# coding=utf-8
"""Core triangulation functions used by various geometry modules.
The functions here are derived from the earcut-python library available at
https://github.com/joshuaskelly/earcut-python
The earcut-python library is, itself, a pure Python port of the earcut JavaScript
triangulation library maintained by Mapbox. The original project can be found at
https://github.com/mapbox/earcut
The version here is based off of the JavaScript earcut 2.1.1 release, and is
functionally identical.
"""
from __future__ import division
[docs]def earcut(data, hole_indices=None, dim=2):
"""Triangulate a list of vertices that make up a shape, either with or without holes.
Args:
data: A flat array of vertex coordinates like [x0,y0, x1,y1, x2,y2, ...].
hole_indices: A flat array of the starting indices for each hole. For example,
[5, 8] for a 12-vertex input would mean one hole with vertices 5-7
and another with 8-11. If a single vertex is passed as as a hole,
Earcut treats it as a Steiner point. If None, no holes will be assumed
for the shape. (Default: None).
dim: An integer for the number of coordinates per vertex in the input
array. For example, 3 means each vertex exists in 3D space with
XX, Y, Z coordinates. (Default: 2 for 2D coordinates).
"""
dim = dim or 2
hasHoles = hole_indices and len(hole_indices)
outerLen = hole_indices[0] * dim if hasHoles else len(data)
outerNode = _linked_list(data, 0, outerLen, dim, True)
triangles = []
if not outerNode:
return triangles
minX = None
minY = None
maxX = None
maxY = None
x = None
y = None
size = None
if hasHoles:
outerNode = _eliminate_holes(data, hole_indices, outerNode, dim)
# if the shape is not too simple, we'll use z-order curve hash later
if (len(data) > 80 * dim): # calculate polygon bbox
minX = maxX = data[0]
minY = maxY = data[1]
for i in range(dim, outerLen, dim):
x = data[i]
y = data[i + 1]
if x < minX:
minX = x
if y < minY:
minY = y
if x > maxX:
maxX = x
if y > maxY:
maxY = y
# minX, minY and size are later used to transform coords into integers
# integers are used for z-order calculation
size = max(maxX - minX, maxY - minY)
_earcut_linked(outerNode, triangles, dim, minX, minY, size)
return triangles
def _linked_list(data, start, end, dim, clockwise):
"""Create a circular doubly linked list from polygon points.
Points will be in the specified winding order.
"""
i = None
last = None
if (clockwise == (_signed_area(data, start, end, dim) > 0)):
for i in range(start, end, dim):
last = _insert_node(i, data[i], data[i + 1], last)
else:
for i in reversed(range(start, end, dim)):
last = _insert_node(i, data[i], data[i + 1], last)
if (last and _equals(last, last.next)):
_remove_node(last)
last = last.next
return last
def _signed_area(data, start, end, dim):
"""Get the signed area from a list of coordinate data."""
sum = 0
j = end - dim
for i in range(start, end, dim):
sum += (data[j] - data[i]) * (data[i + 1] + data[j + 1])
j = i
return sum
def _filter_points(start, end=None):
"""Eliminate colinear or duplicate points."""
if not start:
return start
if not end:
end = start
p = start
again = True
while again or p != end:
again = False
if (not p.steiner and (_equals(p, p.next) or _area(p.prev, p, p.next) == 0)):
_remove_node(p)
p = end = p.prev
if (p == p.next):
return None
again = True
else:
p = p.next
return end
def _earcut_linked(ear, triangles, dim, minX, minY, size, _pass=None):
"""Main ear slicing loop which triangulates a polygon (given as a linked list)."""
if not ear:
return
# interlink polygon nodes in z-order
if not _pass and size:
_index_curve(ear, minX, minY, size)
stop = ear
prev = None
next = None
# iterate through ears, slicing them one by one
while ear.prev != ear.next:
prev = ear.prev
next = ear.next
if _is_ear_hashed(ear, minX, minY, size) if size else _is_ear(ear):
# cut off the triangle
triangles.append(prev.i // dim)
triangles.append(ear.i // dim)
triangles.append(next.i // dim)
_remove_node(ear)
# skipping the next vertex leads to less sliver triangles
ear = next.next
stop = next.next
continue
ear = next
# if we looped through the whole remaining polygon and can't find any more ears
if ear == stop:
# try filtering points and slicing again
if not _pass:
_earcut_linked(_filter_points(ear), triangles, dim, minX, minY, size, 1)
# if this didn't work, try curing all small self-intersections locally
elif _pass == 1:
ear = _cure_local_intersections(ear, triangles, dim)
_earcut_linked(ear, triangles, dim, minX, minY, size, 2)
# as a last resort, try splitting the remaining polygon into two
elif _pass == 2:
_split_earcut(ear, triangles, dim, minX, minY, size)
break
def _is_ear(ear):
"""Check whether a polygon node forms a valid ear with adjacent nodes."""
