SDF (modified, but zeroes are preserved) based on point
11
Uitdrukking 12: "d" Subscript, "p" , Baseline left parenthesis, "p" , right parenthesis equals "q" Subscript, 3 , Baseline left parenthesis, "p" minus "s" , right parenthesis minus "r" squareddpp=q3p−s−r2
12
SDF (modified similarly) based on ray angle
13
Uitdrukking 14: "d" left parenthesis, "x" , right parenthesis equals "d" Subscript, "p" , Baseline left parenthesis, "a" "x" , right parenthesisdx=dpax
14
Solving the SDF
15
Uitdrukking 16: 0 equals "d" left parenthesis, "x" , right parenthesis0=dx
16
Uitdrukking 17: "q" Subscript, 3 , Baseline left parenthesis, "a" , right parenthesis "x" squared minus 2 "p" Subscript, 3 , Baseline left parenthesis, "a" , "s" , right parenthesis "x" plus "q" Subscript, 3 , Baseline left parenthesis, "s" , right parenthesis minus "r" squaredq3ax2−2p3a,sx+q3s−r2
17
Uitdrukking 18: "x" equals StartFraction, 2 "p" Subscript, 3 , Baseline left parenthesis, "a" , "s" , right parenthesis plus StartRoot, 4 "p" Subscript, 3 , Baseline left parenthesis, "a" , "s" , right parenthesis squared minus 4 left parenthesis, "q" Subscript, 3 , Baseline left parenthesis, "s" , right parenthesis minus "r" squared , right parenthesis , EndRoot Over 2 , EndFractionx=2p3a,s+4p3a,s2−4q3s−r22
equals=
4.4 2 0 8 9 7 5 8 6 1 64.42089758616
18
Uitdrukking 19: "x" equals "p" Subscript, 3 , Baseline left parenthesis, "a" , "s" , right parenthesis plus StartRoot, "p" Subscript, 3 , Baseline left parenthesis, "a" , "s" , right parenthesis squared minus "q" Subscript, 3 , Baseline left parenthesis, "s" , right parenthesis plus "r" squared , EndRootx=p3a,s+p3a,s2−q3s+r2
equals=
4.4 2 0 8 9 7 5 8 6 1 64.42089758616
19
Uitdrukking 20: "x" equals "p" Subscript, 3 , Baseline left parenthesis, "a" , "s" , right parenthesis minus StartRoot, "p" Subscript, 3 , Baseline left parenthesis, "a" , "s" , right parenthesis squared minus "q" Subscript, 3 , Baseline left parenthesis, "s" , right parenthesis plus "r" squared , EndRootx=p3a,s−p3a,s2−q3s+r2
equals=
2.7 8 4 5 0 2 4 1 3 8 42.78450241384
20
The raycasting formula's a-dot-s / plus or minus the square root / of a-dot-s, squared, minus s-dot-s / minus radius squared
