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import numpy as np
def psign(p1, p2, p3):
return (p1[0] - p3[0]) * (p2[1] - p3[1]) - (p2[0] - p3[0]) * (p1[1] - p3[1]);
def PointInTriangle(pt, v1, v2, v3):
d1 = psign(pt, v1, v2)
d2 = psign(pt, v2, v3)
d3 = psign(pt, v3, v1)
has_neg = (d1 < 0) | (d2 < 0) | (d3 < 0)
has_pos = (d1 > 0) | (d2 > 0) | (d3 > 0)
return not (has_neg & has_pos)
def get_edges(tri):
return [(tri[0], tri[1]), (tri[0], tri[2]), (tri[1], tri[2])]
def default_add(d, e, V, ti):
if e in d:
d[e][1].append(ti)
else:
a, b = e
length = np.linalg.norm(V[a] - V[b])
if a == b:
raise ValueError('Adding edge with same index:', e, ti)
d[e] = [length, [ti]]
return d
def get_all_edges(V, T):
E_dict = {}
for n, tri in enumerate(T):
E = get_edges(tri) # outputs list of edges with sorted vertex indices. eg [(4,7),(4,8),(7,8)]
for e in E:
a, b = e
length = np.linalg.norm(V[a] - V[b])
E_dict = default_add(E_dict, e, V, n)
return E_dict
def split_long_edges(V, T):
E_dict = get_all_edges(V, T)
target_length = np.median([length for length, _ in E_dict.values()])
run = True
n_init_V = len(V)
while run:
run = False
max_length = 0
for cur_e in E_dict.keys():
cur_length, cur_Ti = E_dict[cur_e]
if cur_length > max_length:
max_length = cur_length
e = cur_e
Ti = cur_Ti
a, b = cur_e
if max_length > target_length:
run = True
# make a new vertex
Vnew = V[a] + V[b]
N = 2.0
Vnew = Vnew / N
new_length = np.linalg.norm(Vnew - V[a])
n = len(V)
V = np.concatenate([V, [Vnew]], axis=0)
for ti in sorted(Ti)[::-1]:
# find the vertex index which a,b is not co-linear with
m = [i for i in T[ti] if i != a and i != b][0]
old = np.pad(np.array([V[ii] for ii in T[ti]]), ((1,0), (0,0)), mode='wrap')+1
new1 = np.pad(np.array([V[ii] for ii in sorted([m, n, a])]), ((1,0), (0,0)), mode='wrap')+2
new2 = np.pad(np.array([V[ii] for ii in sorted([m, n, b])]), ((1,0), (0,0)), mode='wrap')
T[ti] = sorted([m, n, a]) # ADD m,n,a REM a,b,m
tj = len(T)
T.append(sorted([m, n, b])) # ADD m,n,b
# add new edges
an = tuple(sorted([n, a]))
am = tuple(sorted([m, a]))
bn = tuple(sorted([n, b]))
bm = tuple(sorted([m, b]))
mn = tuple(sorted([m, n]))
E_dict[tuple(sorted([m, b]))][1] = [tk for tk in E_dict[tuple(sorted([m, b]))][1] if tk != ti]
cross_length = np.linalg.norm(V[m] - V[n])
E_dict = default_add(E_dict, an, V, ti) # ADD ti to an
E_dict = default_add(E_dict, bn, V, tj) # ADD tj to bn
E_dict = default_add(E_dict, bm, V, tj) # ADD tj to bm
E_dict = default_add(E_dict, mn, V, ti) # ADD ti to mn
E_dict = default_add(E_dict, mn, V, tj) # ADD tj to mn
del E_dict[e]
interior_vertex_indices = set(range(len(V)))
for e in E_dict.keys():
if len(E_dict[e][1]) == 1:
interior_vertex_indices = interior_vertex_indices - set(e)
# Smooth interior triangulated points
for _ in range(100):
for i in interior_vertex_indices:
Vs = []
for e in E_dict.keys():
if i in e:
vn = [j for j in e if j != i][0]
Vs.append(V[vn])
Vnew = np.mean(Vs, axis=0)
V[i] = 0.7 * V[i] + 0.3 * Vnew
return V, T
def polygon_triangulate(P):
N = len(P)
P3 = np.pad(P, ((0,0), (0,1)))
angle_sum = 0
angles = []
exterior_angles = []
T = []
for p1, p2, p3 in zip(P3, np.roll(P3, -1, axis=0), np.roll(P3, -2, axis=0)):
v1 = p2 - p1
v2 = p3 - p2
cross = np.cross(v1, v2)
dot = np.clip(np.dot(v1, v2) / (np.linalg.norm(v1) * np.linalg.norm(v2)), -1, 1)
angle = np.sign(cross[2])*np.arccos(dot)
exterior_angles.append(angle/(2*np.pi)*360)
indices = np.arange(N)
if np.sum(exterior_angles) < 0:
indices = indices[::-1]
# iteratively add a trinagle and remove the index to the point which can no longer be part of another triangle
while len(indices) > 2:
best_i = 0
best_j = 0
best_k = 0
best_n = 0
best_length = np.inf
# iterate over the remaining points to find the best candidate
for n, (i,j,k), in enumerate(zip(np.roll(indices, 1), indices, np.roll(indices, -1))):
v1 = P3[j] - P3[i]
v2 = P3[k] - P3[j]
cross = np.cross(v1, v2)
angle = np.arcsin(cross[2] / (np.linalg.norm(v1) * np.linalg.norm(v2)))
length = np.linalg.norm(v1) + np.linalg.norm(v2)
# pick the shortest triangle which is interior
if angle > 0 and length < best_length:
# check all point to see if one lies inside the triangle, skip in that case
contains_point = False
for nn in range(N):
if nn == i or nn == j or nn == k:
continue
if PointInTriangle(P[nn], P[i], P[j], P[k]):
contains_point = True
break
if contains_point:
continue
best_length = length
best_i = i
best_j = j
best_k = k
best_n = n
# add triangle and remove index to point which can no longer be part of another triangle
T.append(sorted([best_i, best_j, best_k]))
indices = np.delete(indices, best_n)
P, T = split_long_edges(P, T)
return exterior_angles, P, T
class PlotTriangulation:
colors = np.random.uniform(0.2,1,(2048, 3))
def __init__(self, V, T, add_labels=True):
import matplotlib.pyplot as plt
V0 = np.copy(V)
if np.argmax(np.max(V0, axis=0)) == 1:
V0 = V0[:,::-1]
margins = (np.max(V0, axis=0) - np.min(V0, axis=0))*0.05
V0 = V0 - np.min(V0, axis=0) + margins
I = np.zeros((np.max(V0[:,1] + 1 + margins[1]).astype(np.int32), np.max(V0[:,0] + 1 + margins[0]).astype(np.int32)))
plt.imshow(I, cmap='gray')
plt.axis('off')
plt.subplots_adjust(bottom=0, top=1, left=0,right=1)
for j in range(len(T)):
tri = T[j]
t1 = plt.Polygon(V0[tri,:], alpha=0.7, color=PlotTriangulation.colors[j%2048])
plt.gca().add_patch(t1)
if add_labels:
plt.text(np.mean(V0[tri,0]), np.mean(V0[tri,1]), str(j))
if add_labels:
for i, v in enumerate(V0):
plt.text(v[0], v[1], str(i), color='orange')