FuncResult change_peripheral_curves(
Triangulation *manifold,
CONST MatrixInt22 change_matrices[])
{
int i,
v,
f,
old_m,
old_l;
double old_m_coef, /* changed from int to double, JRW 2000/01/18 */
old_l_coef;
Tetrahedron *tet;
Cusp *cusp;
Complex old_Hm,
old_Hl;
/*
* First make sure all the change_matrices have determinant +1.
*/
for (i = 0; i < manifold->num_cusps; i++)
if (DET2(change_matrices[i]) != +1)
return func_bad_input;
/*
* The change_matrices for Klein bottle cusps must have zeros in the
* off-diagonal entries. (Nothing else makes sense topologically.)
*/
for (cusp = manifold->cusp_list_begin.next;
cusp != &manifold->cusp_list_end;
cusp = cusp->next)
if (cusp->topology == Klein_cusp)
for (i = 0; i < 2; i++)
if (change_matrices[cusp->index][i][!i] != 0)
uFatalError("change_peripheral_curves", "change_peripheral_curves");
/*
* Change the peripheral curves according to the change_matrices.
* As stated at the top of this file, the transformation rule is
*
* | new m | | | | old m |
* | | = | change_matrices[i] | | |
* | new l | | | | old l |
*/
for (tet = manifold->tet_list_begin.next;
tet != &manifold->tet_list_end;
tet = tet->next)
for (i = 0; i < 2; i++) /* which orientation */
for (v = 0; v < 4; v++) /* which vertex */
for (f = 0; f < 4; f++) /* which side */
{
old_m = tet->curve[M][i][v][f];
old_l = tet->curve[L][i][v][f];
tet->curve[M][i][v][f]
= change_matrices[tet->cusp[v]->index][0][0] * old_m
+ change_matrices[tet->cusp[v]->index][0][1] * old_l;
tet->curve[L][i][v][f]
= change_matrices[tet->cusp[v]->index][1][0] * old_m
+ change_matrices[tet->cusp[v]->index][1][1] * old_l;
}
/*
* Change the Dehn filling coefficients to reflect the new
* peripheral curves. That is, we want the Dehn filling curves
* to be topologically the same as before, even though the
* coefficients will be different because we changed the basis.
*
* To keep our thinking straight, let's imagine all peripheral
* curves -- old and new -- in terms of some arbitrary but
* fixed basis for pi_1(T^2) = Z + Z. We never actually compute
* such a basis, but it helps keep our thinking straight.
* Relative to this fixed basis, we have
*
* old m = (old m [0], old m [1])
* old l = (old l [0], old l [1])
* new m = (new m [0], new m [1])
* new l = (new l [0], new l [1])
*
* Note that these m's and l's are curves, not coefficients!
* They are elements of pi_1(T^2) = Z + Z.
*
* We can then rewrite the above transformation rule, with
* each peripheral curve (old m, old l, new m and new l)
* appearing as a row in a 2 x 2 matrix:
*
* | <--new m--> | | | | <--old m--> |
* | | = | change_matrices[i] | | |
* | <--new l--> | | | | <--old l--> |
*
* We can invert the change_matrix to solve for the old curves
* in terms of the new ones:
* -1
* | <--old m--> | | | | <--new m--> |
* | | = | change_matrices[i] | | |
* | <--old l--> | | | | <--new l--> |
*
* The Dehn filling curve is
*
* old_m_coef * old_m + old_l_coef * old_l
*
* = old_m_coef * [ change_matrices[i][1][1] * new_m
* - change_matrices[i][0][1] * new_l]
* + old_l_coef * [- change_matrices[i][1][0] * new_m
* + change_matrices[i][0][0] * new_l]
*
* = new_m * [ old_m_coef * change_matrices[i][1][1]
* - old_l_coef * change_matrices[i][1][0] ]
* + new_l * [- old_m_coef * change_matrices[i][0][1]
* + old_l_coef * change_matrices[i][0][0] ]
*
* Therefore
*
* new_m_coef = old_m_coef * change_matrices[i][1][1]
* - old_l_coef * change_matrices[i][1][0]
* new_l_coef = - old_m_coef * change_matrices[i][0][1]
* + old_l_coef * change_matrices[i][0][0]
*/
for (cusp = manifold->cusp_list_begin.next;
cusp != &manifold->cusp_list_end;
cusp = cusp->next)
if (cusp->is_complete == FALSE)
{
old_m_coef = cusp->m;
old_l_coef = cusp->l;
cusp->m = old_m_coef * change_matrices[cusp->index][1][1]
- old_l_coef * change_matrices[cusp->index][1][0];
cusp->l = - old_m_coef * change_matrices[cusp->index][0][1]
+ old_l_coef * change_matrices[cusp->index][0][0];
}
/*
* Update the holonomies according to the rule
*
* | new H(m) | | | | old H(m) |
* | | = | change_matrices[i] | | |
* | new H(l) | | | | old H(l) |
*
* (These are actually logs of holonomies, so it's correct to
* add -- not multiply -- them.)
