscript, "c" "a" "m" "e" "r" "a" "P" "i" "t" "c" "h" , Baseline , "v" Subscript, "r" "o" "t" "a" "t" "e" "Y" "M" "a" "n" "y" , Baseline left parenthesis, negative "p" Subscript, "c" "a" "m" "e" "r" "a" "Y" "a" "w" , Baseline , "v" Subscript, "t" "r" "n" "s" "l" "a" "t" "e" "M" "a" "n" "y" , Baseline left parenthesis, "p" Subscript, "w" "o" "r" "l" "d" "S" "p" "a" "c" "e" , Baseline , negative "p" Subscript, "c" "a" "m" "e" "r" "a" "X" , Baseline , negative "p" Subscript, "c" "a" "m" "e" "r" "a" "Y" , Baseline , negative "p" Subscript, "c" "a" "m" "e" "r" "a" "Z" , Baseline , right parenthesis , right parenthesis , right parenthesispviewSpace=vrotateXMany−pcameraPitch,vrotateYMany−pcameraYaw,vtrnslateManypworldSpace,−pcameraX,−pcameraY,−pcameraZ
Expression 37: "p" Subscript, "p" "r" "o" "j" "e" "c" "t" "e" "d" "F" "a" "s" "t" "D" "o" "N" "o" "t" "U" "s" "e" , Baseline equals "v" Subscript, "n" "o" "r" "m" "a" "l" "i" "z" "e" "M" "a" "n" "y" , Baseline left parenthesis, "v" Subscript, "a" "l" "l" "T" "f" "o" "r" "m" "s" , Baseline left parenthesis, "p" Subscript, "m" "d" "l" , Baseline , 10 , 1 , negative "p" Subscript, "c" "a" "m" "e" "r" "a" "Y" "a" "w" , Baseline , negative "p" Subscript, "c" "a" "m" "e" "r" "a" "P" "i" "t" "c" "h" , Baseline , negative "p" Subscript, "c" "a" "m" "e" "r" "a" "X" , Baseline , negative "p" Subscript, "c" "a" "m" "e" "r" "a" "Y" , Baseline , negative "p" Subscript, "c" "a" "m" "e" "r" "a" "Z" , Baseline , right parenthesis , right parenthesispprojectedFastDoNotUse=vnormalizeManyvallTformspmdl,10,1,−pcameraYaw,−pcameraPitch,−pcameraX,−pcameraY,−pcameraZ
equals=
0.7 2 1 5 2 1 6 1 8 5 3 60.721521618536
negative 0.1 6 7 6 1 1 1 1 6 4 5 5−0.167611116455
negative 0.9−0.9
11
negative 1.2 9 0 6 0 2 3 6 2 8 5−1.29060236285
0.3 3 0 8 0 5 8 7 9 0 2 30.330805879023
negative 0.9−0.9
11
negative 0.9 8 6 1 5 2 9 6 3 6 4 7−0.986152963647
negative 1.1 8 0 5 5 6 3 5 8 6 6−1.18055635866
negative 0.9−0.9
11
0.4 8 6 3 9 8 2 6 1 4 6 80.486398261468
negative 2.0 9 3 0 0 7 7 1 3 5 3−2.09300771353
negative 0.9−0.9
64 element list
37
Expression 38: "l" Subscript, "s" "u" "b" "s" "e" "t" , Baseline left parenthesis, "l" , "o" , right parenthesis equals Start sum from "n" equals left bracket, 1 , ... , StartFraction, length left parenthesis, "l" , right parenthesis Over 4 , EndFraction , right bracket to left bracket, 1 , ... , StartFraction, length left parenthesis, "l" , right parenthesis Over 4 , EndFraction , right bracket , end sum, "l" left bracket, 4 left parenthesis, "n" minus 1 , right parenthesis plus "o" , right bracketlsubsetl,o=1,...,lengthl4∑n=1,...,lengthl4l4n−1+o
38
Finally, the rendered points and the user interface.
