public static Matrix4d buildCanonicalOrientation(Point4d a, Point4d b)
{
/* local scratch variables; will be transformed */
Point4d pa = new Point4d(a.x, a.y, a.z, a.w);
Point4d pb = new Point4d(b.x, b.y, b.z, b.w);
Point4d pivot = findPivotPoint(pa, pb);
Matrix4d retval = buildTranslation(ORIGIN4, pivot);
Matrix4d t1 = buildTranslation(pivot, ORIGIN4);
t1.transform(pa);
t1.transform(pb);
retval *= buildTranslation(ORIGIN4, pa);
Matrix4d t2 = buildTranslation(pa, ORIGIN4);
t2.transform(pa);
t2.transform(pb);
/* calculate spherical coordinates (rho, phi, theta) of pb */
// Projection to affine coordinates is necessary so that we can
// directly reference the x, y, and z components in the following
// calculations.
pb.project();
float rho = HyperbolicMath.vectorLength(pb);
float phi = Mathf.Acos(pb.x / rho);
float theta = Mathf.Atan2(pb.z, pb.y);
if (phi == 0.0f)
{
/* rotate line to achieve alignment on positive x-axis */
retval *= buildXRotation(theta);
retval *= buildZRotation(phi);
}
return(retval);
}