*
* For complete Cusps the holonomies should be zero, but that's OK.
*/
for (cusp = manifold->cusp_list_begin.next;
cusp != &manifold->cusp_list_end;
cusp = cusp->next)
if ( cusp->topology == torus_cusp || cusp->topology == Klein_cusp ) /* DJH */
for (i = 0; i < 2; i++) /* i = ultimate, penultimate */
{
old_Hm = cusp->holonomy[i][M];
old_Hl = cusp->holonomy[i][L];
cusp->holonomy[i][M] = complex_plus(
complex_real_mult(
change_matrices[cusp->index][0][0],
old_Hm
),
complex_real_mult(
change_matrices[cusp->index][0][1],
old_Hl
)
);
cusp->holonomy[i][L] = complex_plus(
complex_real_mult(
change_matrices[cusp->index][1][0],
old_Hm
),
complex_real_mult(
change_matrices[cusp->index][1][1],
old_Hl
)
);
}
/*
* Update the cusp_shapes.
*/
for (cusp = manifold->cusp_list_begin.next;
cusp != &manifold->cusp_list_end;
cusp = cusp->next)
if ( cusp->topology == torus_cusp || cusp->topology == Klein_cusp ) /* DJH */
{
cusp->cusp_shape[initial] = transformed_cusp_shape(
cusp->cusp_shape[initial],
change_matrices[cusp->index]);
if (cusp->is_complete == TRUE)
cusp->cusp_shape[current] = transformed_cusp_shape(
cusp->cusp_shape[current],
change_matrices[cusp->index]);
/*
* else cusp->cusp_shape[current] == Zero, and needn't be changed.
*/
}
return func_OK;
}
static void compute_derivative(
Triangulation *manifold)
{
Tetrahedron *tet;
Complex z[3],
d[3],
*eqn_coef = NULL,
dz[2];
EdgeIndex e;
VertexIndex v;
FaceIndex initial_side,
terminal_side;
int init[2][2],
term[2][2];
double m,
l,
a,
b,
*eqn_coef_00 = NULL,
*eqn_coef_01 = NULL,
*eqn_coef_10 = NULL,
*eqn_coef_11 = NULL;
int i,
j;
for (tet = manifold->tet_list_begin.next;
tet != &manifold->tet_list_end;
tet = tet->next)
{
/*
* Note the three edge parameters.
*/
for (i = 0; i < 3; i++)
z[i] = tet->shape[filled]->cwl[ultimate][i].rect;
/*
* Set the derivatives of log(z0), log(z1) and log(z2)
* with respect to the given coordinate system, as
* indicated by the above table.
*/
switch (tet->coordinate_system)
{
case 0:
d[0] = One;
d[1] = complex_div(MinusOne, z[2]);
d[2] = complex_minus(Zero, z[1]);
break;
case 1:
d[0] = complex_minus(Zero, z[2]);
d[1] = One;
d[2] = complex_div(MinusOne, z[0]);
break;
case 2:
d[0] = complex_div(MinusOne, z[1]);
d[1] = complex_minus(Zero, z[0]);
d[2] = One;
break;
}
/*
* Record this tetrahedron's contribution to the edge equations.
