def pseudo_quad_split(mesh, fkey):
face_vertices = mesh.face_vertices(fkey)
if len(face_vertices) != 4:
return None
pole = None
for i in range(len(face_vertices) - 1):
if face_vertices[i - 1] == face_vertices[i]:
pole = face_vertices[i]
if pole is None:
return None
face_vertices.remove(pole)
idx = face_vertices.index(pole)
a = pole
b = mesh.face_vertex_descendant(fkey, a)
c = mesh.face_vertex_descendant(fkey, b)
mesh.delete_face(fkey)
d = add_vertex_from_vertices(mesh, [a, b], [1, 1])
e = add_vertex_from_vertices(mesh, [b, c], [1, 1])
f = add_vertex_from_vertices(mesh, [c, a], [1, 1])
g = add_vertex_from_vertices(mesh, [a, b, c], [1, 1, 1])
fkey_1 = mesh.add_face([a, d, g, f])
fkey_2 = mesh.add_face([b, e, g, d])
fkey_3 = mesh.add_face([c, f, g, e])
insert_vertices_in_halfedge(mesh, b, a, [d])
insert_vertices_in_halfedge(mesh, c, b, [e])
insert_vertices_in_halfedge(mesh, a, c, [f])
return fkey_1, fkey_2, fkey_3
def vertex_pole(mesh, fkey, pole):
if len(mesh.face_vertices(fkey)) != 4:
return None
if pole not in mesh.face_vertices(fkey):
return None
a = pole
b = mesh.face_vertex_descendant(fkey, a)
c = mesh.face_vertex_descendant(fkey, b)
d = mesh.face_vertex_descendant(fkey, c)
e = add_vertex_from_vertices(mesh, [a, b], [1, 2])
f = add_vertex_from_vertices(mesh, [b, c], [1, 2])
g = add_vertex_from_vertices(mesh, [c, d], [2, 1])
h = add_vertex_from_vertices(mesh, [d, a], [2, 1])
i = add_vertex_from_vertices(mesh, [a, b, c, d], [1, 1, 2, 1])
mesh.delete_face(fkey)
fkey_1 = mesh.add_face([a, a, i, h])
fkey_2 = mesh.add_face([a, a, e, i])
fkey_3 = mesh.add_face([e, b, f, i])
fkey_4 = mesh.add_face([h, i, g, d])
fkey_5 = mesh.add_face([i, f, c, g])
insert_vertices_in_halfedge(mesh, b, a, [e])
insert_vertices_in_halfedge(mesh, c, b, [f])
insert_vertices_in_halfedge(mesh, d, c, [g])
insert_vertices_in_halfedge(mesh, a, d, [h])
return fkey_1, fkey_2, fkey_3, fkey_4, fkey_5
def flat_corner_3(mesh, fkey, corner):
if len(mesh.face_vertices(fkey)) != 4:
return None
if corner not in mesh.face_vertices(fkey):
return None
a = corner
b = mesh.face_vertex_descendant(fkey, a)
c = mesh.face_vertex_descendant(fkey, b)
d = mesh.face_vertex_descendant(fkey, c)
e = add_vertex_from_vertices(mesh, [b, c], [1, 1])
f = add_vertex_from_vertices(mesh, [c, d], [1, 1])
g = add_vertex_from_vertices(mesh, [a, b, c, d], [1, 1, 1, 1])
mesh.delete_face(fkey)
fkey_1 = mesh.add_face([a, b, e, g])
fkey_2 = mesh.add_face([e, c, f, g])
fkey_3 = mesh.add_face([d, a, g, f])
insert_vertices_in_halfedge(mesh, c, b, [e])
insert_vertices_in_halfedge(mesh, d, c, [f])
return fkey_1, fkey_2, fkey_3
def singular_boundary_minus_1(mesh, fkey, vkey):
if vkey not in mesh.face_vertices(fkey):
return None
a = vkey
b = mesh.face_vertex_descendant(fkey, a)
c = mesh.face_vertex_descendant(fkey, b)
d = mesh.face_vertex_descendant(fkey, c)
fkey_1 = mesh.halfedge[c][b]
e = mesh.face_vertex_descendant(fkey_1, b)
f = mesh.face_vertex_descendant(fkey_1, e)
fkey_2 = mesh.halfedge[d][c]
g = mesh.face_vertex_descendant(fkey_2, c)
h = mesh.face_vertex_descendant(fkey_2, g)
mesh.delete_face(fkey)
mesh.delete_face(fkey_1)
mesh.delete_face(fkey_2)
i = add_vertex_from_vertices(mesh, [c, f], [1, 1])
j = add_vertex_from_vertices(mesh, [c, g], [1, 2])
fkey_1 = mesh.add_face([a, c, j, d])
fkey_2 = mesh.add_face([a, b, i, c])
fkey_3 = mesh.add_face([d, j, g, h])
fkey_4 = mesh.add_face([b, e, f, i])
insert_vertices_in_halfedge(mesh, c, f, [i])
