void choose_generators(
Triangulation *manifold,
Boolean compute_corners,
Boolean centroid_at_origin)
{
/*
* To compute the corners we need some sort of geometric structure.
*/
if (compute_corners == TRUE
&& manifold->solution_type[filled] == not_attempted)
uFatalError("choose_generators", "choose_generators.c");
/*
* For each Tetrahedron tet, set tet->flag to unknown_orientation
* to indicate that the Tetrahedron has not yet been visited, and
* set each tet->generator_status[i] to unassigned_generator to
* indicate that no generator has yet been assigned to any face.
*/
initialize_flags(manifold);
/*
* Start a recursion which visits each tetrahedron, assigns
* generators to its faces, and recursively visits any unvisited
* neighbors.
*/
visit_tetrahedra(manifold, compute_corners, centroid_at_origin);
/*
* The number_of_generators should be one plus the number of tetrahedra.
*/
if (manifold->num_generators != manifold->num_tetrahedra + 1)
uFatalError("choose_generators", "choose_generators");
/*
* At this point we have a valid set of generators, but it's
* not as simple as it might be. We'll perform two types of
* simplifications. First we need to count how many of the
* faces incident to each EdgeClass correspond to active generators.
* Initialize all the active_relation flags to TRUE while we're at it.
*/
count_incident_generators(manifold);
/* Now look for EdgeClasses in the Triangulation (2-cells in the
* dual complex) which show that a single generator is homotopically
* trivial, and eliminate the trivial generator. Topologically, this
* corresponds to folding together two adjacent triangular faces
* on the boundary of the fundamental domain (the close-the-book
* move). Geometrically, this corresponds to realizing that two
* faces of the (geometric) fundamental domain are in fact already
* superimposed on each other. In Heegaard terms, it's a handle
* cancellation.
*/
eliminate_trivial_generators(manifold);
/*
* At this point the boundary of the fundamental domain is likely
* to contain groups of faces which are essentially n-gons (n > 3)
* arbitrarily divided into triangles. The generators for such
* triangular faces are all equivalent, and can be merged. They
* can be recognized by looking for EdgeClasses with exactly two
* incident (and distinct) generators.
*/
merge_equivalent_generators(manifold);
}
void choose_generators(
Triangulation *manifold,
Boolean compute_corners,
Boolean centroid_at_origin)
{
/*
* To compute the corners we need some sort of geometric structure.
*/
if (compute_corners == TRUE
&& manifold->solution_type[filled] == not_attempted)
uFatalError("choose_generators", "choose_generators.c");
/*
* For each Tetrahedron tet, set tet->flag to unknown_orientation
* to indicate that the Tetrahedron has not yet been visited, and
* set each tet->generator_status[i] to unassigned_generator to
* indicate that no generator has yet been assigned to any face.
*/
initialize_flags(manifold);
/*
* Start a recursion which visits each tetrahedron, assigns
* generators to its faces, and recursively visits any unvisited
* neighbors.
*/
visit_tetrahedra(manifold, compute_corners, centroid_at_origin);
/*
* The number_of_generators should be one plus the number of tetrahedra.
*/
if (manifold->num_generators != manifold->num_tetrahedra + 1)
uFatalError("choose_generators", "choose_generators.c");
/*
* At this point we have a valid set of generators, but it's
* not as simple as it might be. We'll perform two types of
* simplifications. First we need to count how many of the
* faces incident to each EdgeClass correspond to active generators.
* Initialize all the active_relation flags to TRUE while we're at it.
*/
count_incident_generators(manifold);
/*
* Now look for EdgeClasses in the Triangulation (2-cells in the
* dual complex) which show that a single generator is homotopically
* trivial, and eliminate the trivial generator. Topologically, this
* corresponds to folding together two adjacent triangular faces
* on the boundary of the fundamental domain (the close-the-book
* move). Geometrically, this corresponds to realizing that two
* faces of the (geometric) fundamental domain are in fact already
* superimposed on each other. In Heegaard terms, it's a handle
* cancellation.
*/
eliminate_trivial_generators(manifold);
/*
* At this point the boundary of the fundamental domain is likely
* to contain groups of faces which are essentially n-gons (n > 3)
* arbitrarily divided into triangles. The generators for such
* triangular faces are all equivalent, and can be merged. They
* can be recognized by looking for EdgeClasses with exactly two
* incident (and distinct) generators.
*/
merge_equivalent_generators(manifold);
/*
* 2008/6/12 JRW
*
* Eliminate relations with zero generators.
*
* How can such relations arise?
*
* Under normal operation, eliminate_trivial_generators() finds an active generator
* whose dual 2-cell is incident to an EdgeClass whose other incident 2-cells are
* all dual to inactive generators (i.e. they lie in the interior
* of the fundamental domain). The EdgeClass's relation (of length 1) cancels
* the generator and all is well. One may visualize this operation as taking
* two adjacent triangles on the boundary of the fundamental domain, which share
* a common edge, and gluing them together via a "close-the-book move".
*
* Now consider two adjacent triangles (still on the boundary of the fundamental
* domain) that share two common edges -- in effect a sort of triangular "pita pocket"
* with two closed edges and one open edge. When eliminate_trivial_generators()
* cancels the generator (dual to the triangular face) against one of the
* incident EdgeClasses, the other EdgeClass is left with zero generators,
* and may be eliminated.
*
* Note: Thinking in term of truncated tetrahedra, the above description seems
* to imply that the boundary component at the tip of the pita pocket
* between the two "closed edges" becomes a spherical boundary component
* of the manifold. This suggests that this case would arise for finite
* triangulations (as opposed to ideal triangulations), or for highly degenerate
* ideal triangulations, but I confess that I haven't thought this through carefully.
*/
eliminate_empty_relations(manifold);
}
