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#
# This program is free software; you can redistribute it and/or
# modify it under the terms of the GNU General Public License
# as published by the Free Software Foundation; either version 2
# of the License, or (at your option) any later version.
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# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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# along with this program; if not, write to the Free Software Foundation,
# Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
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#
from . import geom
import math
import random
from math import sqrt
# Points are 3-tuples or 2-tuples of reals: (x,y,z) or (x,y)
# Faces are lists of integers (vertex indices into coord lists)
# After triangulation/quadrangulation, the tris and quads will
# be tuples instead of lists.
# Vmaps are lists taking vertex index -> Point
TOL = 1e-7 # a tolerance for fuzzy equality
GTHRESH = 75 # threshold above which use greedy to _Quandrangulate
ANGFAC = 1.0 # weighting for angles in quad goodness measure
DEGFAC = 10.0 # weighting for degree in quad goodness measure
# Angle kind constants
Ang0 = 1
Angconvex = 2
Angreflex = 3
Angtangential = 4
Ang360 = 5
def TriangulateFace(face, points):
"""Triangulate the given face.
Uses an easy triangulation first, followed by a constrained delauney
triangulation to get better shaped triangles.
Args:
face: list of int - indices in points, assumed CCW-oriented
points: geom.Points - holds coordinates for vertices
Returns:
list of (int, int, int) - 3-tuples are CCW-oriented vertices of
triangles making up the triangulation
"""
if len(face) <= 3:
return [tuple(face)]
tris = EarChopTriFace(face, points)
bord = _BorderEdges([face])
triscdt = _CDT(tris, bord, points)
return triscdt
def TriangulateFaceWithHoles(face, holes, points):
"""Like TriangulateFace, but with holes inside the face.
Works by making one complex polygon that has segments to
and from the holes ("islands"), and then using the same method
as TriangulateFace.
Args:
face: list of int - indices in points, assumed CCW-oriented
holes: list of list of int - each sublist is like face
but CW-oriented and assumed to be inside face
points: geom.Points - holds coordinates for vertices
Returns:
list of (int, int, int) - 3-tuples are CCW-oriented vertices of
triangles making up the triangulation
"""
if len(holes) == 0:
return TriangulateFace(face, points)
allfaces = [face] + holes
sholes = [_SortFace(h, points) for h in holes]
joinedface = _JoinIslands(face, sholes, points)
tris = EarChopTriFace(joinedface, points)
bord = _BorderEdges(allfaces)
triscdt = _CDT(tris, bord, points)
return triscdt
def QuadrangulateFace(face, points):
"""Quadrangulate the face (subdivide into convex quads and tris).
Like TriangulateFace, but after triangulating, join as many pairs
of triangles as possible into convex quadrilaterals.
Args:
face: list of int - indices in points, assumed CCW-oriented
points: geom.Points - holds coordinates for vertices
Returns:
list of 3-tuples or 4-tuples of ints - CCW-oriented vertices of
quadrilaterals and triangles making up the quadrangulation.
"""
if len(face) <= 3:
return [tuple(face)]
tris = EarChopTriFace(face, points)
bord = _BorderEdges([face])
triscdt = _CDT(tris, bord, points)
qs = _Quandrangulate(triscdt, bord, points)
return qs
def QuadrangulateFaceWithHoles(face, holes, points):
"""Like QuadrangulateFace, but with holes inside the faces.
Args:
face: list of int - indices in points, assumed CCW-oriented
holes: list of list of int - each sublist is like face
but CW-oriented and assumed to be inside face
points: geom.Points - holds coordinates for vertices
Returns:
list of 3-tuples or 4-tuples of ints - CCW-oriented vertices of
quadrilaterals and triangles making up the quadrangulation.
"""
if len(holes) == 0:
return QuadrangulateFace(face, points)
allfaces = [face] + holes
sholes = [_SortFace(h, points) for h in holes]
joinedface = _JoinIslands(face, sholes, points)
tris = EarChopTriFace(joinedface, points)
bord = _BorderEdges(allfaces)
triscdt = _CDT(tris, bord, points)
qs = _Quandrangulate(triscdt, bord, points)
return qs
def _SortFace(face, points):
"""Rotate face so leftmost vertex is first, where face is
list of indices in points."""
n = len(face)
if n <= 1:
return face
lefti = 0
leftv = face[0]
for i in range(1, n):
# following comparison is lexicographic on n-tuple
# so sorts on x first, using lower y as tie breaker.
if points.pos[face[i]] < points.pos[leftv]:
lefti = i
leftv = face[i]
return face[lefti:] + face[0:lefti]
def EarChopTriFace(face, points):
"""Triangulate given face, with coords given by indexing into points.
