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.
All four functions are very good except with parallel and anti parallel vectors.
The rotation found is not always what you expect.

The rotation will point a vector in the right direction, i.e. the rotation's forward is the vector direction.

The rotations left direction is in the plane given by the two vectors, which is not always what you want or expect

Function 4 is different, it returns a rotation with left in the horizontal plane

Scripts and functions

Function 1, llRotBetween( vector a, vector b)

This is the built in function, see llRotBetween

Returns a rotation with left in the plane given by a and b, when such a plane can be found

Function 2, RotBetween( vector a, vector b)

This is the suggested replacement for llRotBetween

Returns a rotation with left in the plane given by a and b, when such a plane can be found

Function 3, rotV2V( vector a, vector b)

<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>

Returns a rotation with left in the plane given by a and b, when such a plane can be found

Function 4, rotbetween( vector a, vector b)

<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>

Returns a rotation with left in the horizontal plane. This is convenient for cameras and prims you don't want to be tilted