*/
for (e = 0; e < 6; e++) /* Look at each of the six edges. */
{
/*
* Find the matrix entry(ies) corresponding to the
* derivative of the edge equation with respect to this
* tetrahedron. If the manifold is oriented it will be
* a single entry in the complex matrix. If the manifold
* is unoriented it will be a 2 x 2 block in the real matrix.
*/
if (manifold->orientability == oriented_manifold || manifold->orientability == oriented_orbifold ) /* DJH */
eqn_coef = &tet->edge_class[e]->complex_edge_equation[tet->index];
else
{
eqn_coef_00 = &tet->edge_class[e]->real_edge_equation_re[2 * tet->index];
eqn_coef_01 = &tet->edge_class[e]->real_edge_equation_re[2 * tet->index + 1];
eqn_coef_10 = &tet->edge_class[e]->real_edge_equation_im[2 * tet->index];
eqn_coef_11 = &tet->edge_class[e]->real_edge_equation_im[2 * tet->index + 1];
}
/*
* Add in the derivative of the log of the edge parameter
* with respect to the chosen coordinate system. Please
* see the comment preceding this function for details.
*/
if (manifold->orientability == oriented_manifold || manifold->orientability == oriented_orbifold ) /* DJH */
*eqn_coef = complex_plus(*eqn_coef, d[edge3[e]]);
else
{
/*
* These are the same a and b as in the comment
* preceding this function.
*/
a = d[edge3[e]].real;
b = d[edge3[e]].imag;
if (tet->edge_orientation[e] == right_handed)
{
*eqn_coef_00 += a;
*eqn_coef_01 -= b;
*eqn_coef_10 += b;
*eqn_coef_11 += a;
}
else
{
*eqn_coef_00 -= a;
*eqn_coef_01 += b;
*eqn_coef_10 += b;
*eqn_coef_11 += a;
}
}
}
/*
* Record this tetrahedron's contribution to the cusp equations.
*/
for (v = 0; v < 4; v++) /* Look at each ideal vertex. */
{
/*
* Note the Dehn filling coefficients on this cusp.
* If the cusp is complete, use m = 1.0 and l = 0.0.
*/
if (tet->cusp[v]->is_complete) /* DJH : not sure ? */
{
m = 1.0;
l = 0.0;
}
else
{
m = tet->cusp[v]->m;
l = tet->cusp[v]->l;
}
/*
* Find the matrix entry(ies) corresponding to the
* derivative of the cusp equation with respect to this
* tetrahedron. If the manifold is oriented it will be
* a single entry in the complex matrix. If the manifold
* is unoriented it will be a 2 x 2 block in the real matrix.
*/
/* DJH */
if ( (manifold->orientability == oriented_manifold || manifold->orientability == oriented_orbifold ) && (
tet->cusp[v]->topology == torus_cusp || tet->cusp[v]->topology == Klein_cusp ) )
eqn_coef = &tet->cusp[v]->complex_cusp_equation[tet->index];
else
{
eqn_coef_00 = &tet->cusp[v]->real_cusp_equation_re[2 * tet->index];
eqn_coef_01 = &tet->cusp[v]->real_cusp_equation_re[2 * tet->index + 1];
eqn_coef_10 = &tet->cusp[v]->real_cusp_equation_im[2 * tet->index];
eqn_coef_11 = &tet->cusp[v]->real_cusp_equation_im[2 * tet->index + 1];
}
/*
* Each ideal vertex contains two triangular cross sections,
* one right_handed and the other left_handed. We want to
* compute the contribution of each angle of each triangle
* to the holonomy. We begin by considering the right_handed
* triangle, looking at each of its three angles. A directed
* angle is specified by its initial and terminal sides.
* We find the number of strands of the Dehn filling curve
* passing from the initial side to the terminal side;
* it is m * (number of strands of meridian)
* + l * (number of strands of longitude), where (m,l) are
* the Dehn filling coefficients (in practice, m and l need
* not be integers, but it's simpler to imagine them to be
* integers as you try to understand the following code).