a = ear.prev
b = ear
c = ear.next
if _area(a, b, c) >= 0:
return False # reflex, can't be an ear
# now make sure we don't have other points inside the potential ear
p = ear.next.next
while p != ear.prev:
if _point_in_triangle(a.x, a.y, b.x, b.y, c.x, c.y, p.x, p.y) and \
_area(p.prev, p, p.next) >= 0:
return False
p = p.next
return True
def _is_ear_hashed(ear, minX, minY, size):
"""Check whether a polygon node forms a valid ear using hashes."""
a = ear.prev
b = ear
c = ear.next
if _area(a, b, c) >= 0:
return False # reflex, can't be an ear
# triangle bbox; min & max are calculated like this for speed
minTX = (a.x if a.x < c.x else c.x) if a.x < b.x else (b.x if b.x < c.x else c.x)
minTY = (a.y if a.y < c.y else c.y) if a.y < b.y else (b.y if b.y < c.y else c.y)
maxTX = (a.x if a.x > c.x else c.x) if a.x > b.x else (b.x if b.x > c.x else c.x)
maxTY = (a.y if a.y > c.y else c.y) if a.y > b.y else (b.y if b.y > c.y else c.y)
# z-order range for the current triangle bbox;
minZ = _z_order(minTX, minTY, minX, minY, size)
maxZ = _z_order(maxTX, maxTY, minX, minY, size)
# first look for points inside the triangle in increasing z-order
p = ear.nextZ
while p and p.z <= maxZ:
if p != ear.prev and p != ear.next and \
_point_in_triangle(a.x, a.y, b.x, b.y, c.x, c.y, p.x, p.y) and \
_area(p.prev, p, p.next) >= 0:
return False
p = p.nextZ
# then look for points in decreasing z-order
p = ear.prevZ
while p and p.z >= minZ:
if p != ear.prev and p != ear.next and \
_point_in_triangle(a.x, a.y, b.x, b.y, c.x, c.y, p.x, p.y) and \
_area(p.prev, p, p.next) >= 0:
return False
p = p.prevZ
return True
def _cure_local_intersections(start, triangles, dim):
"""Go through all polygon nodes and cure small local self-intersections."""
do = True
p = start
while do or p != start:
do = False
a = p.prev
b = p.next.next
if not _equals(a, b) and _intersects(a, p, p.next, b) and \
_locally_inside(a, b) and _locally_inside(b, a):
triangles.append(a.i // dim)
triangles.append(p.i // dim)
triangles.append(b.i // dim)
# remove two nodes involved
_remove_node(p)
_remove_node(p.next)
p = start = b
p = p.next
return p
def _split_earcut(start, triangles, dim, minX, minY, size):
"""try splitting polygon into two and triangulate them independently."""
# look for a valid diagonal that divides the polygon into two
do = True
a = start
while do or a != start:
do = False
b = a.next.next
while b != a.prev:
if a.i != b.i and _is_valid_diagonal(a, b):
# split the polygon in two by the diagonal
c = _split_polygon(a, b)
# filter colinear points around the cuts
a = _filter_points(a, a.next)
c = _filter_points(c, c.next)
# run earcut on each half
_earcut_linked(a, triangles, dim, minX, minY, size)
_earcut_linked(c, triangles, dim, minX, minY, size)
return
b = b.next
a = a.next
def _eliminate_holes(data, hole_indices, outerNode, dim):
"""Link holes into the outer loop, producing a single-ring polygon without holes."""
queue = []
i = None
_len = len(hole_indices)
start = None
end = None
_list = None
for i in range(len(hole_indices)):
start = hole_indices[i] * dim
end = hole_indices[i + 1] * dim if i < _len - 1 else len(data)
_list = _linked_list(data, start, end, dim, False)
if (_list == _list.next):
_list.steiner = True
queue.append(_get_leftmost(_list))
queue = sorted(queue, key=lambda i: i.x)
# process holes from left to right
for i in range(len(queue)):
_eliminate_hole(queue[i], outerNode)
outerNode = _filter_points(outerNode, outerNode.next)
return outerNode
def _eliminate_hole(hole, outerNode):
"""Find a bridge between vertices that connects hole with an outer ring.
