def isenclosedBy(self, points):
'''Check whether this point is enclosed by a (convex) polygon.
@param points: The polygon points (L{LatLon}[]).
@return: C{True} if the polygon encloses this point,
C{False} otherwise.
@raise TypeError: Some B{C{points}} are not L{LatLon}.
@raise ValueError: Insufficient number of B{C{points}}.
@example:
>>> b = LatLon(45,1), LatLon(45,2), LatLon(46,2), LatLon(46,1)
>>> p = LatLon(45.1, 1.1)
>>> inside = p.isEnclosedBy(b) # True
@JSname: I{enclosedBy}.
'''
n, points = self.points2(points, closed=True)
# use normal vector to this point for sign of α
n0 = self.toNvector()
if iterNumpy2(points):
def _subtangles(n, points, n0): # iterate
vs = n0.minus(points[n - 1].toNvector())
for i in range(n):
vs1 = n0.minus(points[i].toNvector())
yield vs.angleTo(vs1, vSign=n0) # PYCHOK false
vs = vs1
# sum subtended angles
s = fsum(_subtangles(n, points, n0))
else:
# get vectors from this to each point
vs = [n0.minus(points[i].toNvector()) for i in range(n)]
# close the polygon
vs.append(vs[0])
# sum subtended angles of each edge (using n0 to determine sign)
s = fsum(vs[i].angleTo(vs[i + 1], vSign=n0) for i in range(n))
# Note, this method uses angle summation test: on a plane,
# angles for an enclosed point will sum to 360°, angles for
# an exterior point will sum to 0°. On a sphere, enclosed
# point angles will sum to less than 360° (due to spherical
# excess), exterior point angles will be small but non-zero.
# XXX are winding number optimisations equally applicable to
# spherical surface?
return abs(s) > PI
def areaOf(points, radius=R_M):
'''Calculate the area of a (spherical) polygon (with great circle
arcs joining consecutive points).
@param points: The polygon points (L{LatLon}[]).
@keyword radius: Optional, mean earth radius (C{meter}).
@return: Polygon area (C{meter}, same units as B{C{radius}}, squared).
@raise TypeError: Some B{C{points}} are not L{LatLon}.
@raise ValueError: Insufficient number of B{C{points}}.
@see: L{pygeodesy.areaOf}, L{sphericalTrigonometry.areaOf} and
L{ellipsoidalKarney.areaOf}.
@example:
>>> b = LatLon(45, 1), LatLon(45, 2), LatLon(46, 2), LatLon(46, 1)
>>> areaOf(b) # 8666058750.718977
'''
n, points = _Nvll.points2(points, closed=True)
# use vector to 1st point as plane normal for sign of α
n0 = points[0].toNvector()
if iterNumpy2(points):
def _interangles(n, points, n0): # iterate
v2 = points[n - 2].toNvector()
v1 = points[n - 1].toNvector()
gc = v2.cross(v1)
for i in range(n):
v2 = points[i].toNvector()
gc1 = v1.cross(v2)
v1 = v2
yield gc.angleTo(gc1, vSign=n0)
gc = gc1
# sum interior angles
s = fsum(_interangles(n, points, n0))
else:
# get great-circle vector for each edge
gc, v1 = [], points[n - 1].toNvector()
for i in range(n):
v2 = points[i].toNvector()
gc.append(v1.cross(v2)) # PYCHOK false, does have .cross
v1 = v2
gc.append(gc[0]) # XXX needed?
# sum interior angles: depending on whether polygon is cw or ccw,
# angle between edges is π−α or π+α, where α is angle between
# great-circle vectors; so sum α, then take n·π − |Σα| (cannot
# use Σ(π−|α|) as concave polygons would fail)
s = fsum(gc[i].angleTo(gc[i + 1], vSign=n0) for i in range(n))
# using Girard’s theorem: A = [Σθᵢ − (n−2)·π]·R²
# (PI2 - abs(s) == (n*PI - abs(s)) - (n-2)*PI)
return abs(PI2 - abs(s)) * radius**2
def areaOf(points, radius=R_M, wrap=True):
'''Calculate the area of a (spherical) polygon (with great circle
arcs joining the points).
@param points: The polygon points (L{LatLon}[]).
@keyword radius: Optional, mean earth radius (C{meter}).
@keyword wrap: Wrap and unroll longitudes (C{bool}).