insert_vertices_in_halfedge(mesh, g, c, [j])
return fkey_1, fkey_2, fkey_3, fkey_4
def double_split(mesh, fkey):
if len(mesh.face_vertices(fkey)) != 4:
return None
a, b, c, d = mesh.face_vertices(fkey)
e = add_vertex_from_vertices(mesh, [a, b], [1, 1])
f = add_vertex_from_vertices(mesh, [b, c], [1, 1])
g = add_vertex_from_vertices(mesh, [c, d], [1, 1])
h = add_vertex_from_vertices(mesh, [d, a], [1, 1])
i = add_vertex_from_vertices(mesh, [a, b, c, d], [1, 1, 1, 1])
mesh.delete_face(fkey)
fkey_1 = mesh.add_face([a, e, i, h])
fkey_2 = mesh.add_face([e, b, f, i])
fkey_3 = mesh.add_face([h, i, g, d])
fkey_4 = mesh.add_face([i, f, c, g])
insert_vertices_in_halfedge(mesh, b, a, [e])
insert_vertices_in_halfedge(mesh, c, b, [f])
insert_vertices_in_halfedge(mesh, d, c, [g])
insert_vertices_in_halfedge(mesh, a, d, [h])
return fkey_1, fkey_2, fkey_3, fkey_4
def simple_split(mesh, fkey, edge):
u, v = edge
if len(mesh.face_vertices(fkey)) != 4:
return None
if (u, v) not in mesh.face_halfedges(fkey) and (v, u) not in mesh.face_halfedges(fkey):
return None
pole = None
face_vertices = mesh.face_vertices(fkey)
for fkey_2 in face_vertices:
if face_vertices.count(fkey_2) == 2:
pole = fkey_2
break
if v in mesh.halfedge[u] and mesh.halfedge[u][v] == fkey:
a = u
else:
a = v
if pole is None:
b = mesh.face_vertex_descendant(fkey, a)
c = mesh.face_vertex_descendant(fkey, b)
d = mesh.face_vertex_descendant(fkey, c)
else:
if pole in edge:
if pole == a:
b = [u, v]
b.remove(pole)
b = b[0]
c = mesh.face_vertex_descendant(fkey, b)
d = mesh.face_vertex_descendant(fkey, c)
else:
b = mesh.face_vertex_descendant(fkey, a)
c = b
d = mesh.face_vertex_ancestor(fkey, a)
else:
b = mesh.face_vertex_descendant(fkey, a)
c = mesh.face_vertex_descendant(fkey, b)
d = c
e = add_vertex_from_vertices(mesh, [a, b], [1, 1])
f = add_vertex_from_vertices(mesh, [c, d], [1, 1])
mesh.delete_face(fkey)
fkey_1 = mesh.add_face([a, e, f, d])
fkey_2 = mesh.add_face([e, b, c, f])
if b != a:
insert_vertices_in_halfedge(mesh, b, a, [e])
if d != c:
insert_vertices_in_halfedge(mesh, d, c, [f])
return fkey_1, fkey_2
def remove_tri(mesh, fkey_tri, fkey_quad, pole, t = .5):
if len(mesh.face_vertices(fkey_tri)) != 3 or len(mesh.face_vertices(fkey_quad)) != 4:
return None
if fkey_tri not in mesh.face_neighbours(fkey_quad):
return None
if pole not in mesh.face_vertices(fkey_tri) or pole not in mesh.face_vertices(fkey_quad):
return None
nbr = mesh.face_vertex_descendant(fkey_quad, pole)
if nbr not in mesh.face_vertices(fkey_tri):
d = pole
e = nbr
a = mesh.face_vertex_descendant(fkey_quad, e)
b = mesh.face_vertex_descendant(fkey_quad, a)
c = mesh.face_vertex_descendant(fkey_tri, b)
mesh.delete_face(fkey_tri)
mesh.delete_face(fkey_quad)
f = add_vertex_from_vertices(mesh, [d, e], [1 - t, t])
fkey_tri = mesh.add_face([a, b, f, e])
fkey_quad = mesh.add_face([b, c, d, f])
insert_vertices_in_halfedge(mesh, e, d, [f])
else:
b = pole
d = nbr
e = mesh.face_vertex_descendant(fkey_quad, d)
a = mesh.face_vertex_descendant(fkey_quad, e)
c = mesh.face_vertex_descendant(fkey_tri, b)
mesh.delete_face(fkey_tri)
mesh.delete_face(fkey_quad)
f = add_vertex_from_vertices(mesh, [b, a], [1 - t, t])
fkey_tri = mesh.add_face([a, f, d, e])
fkey_quad = mesh.add_face([b, c, d, f])
insert_vertices_in_halfedge(mesh, b, a, [f])
return fkey_tri, fkey_quad
def primitive_add_vertex(mesh, edge):
"""Add a vertex on an edge adjacent to triangles.