Return a shape with the hole linked into it."""
outerNode = _find_hole_bridge(hole, outerNode)
if outerNode:
b = _split_polygon(outerNode, hole)
_filter_points(b, b.next)
def _find_hole_bridge(hole, outerNode):
"""David Eberly's algorithm for finding a bridge between hole and outer polygon."""
do = True
p = outerNode
hx = hole.x
hy = hole.y
qx = float('-inf')
m = None
# find a segment intersected by a ray from the hole's leftmost point to the left;
# segment's endpoint with lesser x will be potential connection point
while do or p != outerNode:
do = False
if hy <= p.y and hy >= p.next.y and p.next.y - p.y != 0:
x = p.x + (hy - p.y) * (p.next.x - p.x) / (p.next.y - p.y)
if x <= hx and x > qx:
qx = x
if (x == hx):
if hy == p.y:
return p
if hy == p.next.y:
return p.next
m = p if p.x < p.next.x else p.next
p = p.next
if not m:
return None
if hx == qx:
return m.prev # hole touches outer segment; pick lower endpoint
# check points inside the triangle of hole point, segment intersection and endpoint
# if there are no points found, we have a valid connection
# otherwise choose the point of the minimum angle with the ray as connection point
stop = m
mx = m.x
my = m.y
tanMin = float('inf')
tan = None
p = m.next
while p != stop:
hx_or_qx = hx if hy < my else qx
qx_or_hx = qx if hy < my else hx
if hx >= p.x and p.x >= mx and \
_point_in_triangle(hx_or_qx, hy, mx, my, qx_or_hx, hy, p.x, p.y):
try:
tan = abs(hy - p.y) / (hx - p.x) # tangential
except ZeroDivisionError:
break
if (tan < tanMin or (tan == tanMin and p.x > m.x)) and \
_locally_inside(p, hole):
m = p
tanMin = tan
p = p.next
return m
def _index_curve(start, minX, minY, size):
"""Interlink polygon nodes in z-order."""
do = True
p = start
while do or p != start:
do = False
if p.z is None:
p.z = _z_order(p.x, p.y, minX, minY, size)
p.prevZ = p.prev
p.nextZ = p.next
p = p.next
p.prevZ.nextZ = None
p.prevZ = None
_sort_linked(p)
def _sort_linked(_list):
"""Simon Tatham's linked list merge sort algorithm.
More information available at https://www.chiark.greenend.org.uk/
"""
do = True
i = None
p = None
q = None
e = None
tail = None
numMerges = None
pSize = None
qSize = None
inSize = 1
while do or numMerges > 1:
do = False
p = _list
_list = None
tail = None
numMerges = 0
while p:
numMerges += 1
q = p
pSize = 0
for i in range(inSize):
pSize += 1
q = q.nextZ
if not q:
break
qSize = inSize
while pSize > 0 or (qSize > 0 and q):
if pSize == 0:
e = q
q = q.nextZ
qSize -= 1
elif (qSize == 0 or not q):
e = p
p = p.nextZ
pSize -= 1
elif (p.z <= q.z):
e = p
p = p.nextZ
pSize -= 1
else:
e = q
q = q.nextZ
qSize -= 1
if tail:
tail.nextZ = e
else:
_list = e
e.prevZ = tail
tail = e
p = q
tail.nextZ = None
inSize *= 2
return _list
def _z_order(x, y, minX, minY, size):
"""Z-order of a point given coords and size of the data bounding box."""
