void precise_generators(
MatrixPairList *gen_list)
{
MatrixPair *matrix_pair;
for (matrix_pair = gen_list->begin.next;
matrix_pair != &gen_list->end;
matrix_pair = matrix_pair->next)
{
precise_matrix(matrix_pair->m[0]);
o31_invert(matrix_pair->m[0], matrix_pair->m[1]);
}
}
void conjugate_matrices(
MatrixPairList *gen_list,
double displacement[3])
{
/*
* We want to conjugate each MatrixPair on the gen_list so as to move
* the basepoint the distance given by the displacement. The
* displacement, which is a translation (dx, dy, dz) in the tangent
* space to H^3 at (1, 0, 0, 0), is a linear approximation to where the
* basepoint ought to be.
*
* It won't do simply to conjugate by the matrix T1 =
*
* 1 dx dy dz
* dx 1 0 0
* dy 0 1 0
* dz 0 0 1
*
* Even though T1 is a linear approximation to the desired translation,
* it's columns are not quite orthonormal (they are orthonormal to a
* first order approximation, but not exactly). We need to use a
* translation matrix which is exactly orthonormal, so that the
* MatrixPairs on the gen_list remain elements of O(3,1) to full
* accuracy.
*
* One approach would be to apply the Gram-Schmidt process to find an
* element of O(3,1) close to T1. The problem here is that the
* Gram-Schmidt process itself may introduce additional error.
*
* Better to start instead with a second order approximation to the
* translation matrix, given by T2 =
*
* 1 + (dx^2 + dy^2 + dz^2)/2 dx dy dz
* dx 1 + (dx^2)/2 (dx dy)/2 (dx dz)/2
* dy (dx dy)/2 1 + (dy^2)/2 (dy dz)/2
* dz (dx dz)/2 (dy dz)/2 1 + (dz^2)/2
*
* Note that T2 is actually orthonormal to third order; that is,
* its columns fail to be orthonormal only by fourth order terms.
*
* We will start with the second order approximation T2 and apply
* Gram-Schmidt to it. The correction terms -- all of fourth order --
* will not significantly affect the closeness of the approximation.
*
* Digression.
*
* You may be wondering where the matrix T2 came from.
* Drop down a dimension, and consider the product of a translation
* (dx, 0) and a translation (0, dy). The result you get depends
* on the order of the factors.
*
* ( 1 dx 0) ( 1 0 dy) ( 1 dx dy )
* ( dx 1 0) ( 0 1 0 ) = ( dx 1 dxdy)
* ( 0 0 1) ( dy 0 1 ) ( dy 0 1 )
*
* ( 1 0 dy) ( 1 dx 0) ( 1 dx dy )
* ( 0 1 0 ) ( dx 1 0) = ( dx 1 0 )
* ( dy 0 1 ) ( 0 0 1) ( dy dxdy 1 )
*
* Averaging the two results leads to the matrix T2 shown above.
*
* End of digression.
*/
O31Matrix t2;
MatrixPair *matrix_pair;
/*
* Initialize t2 to be the second order approximation shown above.
*/
initialize_t2(displacement, t2);
/*
* Apply the Gram-Schmidt process to bring t2 to a nearby element
* of O(3,1).
*/
o31_GramSchmidt(t2);
/*
* For each MatrixPair on gen_list . . .
*/
for (matrix_pair = gen_list->begin.next;
matrix_pair != &gen_list->end;
matrix_pair = matrix_pair->next)
{
/*
* Conjugate m[0] by t2.
* That is, replace m[0] by (t2^-1) m[0] t2.
*/
o31_conjugate(matrix_pair->m[0], t2, matrix_pair->m[0]);
/*
* Recompute m[1] as the inverse of m[0].
*/
o31_invert(matrix_pair->m[0], matrix_pair->m[1]);
/*
* Set the height.
*/
matrix_pair->height = matrix_pair->m[0][0][0];
}
}
