User:Dora Gustafson/llRotBetween alternatives

llRotBetween, some alternatives and considerations

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 are no single solutions, but infinite numbers of solutions. All 4 functions provide valid, but different solutions. So the sensible thing to do is to test how the solution rotates 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>

Errors / test cycles	Function 1	Function 2	Function 3	Function 4
	llRotBetween	RotBetween	rotV2V	rotbetween
abs error > 1e-7	10 / 13	10 / 18	10 / 15	10 / 16
	10 / 14	10 / 21	10 / 22	10 / 23
		10 / 27	10 / 14	10 / 13
abs error > 1e-6	7 / 1000	0 / 1000	0 / 1000	0 / 1000
	3 / 1000	0 / 10000	1 / 10000	0 / 10000
	8 / 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>

Errors / test cycles	Function 1	Function 2	Function 3	Function 4
	llRotBetween	RotBetween	rotV2V	rotbetween
abs error > 1e-3	10 / 16	10 / 98	10 / 222	10 / 192
	10 / 15	10 / 96	10 / 89	10 / 164
	10 / 27	10 / 109	10 / 99	10 / 75
abs error > 1e-2	0 / 1000	0 / 1000	0 / 1000	0 / 1000
	0 / 10000	0 / 10000	0 / 10000	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>