Quaternion "lookAt" function - c

Question

Im struggling with the following problem. Im working with bone animation and I want (ie) the head of the player to follow an another object in space. My up axis is +Z my forward axis is +Y, and the magnitude of the quaternion is in W. I tried to use the mesa code for gluLookAt and use the 3x3 matrix to transform to a quaternion but it doesn't work as expected so I go in another direction...
So far I got the following code that is "almost" working at least the head of the player is rotating (however the X angle seems to affect the Y rotation axis) in the good direction but its looking straight up instead on following an object on the floor at about 65 degree:
qt LookRotation( v3 lookAt, v3 upDirection )
{
qt t;
v3 forward = lookAt;
v3 up = upDirection;
OrthoNormalize( &forward, &up );
v3 right = v3_cross( up, forward );
mat3 m = mat3_make( right.x, up.x, forward.x,
right.y, up.y, forward.y,
right.z, up.z, forward.z );
t.w = sqrtf( 1.0f +
m.r[ 0 ].x +
m.r[ 1 ].y +
m.r[ 2 ].z ) * 0.5f;
float w4_recip = 1.0f / ( 4.0f * t.w );
t.x = ( m.r[ 2 ].y - m.r[ 1 ].z ) * w4_recip;
t.y = ( m.r[ 0 ].z - m.r[ 2 ].x ) * w4_recip;
t.z = ( m.r[ 1 ].x - m.r[ 0 ].y ) * w4_recip;
t = qt_normalize( t );
return t;
}
...
...
...
v3 v = v3_sub( vec4_to_v3( transform.world.r[ 3 ] /* The object XYZ location in the world */),
skeleton->final_pose.location[ i ] /* i = The head joint location */ );
v = v3_normalize( v );
qt q = LookRotation( v,
v3_make( 0.0f, 0.0f, 1.0f ) );
Can someone help me figuring out this problem... Im kinda new with quaternions and don't really know where I could have messed up. After quite some research basically what I want to do is something like the Unity API: http://docs.unity3d.com/Documentation/ScriptReference/Quaternion.LookRotation.html

Answer 1

I think this function will do what you need:
/// <summary>
/// Evaluates a rotation needed to be applied to an object positioned at sourcePoint to face destPoint
/// </summary>
/// <param name="sourcePoint">Coordinates of source point</param>
/// <param name="destPoint">Coordinates of destionation point</param>
/// <returns></returns>
public static Quaternion LookAt(Vector3 sourcePoint, Vector3 destPoint)
{
Vector3 forwardVector = Vector3.Normalize(destPoint - sourcePoint);
float dot = Vector3.Dot(Vector3.forward, forwardVector);
if (Math.Abs(dot - (-1.0f)) < 0.000001f)
{
return new Quaternion(Vector3.up.x, Vector3.up.y, Vector3.up.z, 3.1415926535897932f);
}
if (Math.Abs(dot - (1.0f)) < 0.000001f)
{
return Quaternion.identity;
}
float rotAngle = (float)Math.Acos(dot);
Vector3 rotAxis = Vector3.Cross(Vector3.forward, forwardVector);
rotAxis = Vector3.Normalize(rotAxis);
return CreateFromAxisAngle(rotAxis, rotAngle);
}
// just in case you need that function also
public static Quaternion CreateFromAxisAngle(Vector3 axis, float angle)
{
float halfAngle = angle * .5f;
float s = (float)System.Math.Sin(halfAngle);
Quaternion q;
q.x = axis.x * s;
q.y = axis.y * s;
q.z = axis.z * s;
q.w = (float)System.Math.Cos(halfAngle);
return q;
}
This code comes from here:
https://gamedev.stackexchange.com/questions/15070/orienting-a-model-to-face-a-target
I just slightly modified it to fit my case, which was implementation of the transform.LookAt without using Unity3D.

Answer 2

Both of current answers have various problems for edge cases. The accepted answer is not correct for other reasons as well including the fact that it sets w=pi for one of the cases and also it doesn't do proper norms. After looking around quite a bit and testing several cases, I also found out that you need front and up vector to do this computation. So without further ado below is the code I'm using:
Quaternion lookAt(const Vector3f& sourcePoint, const Vector3f& destPoint, const Vector3f& front, const Vector3f& up)
{
Vector3f toVector = (destPoint - sourcePoint).normalized();
//compute rotation axis
Vector3f rotAxis = front.cross(toVector).normalized();
if (rotAxis.squaredNorm() == 0)
rotAxis = up;
//find the angle around rotation axis
float dot = VectorMath::front().dot(toVector);
float ang = std::acosf(dot);
//convert axis angle to quaternion
return Eigen::AngleAxisf(rotAxis, ang);
}
Bove uses popular Eigen library. If you don't want to use that then you might need following replacement for Eigen::AngleAxisf:
//Angle-Axis to Quaternion
Quaternionr angleAxisf(const Vector3r& axis, float angle) {
auto s = std::sinf(angle / 2);
auto u = axis.normalized();
return Quaternionr(std::cosf(angle / 2), u.x() * s, u.y() * s, u.z() * s);
}
Note that special cases for dot product 0 or 1 or -1 gets automatically handled because normalized() returns 0 for the zero vector in Eigen library.
On the side note, for all your conversions worries, this is a great document to go to.

Answer 3

You don't need to use acos and axis angle (which will in turn do 2 more trig functions) to get the quaternion from 2 vectors:
public static Quaternion LookAt(Vector3 sourcePoint, Vector3 destPoint)
{
Vector3 forwardVector = Vector3.Normalize(destPoint - sourcePoint);
Vector3 rotAxis = Vector3.Cross(Vector3.forward, forwardVector);
float dot = Vector3.Dot(Vector3.forward, forwardVector);
Quaternion q;
q.x = rotAxis.x;
q.y = rotAxis.y;
q.z = rotAxis.z;
q.w = dot+1;
return q.normalize();
}
The reason for the dot+1 and subsequent normalize is because if you don't, you'll get the quaternion for the double rotation. Those 2 steps will effectively do slerp(identity, q, 0.5) which will be the proper quaternion.