Maths - Affine Conversion - Quaternion/Vector to Matrix

Definition of terms:

It you want to consider the rotation only then the code for this is shown here.

Alternatively if we want to consider the possibility that there is also a translation from the centre, and that the rotation may not be about the centre but may be about some arbitrary point, then we need to extend the notation as follows:

 

Rotation about a point other than origin

For the derivation of how to rotate about a point see this page.


Quaternions and 3x3 matrices alone can only represent rotations about the origin. But if we include a 3D vector with the quaternion we can use this to represent the point about which we are rotating. Also if we use a 4x4 matrix then this can hold a translation and therefore can specify a rotation about a point.

The following code generates a 4x4 matrix from a quaternion and a vector. The derivation is given here.

void setRotate(sfquat q,sfvec3f centre) {
   double sqw = q.w*q.w;
   double sqx = q.x*q.x;
   double sqy = q.y*q.y;
   double sqz = q.z*q.z;
   m00 = sqx - sqy - sqz + sqw; // since sqw + sqx + sqy + sqz =1
   m11 = -sqx + sqy - sqz + sqw;
   m22 = -sqx - sqy + sqz + sqw;
   
   double tmp1 = q.x*q.y;
   double tmp2 = q.z*q.w;
   m01 = 2.0 * (tmp1 + tmp2);
   m10 = 2.0 * (tmp1 - tmp2);
   
   tmp1 = q.x*q.z;
   tmp2 = q.y*q.w;
   m02 = 2.0 * (tmp1 - tmp2);
   m20 = 2.0 * (tmp1 + tmp2);
   
   tmp1 = q.y*q.z;
   tmp2 = q.x*q.w;
   m12 = 2.0 * (tmp1 + tmp2);
   m21 = 2.0 * (tmp1 - tmp2);
   
  double a1,a2,a3;
  if (centre == null) {
    a1=a2=a3=0;
  } else {
    a1 = centre.x;
    a2 = centre.y;
    a3 = centre.z;
  }
  m03 = a1 - a1 * m00 - a2 * m01 - a3 * m02;
  m13 = a2 - a1 * m10 - a2 * m11 - a3 * m12;
  m23 = a3 - a1 * m20 - a2 * m21 - a3 * m22;
  m30 = m31 = m32 = 0.0;
  m33 = 1.0;
}

Rotation about a point - using Quaternion

void setRotate(Quat4d q,sfvec3f centre) {
  double sqw = q.w*q.w;
  double sqx = q.x*q.x;
  double sqy = q.y*q.y;
  double sqz = q.z*q.z;
  m00 =  sqx - sqy - sqz + sqw; // since sqw + sqx + sqy + sqz =1
  m11 = -sqx + sqy - sqz + sqw;
  m22 = -sqx - sqy + sqz + sqw;
  
  double tmp1 = q.x*q.y;
  double tmp2 = q.z*q.w;
  m01 = 2.0 * (tmp1 + tmp2);
  m10 = 2.0 * (tmp1 - tmp2);
  
  tmp1 = q.x*q.z;
  tmp2 = q.y*q.w;
  m02 = 2.0 * (tmp1 - tmp2);
  m20 = 2.0 * (tmp1 + tmp2);
  tmp1 = q.y*q.z;
  tmp2 = q.x*q.w;
  m12 = 2.0 * (tmp1 + tmp2);
  m21 = 2.0 * (tmp1 - tmp2);

  double a1,a2,a3;
  if (centre == null) {
    a1=a2=a3=0;
  } else {
  a1 = centre.x; a2 = centre.y; a3 = centre.z;
  }

  m03 = a1 - a1 * m00 - a2 * m01 - a3 * m02;
  m13 = a2 - a1 * m10 - a2 * m11 - a3 * m12;
  m23 = a3 - a1 * m20 - a2 * m21 - a3 * m22;
  m30 = m31 = m32 = 0.0;
  m33 = 1.0;
}

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Correspondence about this page

Book Shop - Further reading.

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cover If you are interested in 3D games, this looks like a good book to have on the shelf. If, like me, you want to have know the theory and how it is derived then there is a lot for you here. Including - Graphics pipeline, scenegraph, picking, collision detection, bezier curves, surfaces, key frame animation, level of detail, terrain, quadtrees & octtrees, special effects, numerical methods. Includes CDROM with code.
3D Game engine design also includes conversions between matrix, quaternions and axis-angle.

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