Return list of faces, each of which will be a triangle.
Use the ear-chopping method."""
# start with lowest coord in 2d space to try
# to get a pleasing uniform triangulation if starting with
# a regular structure (like a grid)
start = _GetLeastIndex(face, points)
ans = []
incr = 1
n = len(face)
while n > 3:
i = _FindEar(face, n, start, incr, points)
vm1 = face[(i - 1) % n]
v0 = face[i]
v1 = face[(i + 1) % n]
face = _ChopEar(face, i)
n = len(face)
incr = - incr
if incr == 1:
start = i % n
else:
start = (i - 1) % n
ans.append((vm1, v0, v1))
ans.append(tuple(face))
return ans
def _GetLeastIndex(face, points):
"""Return index of coordinate that is leftmost, lowest in face."""
bestindex = 0
bestpos = points.pos[face[0]]
for i in range(1, len(face)):
pos = points.pos[face[i]]
if pos[0] < bestpos[0] or \
(pos[0] == bestpos[0] and pos[1] < bestpos[1]):
bestindex = i
bestpos = pos
return bestindex
def _FindEar(face, n, start, incr, points):
"""An ear of a polygon consists of three consecutive vertices
v(-1), v0, v1 such that v(-1) can connect to v1 without intersecting
the polygon.
Finds an ear, starting at index 'start' and moving
in direction incr. (We attempt to alternate directions, to find
'nice' triangulations for simple convex polygons.)
Returns index into faces of v0 (will always find one, because
uses a desperation mode if fails to find one with above rule)."""
angk = _ClassifyAngles(face, n, points)
for mode in range(0, 5):
i = start
while True:
if _IsEar(face, i, n, angk, points, mode):
return i
i = (i + incr) % n
if i == start:
break # try next higher desperation mode
def _IsEar(face, i, n, angk, points, mode):
"""Return true, false depending on ear status of vertices
with indices i-1, i, i+1.
mode is amount of desperation: 0 is Normal mode,
mode 1 allows degenerate triangles (with repeated vertices)
mode 2 allows local self crossing (folded) ears
mode 3 allows any convex vertex (should always be one)
mode 4 allows anything (just to be sure loop terminates!)"""
k = angk[i]
vm2 = face[(i - 2) % n]
vm1 = face[(i - 1) % n]
v0 = face[i]
v1 = face[(i + 1) % n]
v2 = face[(i + 2) % n]
if vm1 == v0 or v0 == v1:
return (mode > 0)
b = (k == Angconvex or k == Angtangential or k == Ang0)
c = _InCone(vm1, v0, v1, v2, angk[(i + 1) % n], points) and \
_InCone(v1, vm2, vm1, v0, angk[(i - 1) % n], points)
if b and c:
return _EarCheck(face, n, angk, vm1, v0, v1, points)
if mode < 2:
return False
if mode == 3:
return SegsIntersect(vm2, vm1, v0, v1, points)
if mode == 4:
return b
return True
def _EarCheck(face, n, angk, vm1, v0, v1, points):
"""Return True if the successive vertices vm1, v0, v1
forms an ear. We already know that it is not a reflex
Angle, and that the local cone containment is ok.
What remains to check is that the edge vm1-v1 doesn't
intersect any other edge of the face (besides vm1-v0
and v0-v1). Equivalently, there can't be a reflex Angle
inside the triangle vm1-v0-v1. (Well, there are
messy cases when other points of the face coincide with
v0 or touch various lines involved in the ear.)"""
for j in range(0, n):
fv = face[j]
k = angk[j]
b = (k == Angreflex or k == Ang360) \
and not(fv == vm1 or fv == v0 or fv == v1)
if b:
# Is fv inside closure of triangle (vm1,v0,v1)?
c = not(Ccw(v0, vm1, fv, points) \
or Ccw(vm1, v1, fv, points) \
or Ccw(v1, v0, fv, points))
fvm1 = face[(j - 1) % n]
fv1 = face[(j + 1) % n]
# To try to deal with some degenerate cases,
# also check to see if either segment attached to fv
# intersects either segment of potential ear.
d = SegsIntersect(fvm1, fv, vm1, v0, points) or \
SegsIntersect(fvm1, fv, v0, v1, points) or \
SegsIntersect(fv, fv1, vm1, v0, points) or \
SegsIntersect(fv, fv1, v0, v1, points)
if c or d:
return False
return True
def _ChopEar(face, i):
"""Return a copy of face (of length n), omitting element i."""
return face[0:i] + face[i + 1:]
def _InCone(vtest, a, b, c, bkind, points):
"""Return true if point with index vtest is in Cone of points with
indices a, b, c, where Angle ABC has AngleKind Bkind.