Parameters
----------
mesh : Mesh
A mesh.
edge: tuple
Edge vertex keys.
Returns
-------
x : int
The new vertex.
Raises
------
-
"""
u, v = edge
# new vertex
x = add_vertex_from_vertices(mesh, [u, v], [1, 1])
# update adjacent faces
insert_vertices_in_halfedge(mesh, u, v, [x])
insert_vertices_in_halfedge(mesh, v, u, [x])
return x
def singular_boundary_2(mesh, edge, vkey):
u, v = edge
if vkey != u and vkey != v:
return None
if mesh.is_edge_on_boundary(u, v):
return None
if len(mesh.face_vertices(mesh.halfedge[u][v])) != 4 or len(mesh.face_vertices(mesh.halfedge[v][u])) != 4:
return None
if vkey == u:
b = u
e = v
else:
b = v
e = u
fkey_1 = mesh.halfedge[b][e]
fkey_2 = mesh.halfedge[e][b]
f = mesh.face_vertex_descendant(fkey_1, e)
a = mesh.face_vertex_descendant(fkey_1, f)
c = mesh.face_vertex_descendant(fkey_2, b)
d = mesh.face_vertex_descendant(fkey_2, c)
mesh.delete_face(fkey_1)
mesh.delete_face(fkey_2)
g = add_vertex_from_vertices(mesh, [d, e], [2, 1])
h = add_vertex_from_vertices(mesh, [d, e], [1, 2])
i = add_vertex_from_vertices(mesh, [e, f], [2, 1])
j = add_vertex_from_vertices(mesh, [e, f], [1, 2])
fkey_1 = mesh.add_face([a, b, j, f])
fkey_2 = mesh.add_face([b, e, i, j])
fkey_3 = mesh.add_face([b, g, h, e])
fkey_4 = mesh.add_face([b, c, d, g])
insert_vertices_in_halfedge(mesh, e, d, [h, g])
insert_vertices_in_halfedge(mesh, f, e, [j, i])
return fkey_1, fkey_2, fkey_3
def edge_pole(mesh, fkey, edge):
u, v = edge
if len(mesh.face_vertices(fkey)) != 4:
return None
if (u, v) not in mesh.face_halfedges(fkey) and (v, u) not in mesh.face_halfedges(fkey):
return None
if v in mesh.halfedge[u] and mesh.halfedge[u][v] == fkey:
a = u
else:
a = v
b = mesh.face_vertex_descendant(fkey, a)
c = mesh.face_vertex_descendant(fkey, b)
d = mesh.face_vertex_descendant(fkey, c)
e = add_vertex_from_vertices(mesh, [a, b], [2, 1])
f = add_vertex_from_vertices(mesh, [a, b], [1, 1])
g = add_vertex_from_vertices(mesh, [a, b], [1, 2])
h = add_vertex_from_vertices(mesh, [b, c], [1, 2])
i = add_vertex_from_vertices(mesh, [c, d], [2, 1])
j = add_vertex_from_vertices(mesh, [c, d], [1, 2])
k = add_vertex_from_vertices(mesh, [d, a], [2, 1])
l = add_vertex_from_vertices(mesh, [a, b, c, d], [1, 1, 1, 2])
m = add_vertex_from_vertices(mesh, [a, b, c, d], [1, 1, 2, 1])
mesh.delete_face(fkey)
fkey_1 = mesh.add_face([a, e, l, k])
fkey_2 = mesh.add_face([e, f, f, l])
fkey_3 = mesh.add_face([l, f, f, m])
fkey_4 = mesh.add_face([f, f, g, m])
fkey_5 = mesh.add_face([g, b, h, m])
fkey_6 = mesh.add_face([k, l, j, d])
fkey_7 = mesh.add_face([l, m, i, j])
fkey_8 = mesh.add_face([m, h, c, i])
insert_vertices_in_halfedge(mesh, b, a, [g, f, e])
insert_vertices_in_halfedge(mesh, c, b, [h])
insert_vertices_in_halfedge(mesh, d, c, [j, i])