@return: Polygon area (C{meter}, same units as I{radius}, squared).
@raise TypeError: Some I{points} are not L{LatLon}.
@raise ValueError: Insufficient number of I{points}.
@note: The area is based on Karney's U{'Area of a spherical polygon'
<http://osgeo-org.1560.x6.nabble.com/Area-of-a-spherical-polygon-td3841625.html>}.
@see: L{pygeodesy.areaOf}, L{sphericalNvector.areaOf} and
L{ellipsoidalKarney.areaOf}.
@example:
>>> b = LatLon(45, 1), LatLon(45, 2), LatLon(46, 2), LatLon(46, 1)
>>> areaOf(b) # 8666058750.718977
>>> c = LatLon(0, 0), LatLon(1, 0), LatLon(0, 1)
>>> areaOf(c) # 6.18e9
'''
n, points = _Trll.points2(points, closed=True)
# Area method due to Karney: for each edge of the polygon,
#
# tan(Δλ/2) · (tan(φ1/2) + tan(φ2/2))
# tan(E/2) = ------------------------------------
# 1 + tan(φ1/2) · tan(φ2/2)
#
# where E is the spherical excess of the trapezium obtained by
# extending the edge to the equator-circle vector for each edge
if iterNumpy2(points):
def _exs(n, points): # iterate over spherical edge excess
a1, b1 = points[n - 1].to2ab()
ta1 = tan_2(a1)
for i in range(n):
a2, b2 = points[i].to2ab()
db, b2 = unrollPI(b1, b2, wrap=wrap)
ta2, tdb = map1(tan_2, a2, db)
yield atan2(tdb * (ta1 + ta2), 1 + ta1 * ta2)
ta1, b1 = ta2, b2
s = fsum(_exs(n, points)) * 2
else:
a1, b1 = points[n - 1].to2ab()
s, ta1 = [], tan_2(a1)
for i in range(n):
a2, b2 = points[i].to2ab()
db, b2 = unrollPI(b1, b2, wrap=wrap)
ta2, tdb = map1(tan_2, a2, db)
s.append(atan2(tdb * (ta1 + ta2), 1 + ta1 * ta2))
ta1, b1 = ta2, b2
s = fsum(s) * 2
if isPoleEnclosedBy(points):
s = abs(s) - PI2
return abs(s * radius**2)
def isenclosedBy(self, points):
'''Check whether a (convex) polygon encloses this point.
@param points: The polygon points (L{LatLon}[]).
@return: C{True} if the polygon encloses this point,
C{False} otherwise.
@raise ValueError: Insufficient number of I{points} or
non-convex polygon.
@raise TypeError: Some I{points} are not L{LatLon}.
@example:
>>> b = LatLon(45,1), LatLon(45,2), LatLon(46,2), LatLon(46,1)
>>> p = LatLon(45,1, 1.1)
>>> inside = p.isEnclosedBy(b) # True
'''
n, points = self.points2(points, closed=True)
n0 = self.toVector3d()
if iterNumpy2(points):
v1 = points[n - 1].toVector3d()
v2 = points[n - 2].toVector3d()
gc1 = v2.cross(v1)
t0 = gc1.angleTo(n0) > PI_2
for i in range(n):
v2 = points[i].toVector3d()
gc = v1.cross(v2)
v1 = v2
ti = gc.angleTo(n0) > PI_2
if ti != t0:
return False # outside
if gc1.angleTo(gc, vSign=n0) < 0:
raise ValueError('non-convex: %r...' % (points[:2], ))
gc1 = gc
else:
# get great-circle vector for each edge
gc, v1 = [], points[n - 1].toVector3d()
for i in range(n):
v2 = points[i].toVector3d()
gc.append(v1.cross(v2))
v1 = v2
# check whether this point on same side of all
# polygon edges (to the left or right depending
# on anti-/clockwise polygon direction)
t0 = gc[0].angleTo(n0) > PI_2 # True if on the right
for i in range(1, n):
ti = gc[i].angleTo(n0) > PI_2
if ti != t0: # different sides of edge i
return False # outside
# check for convex polygon (otherwise
# the test above is not reliable)
gc1 = gc[n - 1]
for gc2 in gc:
# angle between gc vectors, signed by direction of n0
if gc1.angleTo(gc2, vSign=n0) < 0:
raise ValueError('non-convex: %r...' % (points[:2], ))
gc1 = gc2
return True # inside