The Cone is the set of points inside the left face defined by
segments ab and bc, disregarding all other segments of polygon for
purposes of inside test."""
if bkind == Angreflex or bkind == Ang360:
if _InCone(vtest, c, b, a, Angconvex, points):
return False
return not((not(Ccw(b, a, vtest, points)) \
and not(Ccw(b, vtest, a, points)) \
and Ccw(b, a, vtest, points))
or
(not(Ccw(b, c, vtest, points)) \
and not(Ccw(b, vtest, c, points)) \
and Ccw(b, a, vtest, points)))
else:
return Ccw(a, b, vtest, points) and Ccw(b, c, vtest, points)
def _JoinIslands(face, holes, points):
"""face is a CCW face containing the CW faces in the holes list,
where each hole is sorted so the leftmost-lowest vertex is first.
faces and holes are given as lists of indices into points.
The holes should be sorted by softface.
Add edges to make a new face that includes the holes (a Ccw traversal
of the new face will have the inside always on the left),
and return the new face."""
while len(holes) > 0:
(hole, holeindex) = _LeftMostFace(holes, points)
holes = holes[0:holeindex] + holes[holeindex + 1:]
face = _JoinIsland(face, hole, points)
return face
def _JoinIsland(face, hole, points):
"""Return a modified version of face that splices in the
vertices of hole (which should be sorted)."""
if len(hole) == 0:
return face
hv0 = hole[0]
d = _FindDiag(face, hv0, points)
newface = face[0:d + 1] + hole + [hv0] + face[d:]
return newface
def _LeftMostFace(holes, points):
"""Return (hole,index of hole in holes) where hole has
the leftmost first vertex. To be able to handle empty
holes gracefully, call an empty hole 'leftmost'.
Assumes holes are sorted by softface."""
assert(len(holes) > 0)
lefti = 0
lefthole = holes[0]
if len(lefthole) == 0:
return (lefthole, lefti)
leftv = lefthole[0]
for i in range(1, len(holes)):
ihole = holes[i]
if len(ihole) == 0:
return (ihole, i)
iv = ihole[0]
if points.pos[iv] < points.pos[leftv]:
(lefti, lefthole, leftv) = (i, ihole, iv)
return (lefthole, lefti)
def _FindDiag(face, hv, points):
"""Find a vertex in face that can see vertex hv, if possible,
and return the index into face of that vertex.
Should be able to find a diagonal that connects a vertex of face
left of v to hv without crossing face, but try two
more desperation passes after that to get SOME diagonal, even if
it might cross some edge somewhere.
First desperation pass (mode == 1): allow points right of hv.
Second desperation pass (mode == 2): allow crossing boundary poly"""
besti = - 1
bestdist = 1e30
for mode in range(0, 3):
for i in range(0, len(face)):
v = face[i]
if mode == 0 and points.pos[v] > points.pos[hv]:
continue # in mode 0, only want points left of hv
dist = _DistSq(v, hv, points)
if dist < bestdist:
if _IsDiag(i, v, hv, face, points) or mode == 2:
(besti, bestdist) = (i, dist)
if besti >= 0:
break # found one, so don't need other modes
assert(besti >= 0)
return besti
def _IsDiag(i, v, hv, face, points):
"""Return True if vertex v (at index i in face) can see vertex hv.
v and hv are indices into points.
(v, hv) is a diagonal if hv is in the cone of the Angle at index i on face
and no segment in face intersects (h, hv).
"""
n = len(face)
vm1 = face[(i - 1) % n]
v1 = face[(i + 1) % n]
k = _AngleKind(vm1, v, v1, points)
if not _InCone(hv, vm1, v, v1, k, points):
return False
for j in range(0, n):
vj = face[j]
vj1 = face[(j + 1) % n]
if SegsIntersect(v, hv, vj, vj1, points):
return False
return True
def _DistSq(a, b, points):
"""Return distance squared between coords with indices a and b in points.
"""
diff = Sub2(points.pos[a], points.pos[b])
return Dot2(diff, diff)
def _BorderEdges(facelist):
"""Return a set of (u,v) where u and v are successive vertex indices
in some face in the list in facelist."""
ans = set()
for i in range(0, len(facelist)):
f = facelist[i]
for j in range(1, len(f)):
ans.add((f[j - 1], f[j]))
ans.add((f[-1], f[0]))
return ans
def _CDT(tris, bord, points):
"""Tris is a list of triangles ((a,b,c), CCW-oriented indices into points)
Bord is a set of border edges (u,v), oriented so that tris
is a triangulation of the left face of the border(s).
