LlRotBetween

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Summary

Function: rotation llRotBetween( vector start, vector end );

Returns a rotation that is the shortest rotation between the direction start and the direction end

• vector start
• vector end

Specification

start and end are directions and are relative to the origin <0.0, 0.0, 0.0>. If you have coordinates relative to a different origin, subtract that origin from the input vectors.

Caveats

  • start * llRotBetween(start, end) == end is only true if start and end have the same magnitude and neither have a magnitude of zero (see #Useful Snippets for a workaround).
    • This of course is ignoring floating point precision errors.
  • The above is true because of vector magnitudes and not a shortcoming of this function. The rotation returned is correct regardless of magnitudes
  • Rotations are from -PI to +PI around each axis.

Important Issues

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   llRotBetween has issues, give us something better

Examples

llRotBetween(<1.0, 0.0, 0.0>, <0.0, -1.0, 0.0>)
// will return <0.00000, 0.00000, -0.70711, 0.70711> (which represents -90 degrees on the z axis)
 
llRotBetween(<0.0, 0.0, 0.0>, <0.0, -1.0, 0.0>)
// will return <0.00000, 0.00000, 0.00000, 1.00000> (which represents a zero angle on all axis)
// because <0.0, 0.0, 0.0> does not convey a direction.

Useful Snippets

This function adjusts the magnitude of the quaternion so start * llRotBetween(start, end) == end is true as long as neither have a magnitude really close to zero. They do not have to have the same magnitude. (If either is too close to zero than this will return an unadjusted quaternion). While this is mathematically correct, it won't help with floating point rounding errors, so it's more accurate to say start * return ≈ end.

rotation RotBetween(vector start, vector end) //adjusts quaternion magnitude so (start * return == end) 
{//Authors note: I have never had a use for this but it's good to know how to do it if I did.
    rotation rot = llRotBetween(start, end);
    float d = start * start;
    if(d)//is 'start' zero?
        if((d = llPow(end * end / d, 0.25)))//is 'end' zero?
            return <rot.x * d, rot.y * d, rot.z * d, rot.s * d>;
    return rot;
}//Strife Onizuka

Notes

Vectors that are near opposite each other in direction may lead to erroneous results.

// First Vector is due north second vector is ALMOST due south.
rotation lRotation = llRotBetween( <0., 1., 0.>, <-0.001, -.1, 0.> );
llSay(0, lRotation );
// Provides a result of <1.00000, 0.00000, 0.00000, 0.00000>.

See Also

Functions

•  llAngleBetween

Deep Notes

Replacement

Due to the annoying quirks of this function Moon Metty wrote a drop in replacement. - SCR-309. The maximum error is reported to be 2.7e-7 @ 2.2 radians.

rotation RotBetween(vector a, vector b)
{
    float aabb = llSqrt((a * a) * (b * b)); // product of the lengths of the arguments
    if (aabb)
    {
        float ab = (a * b) / aabb; // normalized dotproduct of the arguments (cosine)
        vector c = <(a.y * b.z - a.z * b.y) / aabb,
                    (a.z * b.x - a.x * b.z) / aabb,
                    (a.x * b.y - a.y * b.x) / aabb >; // normalized crossproduct of the arguments
        float cc = c * c; // squared length of the normalized crossproduct (sine)
        if (cc) // test if the arguments are not (anti)parallel
        {
            float s;
            if (ab > -0.707107)
                s = 1 + ab; // use the cosine to adjust the s-element
            else 
                s = cc / (1 + llSqrt(1 - cc)); // use the sine to adjust the s-element
            float m = llSqrt(cc + s * s); // the magnitude of the quaternion
            return <c.x / m, c.y / m, c.z / m, s / m>; // return the normalized quaternion
        }
        if (ab > 0) 
            return ZERO_ROTATION; // the arguments are parallel, or anti-parallel if not true:
        float m = llSqrt(a.x * a.x + a.y * a.y); // the length of one argument projected on the XY-plane
        if (m) 
            return <a.y / m, -a.x / m, 0, 0>; // return a rotation with the axis in the XY-plane
        return <1, 0, 0, 0>; // the arguments are parallel to the Z-axis, rotate around the X-axis
    }
    return ZERO_ROTATION; // the arguments are too small, return zero rotation
}//Written by Moon Metty, optimized by Strife Onizuka
 
// This version keeps the axis in the XY-plane, in case of anti-parallel vectors (unlike the current LL implementation). -- Moon Metty

Bad Reference Implementation

//Loosely based on SL source code, like SL's version, it's not very accurate. You want to use the above version.
rotation llRotBetween(vector start, vector end) {
    vector v1 = llVecNorm(start);
    vector v2 = llVecNorm(end);
    float dot = v1 * v2;
    vector axis = v1 % v2;
    if (dot < -0.9999999) {
        // 180 degrees or there abouts
        vector ortho_axis = llVecNorm(<1.f, 0.f, 0.f> - (v1 * (v1.x / (v1 * v1))));
        if (ortho_axis)
            return < ortho_axis.x, ortho_axis.y, ortho_axis.z, 0.f>;
        return <0.0, 0.0, 1.0, 0.0>;
    }
    else if(dot > 0.9999999) {
        //parallel
        return ZERO_ROTATION;
    }
    dot = dot + 1.0;
    float m = llPow((axis * axis) + (dot * dot), -0.5);
    return <axis.x * m, axis.y * m, axis.z * m, dot * m>;
}

llRotBetween alternatives

Alternatives and considerations

All Issues

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   llRotBetween sometimes gives erroneous results.
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Source

Signature

function rotation llRotBetween( vector start, vector end );