User:Dora Gustafson/llRotBetween alternatives

llRotBetween, some alternatives and considerations

Under construction

Scope

Four functions are covered, They are

1. llRotBetween( vector a, vector b); // The built in function
2. RotBetween( vector a, vector b); // The replacement outlined in the wiki
3. rotV2V( vector a, vector b); // The shortest replacement
4. rotbetween( vector a, vector b); // Using a Vector to rotation function

Tests

First the rotation solutions were compared and they didn't compare at all. Testing rotations is a bad idea since there is no single solution, but an infinite number of solutions. All 4 functions provide different valid solutions. So the sensible thing to do is to test how the solution affects a vector:

Test with random, normalized vectors

<lsl> cycles = 0; record = []; while ( cycles++ < nLimit && llGetListLength( record) < 30) {

   u = llVecNorm(< llFrand( 2.0)-1.0, llFrand( 2.0)-1.0, llFrand( 2.0)-1.0 >);
   v = llVecNorm(< llFrand( 2.0)-1.0, llFrand( 2.0)-1.0, llFrand( 2.0)-1.0 >);
   solution = rotbetween( u, v);
   if ( llVecDist( u*solution, v) > epsilon ) record += [cycles, u, v];

}</lsl>

Results for function 1. llRotBetween

Errors > 1E-7 / number of test cycles

10/13, 10/14

Errors > 1E-6 / number of test cycles

7/1000, 3/1000, 8/1000

Results for function 2. RotBetween

Errors > 1E-7 / number of test cycles

10/18, 10/21, 10/27

Errors > 1E-6 / number of test cycles

0/1000, 0/10000

Results for function 3. rotV2V

Errors > 1E-7 / number of test cycles

10/15, 10/22, 10/14

Errors > 1E-6 / number of test cycles

0/1000, 1/10000, 0/10000

Results for function 4. rotbetween

Errors > 1E-7 / number of test cycles

10/16, 10/23, 10/13

Errors > 1E-6 / number of test cycles

0/1000, 0/10000

Test with close to parallel and anti parallel vectors

<lsl> cycles = 0; record = []; while ( cycles++ < nLimit && llGetListLength( record) < 30) {

   u = llVecNorm(< llFrand( 2.0)-1.0, llFrand( 2.0)-1.0, llFrand( 2.0)-1.0 >);
   v = < llFrand( 2E-3)-1E-3, llFrand( 2E-3)-1E-3, llFrand( 2E-3)-1E-3 >;
   if ( llFrand(1.0) < 0.5 ) v -= u;
   else v += u;
   solution = rotbetween( u, v);
   if ( llVecDist( u*solution, v) > epsilon ) record += [cycles, u, v];

}</lsl>

Results for function 1. llRotBetween

Errors > 1E-3 / number of test cycles

10/16, 10/15, 10/27

Errors > 1E-2 / number of test cycles

0/1000, 0/10000

Results for function 2. RotBetween

Errors > 1E-3 / number of test cycles

10/98, 10/96, 10/109

Errors > 1E-2 / number of test cycles

0/1000, 0/10000

Results for function 3. rotV2V

Errors > 1E-3 / number of test cycles

10/222, 10/89, 10/99

Errors > 1E-2 / number of test cycles

0/1000, 0/10000

Results for function 4. rotbetween

Errors > 1E-3 / number of test cycles

10/192, 10/164, 10/75

Errors > 1E-2 / number of test cycles

0/1000, 0/10000

Conclusion

Functions 2, 3 and 4 have similar performances. Function 1, the built-in, has a lower performance

Scripts and functions

Function 1, llRotBetween()

This is the built in function, see llRotBetween

Function 2, RotBetween()

This is the suggested replacement for llRotBetween

Function 3, rotV2V()

<lsl>rotation rotV2V( vector a, vector b) {

   a = llVecNorm(a);
   b = llVecNorm(b);
   vector c = b;
   while ( llFabs(a*c) > 0.999999 ) c = llVecNorm(< llFrand( 2.0)-1.0, llFrand( 2.0)-1.0, llFrand( 2.0)-1.0 >);
   c = llVecNorm(a%c);
   return ZERO_ROTATION/llAxes2Rot( a, c, a%c)*llAxes2Rot( b, c, b%c);

} </lsl>

Function 4, rotbetween()

<lsl>rotation Vec2Rot( vector a) {

   a = llVecNorm(a);
   vector  c = < 0.0, 0.0, 1.0 >;
   if ( llFabs(a*c) > 0.999999 ) c = llVecNorm(< llFrand( 2.0)-1.0, llFrand( 2.0)-1.0, 0.0 >);
   vector b = llVecNorm(c%a);
   return llAxes2Rot( a, b, a%b);

}

rotation rotbetween( vector a, vector b) {

   return ZERO_ROTATION/Vec2Rot(a)*Vec2Rot(b);

} </lsl>