Make the triangulation "Constrained Delaunay" by flipping "reversed"
quadrangulaterals until can flip no more.
Return list of triangles in new triangulation."""
td = _TriDict(tris)
re = _ReveresedEdges(tris, td, bord, points)
ts = set(tris)
# reverse the reversed edges until done.
# reversing and edge adds new edges, which may or
# may not be reversed or border edges, to re for
# consideration, but the process will stop eventually.
while len(re) > 0:
(a, b) = e = re.pop()
if e in bord or not _IsReversed(e, td, points):
continue
# rotate e in quad adbc to get other diagonal
erev = (b, a)
tl = td.get(e)
tr = td.get(erev)
if not tl or not tr:
continue # shouldn't happen
c = _OtherVert(tl, a, b)
d = _OtherVert(tr, a, b)
if c is None or d is None:
continue # shouldn't happen
newt1 = (c, d, b)
newt2 = (c, a, d)
del td[e]
del td[erev]
td[(c, d)] = newt1
td[(d, b)] = newt1
td[(b, c)] = newt1
td[(d, c)] = newt2
td[(c, a)] = newt2
td[(a, d)] = newt2
if tl in ts:
ts.remove(tl)
if tr in ts:
ts.remove(tr)
ts.add(newt1)
ts.add(newt2)
re.extend([(d, b), (b, c), (c, a), (a, d)])
return list(ts)
def _TriDict(tris):
"""tris is a list of triangles (a,b,c), CCW-oriented indices.
Return dict mapping all edges in the triangles to the containing
triangle list."""
ans = dict()
for i in range(0, len(tris)):
(a, b, c) = t = tris[i]
ans[(a, b)] = t
ans[(b, c)] = t
ans[(c, a)] = t
return ans
def _ReveresedEdges(tris, td, bord, points):
"""Return list of reversed edges in tris.
Only want edges not in bord, and only need one representative
of (u,v)/(v,u), so choose the one with u < v.
td is dictionary from _TriDict, and is used to find left and right
triangles of edges."""
ans = []
for i in range(0, len(tris)):
(a, b, c) = tris[i]
for e in [(a, b), (b, c), (c, a)]:
if e in bord:
continue
(u, v) = e
if u < v:
if _IsReversed(e, td, points):
ans.append(e)
return ans
def _IsReversed(e, td, points):
"""If e=(a,b) is a non-border edge, with left-face triangle tl and
right-face triangle tr, then it is 'reversed' if the circle through
a, b, and (say) the other vertex of tl containts the other vertex of tr.
td is a _TriDict, for finding triangles containing edges, and points
gives the coordinates for vertex indices used in edges."""
tl = td.get(e)
if not tl:
return False
(a, b) = e
tr = td.get((b, a))
if not tr:
return False
c = _OtherVert(tl, a, b)
d = _OtherVert(tr, a, b)
if c is None or d is None:
return False
return InCircle(a, b, c, d, points)
def _OtherVert(tri, a, b):
"""tri should be a tuple of 3 vertex indices, two of which are a and b.
Return the third index, or None if all vertices are a or b"""
for v in tri:
if v != a and v != b:
return v
return None
def _ClassifyAngles(face, n, points):
"""Return vector of anglekinds of the Angle around each point in face."""
return [_AngleKind(face[(i - 1) % n], face[i], face[(i + 1) % n], points) \
for i in list(range(0, n))]
def _AngleKind(a, b, c, points):
"""Return one of the Ang... constants to classify Angle formed by ABC,
in a counterclockwise traversal of a face,
where a, b, c are indices into points."""
if Ccw(a, b, c, points):
return Angconvex
elif Ccw(a, c, b, points):
return Angreflex
else:
vb = points.pos[b]
udotv = Dot2(Sub2(vb, points.pos[a]), Sub2(points.pos[c], vb))
if udotv > 0.0:
return Angtangential
else:
return Ang0 # to fix: return Ang360 if "inside" spur
def _Quandrangulate(tris, bord, points):
"""Tris is list of triangles, forming a triangulation of region whose
border edges are in set bord.