* The number of strands of the Dehn filling curves passing
* from the initial to the terminal side is multiplied by
* the derivative of the log of the complex angle, to yield
* the contribution to the derivative matrix. If the manifold
* is oriented, that complex number is added directly to
* the relevant matrix entry. If the manifold is unoriented,
* we convert the complex number to a 2 x 2 real matrix
* (cf. the comments preceding this function) and add it to
* the appropriate 2 x 2 block of the real derivative matrix.
* The 2 x 2 matrix for the left_handed triangle is modified
* to account for the fact that although the real part of the
* derivative of the log (i.e. the compression/expansion
* factor) is the same, the imaginary part (i.e. the rotation)
* is negated. [Note that in computing the edge equations
* the real part was negated, while for the cusp equations
* the imaginary part is negated. I will leave an explanation
* of the difference as an exercise for the reader.]
*
* Note that we cannot possibly handle curves on the
* left_handed sheet of the orientation double cover of
* a cusp of an oriented manifold. The reason is that the
* log of the holonomy of the Dehn filling curve is not
* a complex analytic function of the shape of the tetrahedron
* (it's the complex conjugate of such a function). I.e.
* it doesn't have a derivative in the complex sense. This
* is why we make the convention that all peripheral curves
* in oriented manifolds lie on the right_handed sheet of
* the double cover.
*/
for (initial_side = 0; initial_side < 4; initial_side++)
{
if (initial_side == v)
continue;
terminal_side = remaining_face[v][initial_side];
/*
* Note the intersection numbers of the meridian and
* longitude with the initial and terminal sides.
*/
for (i = 0; i < 2; i++) { /* which curve */
for (j = 0; j < 2; j++) { /* which sheet */
init[i][j] = tet->curve[i][j][v][initial_side];
term[i][j] = tet->curve[i][j][v][terminal_side];
}
}
/*
* For each triangle (right_handed and left_handed),
* multiply the number of strands of the Dehn filling
* curve running from initial_side to terminal_side
* by the derivative of the log of the edge parameter.
*/
for (i = 0; i < 2; i++) /* which sheet */
dz[i] = complex_real_mult(
m * FLOW(init[M][i],term[M][i]) + l * FLOW(init[L][i],term[L][i]),
d[ edge3_between_faces[initial_side][terminal_side] ]
);
/*
* If the manifold is oriented, the Dehn filling curve
* must lie of the right_handed sheet of the orientation
* double cover (cf. above). Add its contributation to
* the cusp equation.
*/
/* DJH */
if (manifold->orientability == oriented_manifold || manifold->orientability == oriented_orbifold )
*eqn_coef = complex_plus(*eqn_coef, dz[right_handed]);
/* "else" follows below */
/*
* If the manifold is unoriented, treat the right_ and
* left_handed sheets separately. Add in the contribution
* of the right_handed sheet normally. For the left_handed
* sheet, we must account for the fact that even though
* the modulus of the derivative (i.e. the expansion/
* contraction factor) is correct, its argument (i.e. the
* angle of rotation) is the negative of what it should be.
*/
else
{
a = dz[right_handed].real;
b = dz[right_handed].imag;
*eqn_coef_00 += a;
*eqn_coef_01 -= b;
*eqn_coef_10 += b;
*eqn_coef_11 += a;
a = dz[left_handed].real;
b = dz[left_handed].imag;
*eqn_coef_00 += a;
*eqn_coef_01 -= b;
*eqn_coef_10 -= b;
*eqn_coef_11 -= a;
}
}
}
}
}
static void compute_rhs(
Triangulation *manifold)
{
EdgeClass *edge;
Cusp *cusp;
Complex desired_holonomy,
current_holonomy,
rhs;
/*
* The right hand side of each equation will be the desired value
* of the edge angle sum or the holonomy (depending on whether it's
* an edge equation or a cusp equation) minus the current value.
* Thus, when the equations are solved and the Shapes of the
* Tetrahedra are updated, the edge angle sums and the holonomies
* will take on their desired values, to the accuracy of the
* linear approximation.
*/
/*
* Set the right hand side (rhs) of each edge equation.
*/
for ( edge = manifold->edge_list_begin.next;
edge != &manifold->edge_list_end;
edge = edge->next)
{
/*
* The desired value of the sum of the logs of the complex
* edge parameters is 2 pi i. The current value is
* edge->edge_angle_sum.
