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 rotations provided by functions were compared and they didn't compare at all.

Testing rotations is a bad idea since there is no single solution to the question, but an infinite number of solutions.

All 4 functions provide valid, but different rotations.

So the sensible thing to do is to test how the solution rotates a vector:

Test with random, normalized vectors

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];
}

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

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];
}

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.

See note about the functions nature

Function 4 is different, the solution will map 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

See note about the functions nature

Function 2, RotBetween( vector a, vector b)

This is the suggested Replacement for llRotBetween

See note about the functions nature

Function 3, rotV2V( vector a, vector b)

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);
}

See note about the functions nature

Function 4, rotbetween( vector a, vector b)

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);
}

Returns a solution that will map a rotation with left in the horizontal plane. This is convenient for cameras and prims you don't want tilted

---

About the function's nature

The function determines a rotation (quaternion) which can map a vector onto another.

It does nothing else, which means that there are no conditions on the other vectors associated with a rotation in space.

When the solution is used to rotate a body in space one should be aware of it’s nature.

For a prim whose rotation is defined by three orthogonal vectors the forward vector mapping will be correct while the other vector mappings are unpredictable.

A prim mapping will normally be rotated about the prim’s forward axis in an unpredictable way.

The prim will be tilted to one side or the other side with respect to the vertical.

There are numerous solutions to the task of finding the rotation between two vectors and all are correct in the sense that they can correctly map one vector onto another.