Combine adjacent triangles to make quads, trying for "good" quads where
possible. Some triangles will probably remain uncombined"""
(er, td) = _ERGraph(tris, bord, points)
if len(er) == 0:
return tris
if len(er) > GTHRESH:
match = _GreedyMatch(er)
else:
match = _MaxMatch(er)
return _RemoveEdges(tris, match)
def _RemoveEdges(tris, match):
"""tris is list of triangles.
er is as returned from _MaxMatch or _GreedyMatch.
Return list of (A,D,B,C) resulting from deleting edge (A,B) causing a merge
of two triangles; append to that list the remaining unmatched triangles."""
ans = []
triset = set(tris)
while len(match) > 0:
(_, e, tl, tr) = match.pop()
(a, b) = e
if tl in triset:
triset.remove(tl)
if tr in triset:
triset.remove(tr)
c = _OtherVert(tl, a, b)
d = _OtherVert(tr, a, b)
if c is None or d is None:
continue
ans.append((a, d, b, c))
return ans + list(triset)
def _ERGraph(tris, bord, points):
"""Make an 'Edge Removal Graph'.
Given a list of triangles, the 'Edge Removal Graph' is a graph whose
nodes are the triangles (think of a point in the center of them),
and whose edges go between adjacent triangles (they share a non-border
edge), such that it would be possible to remove the shared edge
and form a convex quadrilateral. Forming a quadrilateralization
is then a matter of finding a matching (set of edges that don't
share a vertex - remember, these are the 'face' vertices).
For better quadrilaterlization, we'll make the Edge Removal Graph
edges have weights, with higher weights going to the edges that
are more desirable to remove. Then we want a maximum weight matching
in this graph.
We'll return the graph in a kind of implicit form, using edges of
the original triangles as a proxy for the edges between the faces
(i.e., the edge of the triangle is the shared edge). We'll arbitrarily
pick the triangle graph edge with lower-index start vertex.
Also, to aid in traversing the implicit graph, we'll keep the left
and right triangle triples with edge 'ER edge'.
Finally, since we calculate it anyway, we'll return a dictionary
mapping edges of the triangles to the triangle triples they're in.
Args:
tris: list of (int, int, int) giving a triple of vertex indices for
triangles, assumed CCW oriented
bord: set of (int, int) giving vertex indices for border edges
points: geom.Points - for mapping vertex indices to coords
Returns:
(list of (weight,e,tl,tr), dict)
where edge e=(a,b) is non-border edge
with left face tl and right face tr (each a triple (i,j,k)),
where removing the edge would form an "OK" quad (no concave angles),
with weight representing the desirability of removing the edge
The dict maps int pairs (a,b) to int triples (i,j,k), that is,
mapping edges to their containing triangles.
"""
td = _TriDict(tris)
dd = _DegreeDict(tris)
ans = []
ctris = tris[:] # copy, so argument not affected
while len(ctris) > 0:
(i, j, k) = tl = ctris.pop()
for e in [(i, j), (j, k), (k, i)]:
if e in bord:
continue
(a, b) = e
# just consider one of (a,b) and (b,a), to avoid dups
if a > b:
continue
erev = (b, a)
tr = td.get(erev)
if not tr:
continue
c = _OtherVert(tl, a, b)
d = _OtherVert(tr, a, b)
if c is None or d is None:
continue
# calculate amax, the max of the new angles that would
# be formed at a and b if tl and tr were combined
amax = max(Angle(c, a, b, points) + Angle(d, a, b, points),
Angle(c, b, a, points) + Angle(d, b, a, points))
if amax > 180.0:
continue
weight = ANGFAC * (180.0 - amax) + DEGFAC * (dd[a] + dd[b])
ans.append((weight, e, tl, tr))
return (ans, td)
def _GreedyMatch(er):
"""er is list of (weight,e,tl,tr).
Find maximal set so that each triangle appears in at most
one member of set"""
# sort in order of decreasing weight
er.sort(key=lambda v: v[0], reverse=True)
match = set()
ans = []
while len(er) > 0:
(_, _, tl, tr) = q = er.pop()
if tl not in match and tr not in match:
match.add(tl)
match.add(tr)
ans.append(q)
return ans
def _MaxMatch(er):
"""Like _GreedyMatch, but use divide and conquer to find best possible set.
Args:
er: list of (weight,e,tl,tr) - see _ERGraph
Returns:
list that is a subset of er giving a maximum weight match
"""
(ans, _) = _DCMatch(er)
return ans
def _DCMatch(er):
"""Recursive helper for _MaxMatch.
Divide and Conquer approach to finding max weight matching.