*/
rhs = complex_minus(TwoPiI_on[ (int) edge->singular_order], edge->edge_angle_sum); /* DJH */
if (manifold->orientability == oriented_manifold || manifold->orientability == oriented_orbifold )
edge->complex_edge_equation[manifold->num_tetrahedra] = rhs;
else
{
edge->real_edge_equation_re[2 * manifold->num_tetrahedra] = rhs.real;
edge->real_edge_equation_im[2 * manifold->num_tetrahedra] = rhs.imag;
}
}
/*
* Set the right hand side (rhs) of each cusp equation.
*/
for (cusp = manifold->cusp_list_begin.next;
cusp != &manifold->cusp_list_end;
cusp = cusp->next) /* DJH */
if ( cusp->topology == torus_cusp || cusp->topology == Klein_cusp )
{
/*
* For complete cusps we want the log of the holonomy of the
* meridian to be zero. For Dehn filled cusps we want the
* log of the holonomy of the Dehn filling curve to be 2 pi i.
*/
if (cusp->is_complete)
{
desired_holonomy = Zero;
current_holonomy = cusp->holonomy[ultimate][M];
}
else
{
desired_holonomy = TwoPiI;
current_holonomy = complex_plus(
complex_real_mult(cusp->m, cusp->holonomy[ultimate][M]),
complex_real_mult(cusp->l, cusp->holonomy[ultimate][L])
);
}
rhs = complex_minus(desired_holonomy, current_holonomy);
if (manifold->orientability == oriented_manifold || manifold->orientability == oriented_orbifold ) /* DJH */
cusp->complex_cusp_equation[manifold->num_tetrahedra] = rhs;
else
{
cusp->real_cusp_equation_re[2 * manifold->num_tetrahedra] = rhs.real;
cusp->real_cusp_equation_im[2 * manifold->num_tetrahedra] = rhs.imag;
}
}
}
void compute_core_geodesic(
Cusp *cusp,
int *singularity_index,
Complex length[2])
{
int i;
long int positive_d,
negative_c;
Real pi_over_n;
/*
* If the Cusp is unfilled or the Dehn filling coefficients aren't
* integers, then just write in some zeros (as explained at the top
* of this file) and return.
*/
if (cusp->is_complete == TRUE
|| Dehn_coefficients_are_integers(cusp) == FALSE)
{
*singularity_index = 0;
length[ultimate] = Zero;
length[penultimate] = Zero;
return;
}
/*
* The euclidean_algorithm() will give the singularity index
* directly (as the g.c.d.), and the coefficients lead to the
* complex length (cf. the explanation at the top of this file).
*/
*singularity_index = euclidean_algorithm(
(long int) cusp->m,
(long int) cusp->l,
&positive_d,
&negative_c);
for (i = 0; i < 2; i++) /* i = ultimate, penultimate */
{
/*
* length[i] = c H(m) + d H(l)
*
* (The holonomies are already in logarithmic form.)
*/
length[i] = complex_plus(
complex_real_mult(
(Real) (double)(- negative_c),
cusp->holonomy[i][M]
),
complex_real_mult(
(Real) (double) positive_d,
cusp->holonomy[i][L]
)
);
/*
* Make sure the length is positive.
*/
if (length[i].real < 0.0)
length[i] = complex_negate(length[i]);
/*
* We want to normalize the torsion to the range
* [-pi/n + epsilon, pi/n + epsilon], where n is
* the order of the singular locus.
*/
pi_over_n = PI / *singularity_index;
while (length[i].imag < - pi_over_n + TORSION_EPSILON)
length[i].imag += 2.0 * pi_over_n;
while (length[i].imag > pi_over_n + TORSION_EPSILON)
length[i].imag -= 2.0 * pi_over_n;
/*
* In the case of a Klein bottle cusp, H(m) will be purely
* rotational and H(l) will be purely translational
* (cf. the documentation at the top of holonomy.c).
* But the longitude used in practice is actually the
* double cover of the true longitude, so we have to
* divide the core_length by two to compensate.
*/
if (cusp->topology == Klein_cusp)
length[i].real /= 2.0;
}
}