insert_vertices_in_halfedge(mesh, a, d, [k])
return fkey_1, fkey_2, fkey_3, fkey_4, fkey_5, fkey_6, fkey_7, fkey_8
def close_opening(mesh, vkeys):
# vkeys of boundary component
# sorted and oriented towards opening
if vkeys[0] == vkeys[-1]:
del vkeys[-1]
if len(vkeys) == 4:
return [mesh.add_face(vkeys)]
else:
g = add_vertex_from_vertices(mesh, vkeys, [1] * len(vkeys))
new_vkeys = []
for i in range(len(vkeys)):
u = vkeys[i - 1]
v = vkeys[i]
a = add_vertex_from_vertices(mesh, [u, v], [1, 1])
new_vkeys.append(a)
insert_vertices_in_halfedge(mesh, v, u, [a])
return [mesh.add_face([new_vkeys[i - 1], vkeys[i - 1], new_vkeys[i], g]) for i in range(len(vkeys))]
def add_opening(mesh, fkey):
if len(mesh.face_vertices(fkey)) != 4:
return None
a, b, c, d = mesh.face_vertices(fkey)
e = add_vertex_from_vertices(mesh, [a, b, c, d], [2, 1, 1, 1])
f = add_vertex_from_vertices(mesh, [a, b, c, d], [1, 2, 1, 1])
g = add_vertex_from_vertices(mesh, [a, b, c, d], [1, 1, 2, 1])
h = add_vertex_from_vertices(mesh, [a, b, c, d], [1, 1, 1, 2])
mesh.delete_face(fkey)
fkey_1 = mesh.add_face([a, b, f, e])
fkey_2 = mesh.add_face([b, c, g, f])
fkey_3 = mesh.add_face([c, d, h, g])
fkey_4 = mesh.add_face([d, a, e, h])
return fkey_1, fkey_2, fkey_3, fkey_4
def flat_corner_2(mesh, fkey, corner):
if len(mesh.face_vertices(fkey)) != 4:
return None
if corner not in mesh.face_vertices(fkey):
return None
a = corner
b = mesh.face_vertex_descendant(fkey, a)
c = mesh.face_vertex_descendant(fkey, b)
d = mesh.face_vertex_descendant(fkey, c)
e = add_vertex_from_vertices(mesh, [a, b, c, d], [1, 1, 1, 1])
mesh.delete_face(fkey)
fkey_1 = mesh.add_face([a, b, c, e])
fkey_2 = mesh.add_face([a, e, c, d])
return fkey_1, fkey_2
def split_35_diag(mesh, fkey, corner):
if len(mesh.face_vertices(fkey)) != 4:
return None
if corner not in mesh.face_vertices(fkey):
return None
a = corner
b = mesh.face_vertex_descendant(fkey, a)
c = mesh.face_vertex_descendant(fkey, b)
d = mesh.face_vertex_descendant(fkey, c)
e = add_vertex_from_vertices(mesh, [a, b], [2, 1])
f = add_vertex_from_vertices(mesh, [a, b], [1, 2])
g = add_vertex_from_vertices(mesh, [b, c], [1, 2])
h = add_vertex_from_vertices(mesh, [c, d], [2, 1])
i = add_vertex_from_vertices(mesh, [d, a], [2, 1])
j = add_vertex_from_vertices(mesh, [d, a], [1, 2])
k = add_vertex_from_vertices(mesh, [a, c], [2, 1])
l = add_vertex_from_vertices(mesh, [a, c], [1, 2])
mesh.delete_face(fkey)
fkey_1 = mesh.add_face([a, e, k, j])
fkey_2 = mesh.add_face([e, f, l, k])
fkey_3 = mesh.add_face([f, b, g, l])
fkey_4 = mesh.add_face([j, k, l, i])
fkey_5 = mesh.add_face([i, l, h, d])
fkey_6 = mesh.add_face([l, g, c, h])
insert_vertices_in_halfedge(mesh, b, a, [f, e])