If we're lucky, there's an edge in er that separates the edge removal
graph into (at least) two separate components. Then the max weight
is either one that includes that edge or excludes it - and we can
use a recursive call to _DCMatch to handle each component separately
on what remains of the graph after including/excluding the separating edge.
If we're not lucky, we fall back on _EMatch (see below).
Args:
er: list of (weight, e, tl, tr) (see _ERGraph)
Returns:
(list of (weight, e, tl, tr), float) - the subset forming a maximum
matching, and the total weight of the match.
"""
if not er:
return ([], 0.0)
if len(er) == 1:
return (er, er[0][0])
match = []
matchw = 0.0
for i in range(0, len(er)):
(nc, comp) = _FindComponents(er, i)
if nc == 1:
# er[i] doesn't separate er
continue
(wi, _, tl, tr) = er[i]
if comp[tl] != comp[tr]:
# case 1: er separates graph
# compare the matches that include er[i] versus
# those that exclude it
(a, b) = _PartitionComps(er, comp, i, comp[tl], comp[tr])
ax = _CopyExcluding(a, tl, tr)
bx = _CopyExcluding(b, tl, tr)
(axmatch, wax) = _DCMatch(ax)
(bxmatch, wbx) = _DCMatch(bx)
if len(ax) == len(a):
wa = wax
amatch = axmatch
else:
(amatch, wa) = _DCMatch(a)
if len(bx) == len(b):
wb = wbx
bmatch = bxmatch
else:
(bmatch, wb) = _DCMatch(b)
w = wa + wb
wx = wax + wbx + wi
if w > wx:
match = amatch + bmatch
matchw = w
else:
match = [er[i]] + axmatch + bxmatch
matchw = wx
else:
# case 2: er not needed to separate graph
(a, b) = _PartitionComps(er, comp, -1, 0, 0)
(amatch, wa) = _DCMatch(a)
(bmatch, wb) = _DCMatch(b)
match = amatch + bmatch
matchw = wa + wb
if match:
break
if not match:
return _EMatch(er)
return (match, matchw)
def _EMatch(er):
"""Exhaustive match helper for _MaxMatch.
This is the case when we were unable to find a single edge
separating the edge removal graph into two components.
So pick a single edge and try _DCMatch on the two cases of
including or excluding that edge. We may be lucky in these
subcases (say, if the graph is currently a simple cycle, so
only needs one more edge after the one we pick here to separate
it into components). Otherwise, we'll end up back in _EMatch
again, and the worse case will be exponential.
Pick a random edge rather than say, the first, to hopefully
avoid some pathological cases.
Args:
er: list of (weight, el, tl, tr) (see _ERGraph)
Returns:
(list of (weight, e, tl, tr), float) - the subset forming a maximum
matching, and the total weight of the match.
"""
if not er:
return ([], 0.0)
if len(er) == 1:
return (er, er[1][1])
i = random.randint(0, len(er) - 1)
eri = (wi, _, tl, tr) = er[i]
# case a: include eri. exlude other edges that touch tl or tr
a = _CopyExcluding(er, tl, tr)
a.append(eri)
(amatch, wa) = _DCMatch(a)
wa += wi
if len(a) == len(er) - 1:
# if a excludes only eri, then er didn't touch anything else
# in the graph, and the best match will always include er
# and we can skip the call for case b
wb = -1.0
bmatch = []
else:
b = er[:i] + er[i + 1:]
(bmatch, wb) = _DCMatch(b)
if wa > wb:
match = amatch
match.append(eri)
matchw = wa
else:
match = bmatch
matchw = wb
return (match, matchw)
def _FindComponents(er, excepti):
"""Find connected components induced by edges, excluding one edge.
Args:
er: list of (weight, el, tl, tr) (see _ERGraph)
excepti: index in er of edge to be excluded
Returns:
(int, dict): int is number of connected components found,
dict maps triangle triple ->
connected component index (starting at 1)
"""
ncomps = 0
comp = dict()
for i in range(0, len(er)):
(_, _, tl, tr) = er[i]
for t in [tl, tr]:
if t not in comp:
ncomps += 1
_FCVisit(er, excepti, comp, t, ncomps)
return (ncomps, comp)
def _FCVisit(er, excepti, comp, t, compnum):
"""Helper for _FindComponents depth-first-search."""
comp[t] = compnum
for i in range(0, len(er)):
if i == excepti:
continue
(_, _, tl, tr) = er[i]
if tl == t or tr == t:
s = tl
if s == t:
s = tr
if s not in comp:
_FCVisit(er, excepti, comp, s, compnum)
def _PartitionComps(er, comp, excepti, compa, compb):
"""Partition the edges of er by component number, into two lists.