# coords are transformed into non-negative 15-bit integer range
x = int(32767 * (x - minX) // size)
y = int(32767 * (y - minY) // size)
x = (x | (x << 8)) & 0x00FF00FF
x = (x | (x << 4)) & 0x0F0F0F0F
x = (x | (x << 2)) & 0x33333333
x = (x | (x << 1)) & 0x55555555
y = (y | (y << 8)) & 0x00FF00FF
y = (y | (y << 4)) & 0x0F0F0F0F
y = (y | (y << 2)) & 0x33333333
y = (y | (y << 1)) & 0x55555555
return x | (y << 1)
def _get_leftmost(start):
"""Find the leftmost node of a polygon ring."""
do = True
p = start
leftmost = start
while do or p != start:
do = False
if p.x < leftmost.x:
leftmost = p
p = p.next
return leftmost
def _point_in_triangle(ax, ay, bx, by, cx, cy, px, py):
"""Check if a point lies within a convex triangle."""
return (cx - px) * (ay - py) - (ax - px) * (cy - py) >= 0 and \
(ax - px) * (by - py) - (bx - px) * (ay - py) >= 0 and \
(bx - px) * (cy - py) - (cx - px) * (by - py) >= 0
def _is_valid_diagonal(a, b):
"""Check if a diagonal between two polygon nodes is valid.
A valid diagonal is defined as one that lies in polygon interior.
"""
return a.next.i != b.i and a.prev.i != b.i and not _intersects_polygon(a, b) and \
_locally_inside(a, b) and _locally_inside(b, a) and _middle_inside(a, b)
def _area(p, q, r):
"""Get the signed area of a triangle."""
return (q.y - p.y) * (r.x - q.x) - (q.x - p.x) * (r.y - q.y)
def _equals(p1, p2):
"""Check if two points are equal."""
return p1.x == p2.x and p1.y == p2.y
def _intersects(p1, q1, p2, q2):
"""Check if two segments intersect."""
if (_equals(p1, q1) and _equals(p2, q2)) or (_equals(p1, q2) and _equals(p2, q1)):
return True
return _area(p1, q1, p2) > 0 != _area(p1, q1, q2) > 0 and \
_area(p2, q2, p1) > 0 != _area(p2, q2, q1) > 0
def _intersects_polygon(a, b):
"""Check if a polygon diagonal intersects any polygon segments."""
do = True
p = a
while do or p != a:
do = False
init_int = p.i != a.i and p.next.i != a.i and p.i != b.i and p.next.i != b.i
if init_int and _intersects(p, p.next, a, b):
return True
p = p.next
return False
def _locally_inside(a, b):
"""Check if a polygon diagonal is locally inside the polygon."""
if _area(a.prev, a, a.next) < 0:
return _area(a, b, a.next) >= 0 and _area(a, a.prev, b) >= 0
else:
return _area(a, b, a.prev) < 0 or _area(a, a.next, b) < 0
def _middle_inside(a, b):
"""Check if the middle point of a polygon diagonal is inside a polygon."""
do = True
p = a
inside = False
px = (a.x + b.x) / 2
py = (a.y + b.y) / 2
while do or p != a:
do = False
if ((p.y > py) != (p.next.y > py)) and \
(px < (p.next.x - p.x) * (py - p.y) / (p.next.y - p.y) + p.x):
inside = not inside
p = p.next
return inside
def _split_polygon(a, b):
"""Link two polygon vertices with a bridge.
If the vertices belong to the same ring, the polygon will be split into two.
If one belongs to the outer ring and another to a hole, the hole will be merged
into a single ring.
"""
a2 = _Node(a.i, a.x, a.y)
b2 = _Node(b.i, b.x, b.y)
an = a.next
bp = b.prev
a.next = b
b.prev = a
a2.next = an
an.prev = a2
b2.next = a2
a2.prev = b2
bp.next = b2
b2.prev = bp
return b2
def _insert_node(i, x, y, last):
"""Create a node and optionally link it with previous one.
Linking is done in a circular doubly linked list.
"""
p = _Node(i, x, y)
if not last:
p.prev = p
p.next = p
else:
p.next = last.next
p.prev = last
last.next.prev = p
last.next = p
return p
def _remove_node(p):
"""Remove a node from a list."""
p.next.prev = p.prev
p.prev.next = p.next
if p.prevZ:
p.prevZ.nextZ = p.nextZ
if p.nextZ:
p.nextZ.prevZ = p.prevZ
class _Node(object):
"""Node within a coordinate array."""
def __init__(self, i, x, y):
# vertex index in coordinates array
self.i = i
# vertex coordinates
self.x = x
self.y = y
# previous and next vertex nodes in a polygon ring
self.prev = None
self.next = None
# z-order curve value
self.z = None
# previous and next nodes in z-order
self.prevZ = None
self.nextZ = None
# indicates whether this is a steiner point
self.steiner = False