left parenthesis, 0.7 4 8 8 , 2.1 5 3 , right parenthesis0.7488,2.153
left parenthesis, negative 1.2 6 1 , 1.8 1 5 , right parenthesis−1.261,1.815
left parenthesis, negative 1.0 1 6 , negative 0.0 5 9 9 2 , right parenthesis−1.016,−0.05992
left parenthesis, 0.4 7 3 1 , negative 0.5 5 4 3 , right parenthesis0.4731,−0.5543
left parenthesis, 0.7 4 8 8 , 2.1 5 3 , right parenthesis0.7488,2.153
left parenthesis, 1.2 4 9 , 1.6 2 6 , right parenthesis1.249,1.626
left parenthesis, negative 0.1 2 1 8 , 1.5 3 3 , right parenthesis−0.1218,1.533
left parenthesis, negative 1.2 6 1 , 1.8 1 5 , right parenthesis−1.261,1.815
left parenthesis, negative 1.0 1 6 , negative 0.0 5 9 9 2 , right parenthesis−1.016,−0.05992
left parenthesis, negative 0.1 2 6 2 , 0.2 9 6 6 , right parenthesis−0.1262,0.2966
left parenthesis, negative 0.1 2 1 8 , 1.5 3 3 , right parenthesis−0.1218,1.533
left parenthesis, 1.2 4 9 , 1.6 2 6 , right parenthesis1.249,1.626
left parenthesis, 0.9 8 8 2 , 0.0 8 0 5 9 , right parenthesis0.9882,0.08059
left parenthesis, 0.4 7 3 1 , negative 0.5 5 4 3 , right parenthesis0.4731,−0.5543
left parenthesis, 0.9 8 8 2 , 0.0 8 0 5 9 , right parenthesis0.9882,0.08059
16 element list
40
Model Rotation: left parenthesis, negative 1.5 plus StartFraction, "p" Subscript, "m" "d" "l" "Y" "a" "w" , Baseline Over 2 pi , EndFraction , negative 1.5 plus StartFraction, negative "p" Subscript, "m" "d" "l" "P" "i" "t" "c" "h" , Baseline Over 2 pi , EndFraction , right parenthesis−1.5+pmdlYaw2π,−1.5+−pmdlPitch2π
Label:
equals=
left parenthesis, negative 1.5 , negative 1.4 9 3 6 3 3 8 , right parenthesis−1.5,−1.4936338
41
Hidden Label: left bracket, left parenthesis, negative 2 , negative 1 , right parenthesis , left parenthesis, negative 1 , negative 1 , right parenthesis , left parenthesis, negative 1 , negative 2 , right parenthesis , left parenthesis, negative 2 , negative 2 , right parenthesis , left parenthesis, negative 2 , negative 1 , right parenthesis , right bracket−2,−1,−1,−1,−1,−2,−2,−2,−2,−1
Label
42
Hidden Label: left parenthesis, negative 1.5 , negative 1.5 , right parenthesis−1.5,−1.5
Label
43
Camera Position: left parenthesis, 1.5 plus StartFraction, "p" Subscript, "c" "a" "m" "e" "r" "a" "X" , Baseline Over 10 , EndFraction , negative 1.5 plus StartFraction, negative "p" Subscript, "c" "a" "m" "e" "r" "a" "Z" , Baseline Over 10 , EndFraction , right parenthesis1.5+pcameraX10,−1.5+−pcameraZ10
Label:
equals=
left parenthesis, 1.6 7 , negative 1.8 , right parenthesis1.67,−1.8
44
Hidden Label: left bracket, left parenthesis, 1 , negative 1 , right parenthesis , left parenthesis, 2 , negative 1 , right parenthesis , left parenthesis, 2 , negative 2 , right parenthesis , left parenthesis, 1 , negative 2 , right parenthesis , left parenthesis, 1 , negative 1 , right parenthesis , right bracket1,−1,2,−1,2,−2,1,−2,1,−1
Label
45
Hidden Label: left parenthesis, 1.5 , negative 1.5 , right parenthesis1.5,−1.5
Label
46
Camera Height: left parenthesis, 2.2 , negative 1.5 plus StartFraction, "p" Subscript, "c" "a" "m" "e" "r" "a" "Y" , Baseline Over 10 , EndFraction , right parenthesis2.2,−1.5+pcameraY10
Label:
equals=
left parenthesis, 2.2 , negative 1.4 5 , right parenthesis2.2,−1.45
47
Hidden Label: left bracket, left parenthesis, 2.2 , negative 1 , right parenthesis , left parenthesis, 2.2 , negative 2 , right parenthesis , right bracket2.2,−1,2.2,−2
Label
48
Camera FOV: left parenthesis, 1.5 plus StartFraction, "p" Subscript, "c" "a" "m" "e" "r" "a" "F" "O" "V" , Baseline minus 10 Over negative 80 , EndFraction plus 0.5 , negative 2.2 , right parenthesis1.5+pcameraFOV−10−80+0.5,−2.2
Label:
equals=
left parenthesis, 1.6 6 6 2 5 , negative 2.2 , right parenthesis1.66625,−2.2
49
Hidden Label: left bracket, left parenthesis, 1 , negative 2.2 , right parenthesis , left parenthesis, 2 , negative 2.2 , right parenthesis , right bracket1,−2.2,2,−2.2
Label
50
Camera Rotation: left parenthesis, 0 plus StartFraction, negative "p" Subscript, "c" "a" "m" "e" "r" "a" "Y" "a" "w" , Baseline Over 2 pi , EndFraction , negative 1.5 plus StartFraction, "p" Subscript, "c" "a" "m" "e" "r" "a" "P" "i" "t" "c" "h" , Baseline Over pi , EndFraction , right parenthesis0+−pcameraYaw2π,−1.5+pcameraPitchπ
Hidden Label: left bracket, left parenthesis, 0.5 , negative 1 , right parenthesis , left parenthesis, 0.5 , negative 2 , right parenthesis , left parenthesis, negative 0.5 , negative 2 , right parenthesis , left parenthesis, negative 0.5 , negative 1 , right parenthesis , left parenthesis, 0.5 , negative 1 , right parenthesis , right bracket0.5,−1,0.5,−2,−0.5,−2,−0.5,−1,0.5,−1
Label
52
Hidden Label: left parenthesis, 0 , negative 1.5 , right parenthesis0,−1.5