Generally, put odd components into first list and even into second,
except that component compa goes in the first and compb goes in the second,
and we ignore edge er[excepti].
Args:
er: list of (weight, el, tl, tr) (see _ERGraph)
comp: dict - mapping triangle triple -> connected component index
excepti: int - index in er to ignore (unless excepti==-1)
compa: int - component to go in first list of answer (unless 0)
compb: int - component to go in second list of answer (unless 0)
Returns:
(list, list) - a partition of er according to above rules
"""
parta = []
partb = []
for i in range(0, len(er)):
if i == excepti:
continue
tl = er[i][2]
c = comp[tl]
if c == compa or (c != compb and (c & 1) == 1):
parta.append(er[i])
else:
partb.append(er[i])
return (parta, partb)
def _CopyExcluding(er, s, t):
"""Return a copy of er, excluding all those involving triangles s and t.
Args:
er: list of (weight, e, tl, tr) - see _ERGraph
s: 3-tuple of int - a triangle
t: 3-tuple of int - a triangle
Returns:
Copy of er excluding those with tl or tr == s or t
"""
ans = []
for e in er:
(_, _, tl, tr) = e
if tl == s or tr == s or tl == t or tr == t:
continue
ans.append(e)
return ans
def _DegreeDict(tris):
"""Return a dictionary mapping vertices in tris to the number of triangles
that they are touch."""
ans = dict()
for t in tris:
for v in t:
if v in ans:
ans[v] = ans[v] + 1
else:
ans[v] = 1
return ans
def PolygonPlane(face, points):
"""Return a Normal vector for the face with 3d coords given by indexing
into points."""
if len(face) < 3:
return (0.0, 0.0, 1.0) # arbitrary, we really have no idea
else:
coords = [points.pos[i] for i in face]
return Normal(coords)
# This Normal appears to be on the CCW-traversing side of a polygon
def Normal(coords):
"""Return an average Normal vector for the point list, 3d coords."""
if len(coords) < 3:
return (0.0, 0.0, 1.0) # arbitrary
(ax, ay, az) = coords[0]
(bx, by, bz) = coords[1]
(cx, cy, cz) = coords[2]
if len(coords) == 3:
sx = (ay - by) * (az + bz) + \
(by - cy) * (bz + cz) + \
(cy - ay) * (cz + az)
sy = (az - bz) * (ax + bx) + \
(bz - cz) * (bx + cx) + \
(cz - az) * (cx + ax)
sz = (ax - bx) * (by + by) + \
(bx - cx) * (by + cy) + \
(cx - ax) * (cy + ay)
return Norm3(sx, sy, sz)
else:
sx = (ay - by) * (az + bz) + (by - cy) * (bz + cz)
sy = (az - bz) * (ax + bx) + (bz - cz) * (bx + cx)
sz = (ax - bx) * (ay + by) + (bx - cx) * (by + cy)
return _NormalAux(coords[3:], coords[0], sx, sy, sz)
def _NormalAux(rest, first, sx, sy, sz):
(ax, ay, az) = rest[0]
if len(rest) == 1:
(bx, by, bz) = first
else:
(bx, by, bz) = rest[1]
nx = sx + (ay - by) * (az + bz)
ny = sy + (az - bz) * (ax + bx)
nz = sz + (ax - bx) * (ay + by)
if len(rest) == 1:
return Norm3(nx, ny, nz)
else:
return _NormalAux(rest[1:], first, nx, ny, nz)
def Norm3(x, y, z):
"""Return vector (x,y,z) normalized by dividing by squared length.
Return (0.0, 0.0, 1.0) if the result is undefined."""
sqrlen = x * x + y * y + z * z
if sqrlen < 1e-100:
return (0.0, 0.0, 1.0)
else:
try:
d = sqrt(sqrlen)
return (x / d, y / d, z / d)
except:
return (0.0, 0.0, 1.0)
# We're using right-hand coord system, where
# forefinger=x, middle=y, thumb=z on right hand.
# Then, e.g., (1,0,0) x (0,1,0) = (0,0,1)
def Cross3(a, b):
"""Return the cross product of two vectors, a x b."""