insert_vertices_in_halfedge(mesh, c, b, [g])
insert_vertices_in_halfedge(mesh, d, c, [h])
insert_vertices_in_halfedge(mesh, a, d, [j, i])
return fkey_1, fkey_2, fkey_3, fkey_4, fkey_5, fkey_6
def insert_partial_pole(mesh, fkey, pole, edge):
u, v = edge
if len(mesh.face_vertices(fkey)) != 4:
return None
if pole not in mesh.face_vertices(fkey):
return None
if (u, v) not in mesh.face_halfedges(fkey) and (v, u) not in mesh.face_halfedges(fkey):
return None
if v not in mesh.halfedge[u] or mesh.halfedge[u][v] != fkey:
u, v = v, u
a = pole
b = mesh.face_vertex_descendant(fkey, a)
c = mesh.face_vertex_descendant(fkey, b)
d = mesh.face_vertex_descendant(fkey, c)
if u != b and u != c:
return None
mesh.delete_face(fkey)
e = add_vertex_from_vertices(mesh, [u, v], [1, 1])
if u == b:
fkey_1 = mesh.add_face([a, a, b, e])
fkey_2 = mesh.add_face([a, e, c, d])
if u == c:
fkey_1 = mesh.add_face([a, b, c, e])
fkey_2 = mesh.add_face([a, a, e, d])
insert_vertices_in_halfedge(mesh, v, u, [e])
return fkey_1, fkey_2
def flat_corner_33(mesh, fkey, corner):
if len(mesh.face_vertices(fkey)) != 4:
return None
if corner not in mesh.face_vertices(fkey):
return None
a = corner
b = mesh.face_vertex_descendant(fkey, a)
c = mesh.face_vertex_descendant(fkey, b)
d = mesh.face_vertex_descendant(fkey, c)
e = add_vertex_from_vertices(mesh, [d, a], [1, 1])
f = add_vertex_from_vertices(mesh, [a, b], [1, 1])
g = add_vertex_from_vertices(mesh, [b, c], [1, 1])
h = add_vertex_from_vertices(mesh, [c, d], [1, 1])
i = add_vertex_from_vertices(mesh, [a, b, c, d], [1, 1, 1, 2])
j = add_vertex_from_vertices(mesh, [a, b, c, d], [1, 2, 1, 1])
k = add_vertex_from_vertices(mesh, [a, b, c, d], [1, 1, 1, 1])
mesh.delete_face(fkey)
fkey_1 = mesh.add_face([a, f, j, k])
fkey_2 = mesh.add_face([f, b, g, j])
fkey_3 = mesh.add_face([j, g, c , k])
fkey_4 = mesh.add_face([a, k, i, e])
fkey_5 = mesh.add_face([k, c, h, i])
fkey_6 = mesh.add_face([e, i, h, d])
insert_vertices_in_halfedge(mesh, b, a, [f])
insert_vertices_in_halfedge(mesh, c, b, [g])
insert_vertices_in_halfedge(mesh, d, c, [h])
insert_vertices_in_halfedge(mesh, a, d, [e])
return fkey_1, fkey_2, fkey_3, fkey_4, fkey_5, fkey_6
def face_pole(mesh, fkey):
if len(mesh.face_vertices(fkey)) != 4:
return None
a, b, c, d = mesh.face_vertices(fkey)
e = add_vertex_from_vertices(mesh, [a, b], [2, 1])
f = add_vertex_from_vertices(mesh, [a, b], [1, 2])
g = add_vertex_from_vertices(mesh, [b, c], [2, 1])
h = add_vertex_from_vertices(mesh, [b, c], [1, 2])
i = add_vertex_from_vertices(mesh, [c, d], [2, 1])
j = add_vertex_from_vertices(mesh, [c, d], [1, 2])
k = add_vertex_from_vertices(mesh, [d, a], [2, 1])
l = add_vertex_from_vertices(mesh, [d, a], [1, 2])
m = add_vertex_from_vertices(mesh, [a, b, c, d], [2, 1, 1, 1])