(ax, ay, az) = a
(bx, by, bz) = b
return (ay * bz - az * by, az * bx - ax * bz, ax * by - ay * bx)
def Dot2(a, b):
"""Return the dot product of two 2d vectors, a . b."""
return a[0] * b[0] + a[1] * b[1]
def Perp2(a, b):
"""Return a sort of 2d cross product."""
return a[0] * b[1] - a[1] * b[0]
def Sub2(a, b):
"""Return difference of 2d vectors, a-b."""
return (a[0] - b[0], a[1] - b[1])
def Add2(a, b):
"""Return the sum of 2d vectors, a+b."""
return (a[0] + b[0], a[1] + b[1])
def Length2(v):
"""Return length of vector v=(x,y)."""
return sqrt(v[0] * v[0] + v[1] * v[1])
def LinInterp2(a, b, alpha):
"""Return the point alpha of the way from a to b."""
beta = 1 - alpha
return (beta * a[0] + alpha * b[0], beta * a[1] + alpha * b[1])
def Normalized2(p):
"""Return vector p normlized by dividing by its squared length.
Return (0.0, 1.0) if the result is undefined."""
(x, y) = p
sqrlen = x * x + y * y
if sqrlen < 1e-100:
return (0.0, 1.0)
else:
try:
d = sqrt(sqrlen)
return (x / d, y / d)
except:
return (0.0, 1.0)
def Angle(a, b, c, points):
"""Return Angle abc in degrees, in range [0,180),
where a,b,c are indices into points."""
u = Sub2(points.pos[c], points.pos[b])
v = Sub2(points.pos[a], points.pos[b])
n1 = Length2(u)
n2 = Length2(v)
if n1 == 0.0 or n2 == 0.0:
return 0.0
else:
costheta = Dot2(u, v) / (n1 * n2)
if costheta > 1.0:
costheta = 1.0
if costheta < - 1.0:
costheta = - 1.0
return math.acos(costheta) * 180.0 / math.pi
def SegsIntersect(ixa, ixb, ixc, ixd, points):
"""Return true if segment AB intersects CD,
false if they just touch. ixa, ixb, ixc, ixd are indices
into points."""
a = points.pos[ixa]
b = points.pos[ixb]
c = points.pos[ixc]
d = points.pos[ixd]
u = Sub2(b, a)
v = Sub2(d, c)
w = Sub2(a, c)
pp = Perp2(u, v)
if abs(pp) > TOL:
si = Perp2(v, w) / pp
ti = Perp2(u, w) / pp
return 0.0 < si < 1.0 and 0.0 < ti < 1.0
else:
# parallel or overlapping
if Dot2(u, u) == 0.0 or Dot2(v, v) == 0.0:
return False
else:
pp2 = Perp2(w, v)
if abs(pp2) > TOL:
return False # parallel, not collinear
z = Sub2(b, c)
(vx, vy) = v
(wx, wy) = w
(zx, zy) = z
if vx == 0.0:
(t0, t1) = (wy / vy, zy / vy)
else:
(t0, t1) = (wx / vx, zx / vx)
return 0.0 < t0 < 1.0 and 0.0 < t1 < 1.0
def Ccw(a, b, c, points):
"""Return true if ABC is a counterclockwise-oriented triangle,
where a, b, and c are indices into points.
Returns false if not, or if colinear within TOL."""
(ax, ay) = (points.pos[a][0], points.pos[a][1])
(bx, by) = (points.pos[b][0], points.pos[b][1])
(cx, cy) = (points.pos[c][0], points.pos[c][1])
d = ax * by - bx * ay - ax * cy + cx * ay + bx * cy - cx * by
return d > TOL
def InCircle(a, b, c, d, points):
"""Return true if circle through points with indices a, b, c
contains point with index d (indices into points).
Except: if ABC forms a counterclockwise oriented triangle
then the test is reversed: return true if d is outside the circle.
Will get false, no matter what orientation, if d is cocircular, with TOL^2.
| xa ya xa^2+ya^2 1 |
| xb yb xb^2+yb^2 1 | > 0
| xc yc xc^2+yc^2 1 |
| xd yd xd^2+yd^2 1 |
"""
(xa, ya, za) = _Icc(points.pos[a])
(xb, yb, zb) = _Icc(points.pos[b])
(xc, yc, zc) = _Icc(points.pos[c])
(xd, yd, zd) = _Icc(points.pos[d])
det = xa * (yb * zc - yc * zb - yb * zd + yd * zb + yc * zd - yd * zc) \
- xb * (ya * zc - yc * za - ya * zd + yd * za + yc * zd - yd * zc) \
+ xc * (ya * zb - yb * za - ya * zd + yd * za + yb * zd - yd * zb) \
- xd * (ya * zb - yb * za - ya * zc + yc * za + yb * zc - yc * zb)
return det > TOL * TOL
def _Icc(p):
(x, y) = (p[0], p[1])
return (x, y, x * x + y * y)