n = add_vertex_from_vertices(mesh, [a, b, c, d], [1, 2, 1, 1])
o = add_vertex_from_vertices(mesh, [a, b, c, d], [1, 1, 2, 1])
p = add_vertex_from_vertices(mesh, [a, b, c, d], [1, 1, 1, 2])
q = add_vertex_from_vertices(mesh, [a, b, c, d], [1, 1, 1, 1])
mesh.delete_face(fkey)
fkey_1 = mesh.add_face([a, e, m, l])
fkey_2 = mesh.add_face([e, f, n, m])
fkey_3 = mesh.add_face([f, b, g, n])
fkey_4 = mesh.add_face([l, m, p, k])
fkey_5 = mesh.add_face([m, q, q, p])
fkey_6 = mesh.add_face([m, n, q, q])
fkey_7 = mesh.add_face([q, q, n, o])
fkey_8 = mesh.add_face([n, g, h, o])
fkey_9 = mesh.add_face([p, q, q, o])
fkey_10 = mesh.add_face([k, p, j, d])
fkey_11 = mesh.add_face([p, o, i, j])
fkey_12 = mesh.add_face([o, h, c, i])
insert_vertices_in_halfedge(mesh, b, a, [f, e])
insert_vertices_in_halfedge(mesh, c, b, [h, g])
insert_vertices_in_halfedge(mesh, d, c, [j, i])
insert_vertices_in_halfedge(mesh, a, d, [l, k])
return fkey_1, fkey_2, fkey_3, fkey_4, fkey_5, fkey_6, fkey_7, fkey_8, fkey_9, fkey_10, fkey_11, fkey_12
def split_35(mesh, fkey, edge):
u, v = edge
if len(mesh.face_vertices(fkey)) != 4:
return None
if (u, v) not in mesh.face_halfedges(fkey) and (v, u) not in mesh.face_halfedges(fkey):
return None
if v in mesh.halfedge[u] and mesh.halfedge[u][v] == fkey:
a = u
else:
a = v
b = mesh.face_vertex_descendant(fkey, a)
c = mesh.face_vertex_descendant(fkey, b)
d = mesh.face_vertex_descendant(fkey, c)
e = add_vertex_from_vertices(mesh, [a, b], [2, 1])
f = add_vertex_from_vertices(mesh, [a, b], [1, 1])
g = add_vertex_from_vertices(mesh, [a, b], [1, 2])
h = add_vertex_from_vertices(mesh, [b, c], [1, 1])
i = add_vertex_from_vertices(mesh, [b, c], [1, 2])
j = add_vertex_from_vertices(mesh, [c, d], [1, 1])
k = add_vertex_from_vertices(mesh, [d, a], [2, 1])
l = add_vertex_from_vertices(mesh, [d, a], [1, 1])
m = add_vertex_from_vertices(mesh, [a, b, c, d], [3, 2, 1, 2])
n = add_vertex_from_vertices(mesh, [a, b, c, d], [2, 2, 1, 1])
o = add_vertex_from_vertices(mesh, [a, b, c, d], [2, 3, 2, 1])
p = add_vertex_from_vertices(mesh, [a, b, c, d], [1, 1, 2, 2])
mesh.delete_face(fkey)
fkey_1 = mesh.add_face([a, e, m, l])
fkey_2 = mesh.add_face([e, f, n, m])
fkey_3 = mesh.add_face([f, g, o, n])
fkey_4 = mesh.add_face([g, b, h, o])
fkey_5 = mesh.add_face([l, m, p, k])
fkey_6 = mesh.add_face([m, n, o, p])
fkey_7 = mesh.add_face([o, h, i, p])
fkey_8 = mesh.add_face([k, p, j, d])
fkey_9 = mesh.add_face([p, i, c, j])
insert_vertices_in_halfedge(mesh, b, a, [g, f, e])
insert_vertices_in_halfedge(mesh, c, b, [i, h])
insert_vertices_in_halfedge(mesh, d, c, [j])
insert_vertices_in_halfedge(mesh, a, d, [l, k])
return fkey_1, fkey_2, fkey_3, fkey_4, fkey_5, fkey_6, fkey_7, fkey_8, fkey_9