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 perimeterOf(points, closed=False, radius=R_M, wrap=True):
'''Compute the perimeter of a (spherical) polygon (with great circle
arcs joining the points).
@param points: The polygon points (L{LatLon}[]).
@keyword closed: Optionally, close the polygon (C{bool}).
@keyword radius: Mean earth radius (C{meter}).
@keyword wrap: Wrap and unroll longitudes (C{bool}).
@return: Polygon perimeter (C{meter}, same units as B{C{radius}}).
@raise TypeError: Some B{C{points}} are not L{LatLon}.
@raise ValueError: Insufficient number of B{C{points}}.
@note: This perimeter is based on the L{haversine} formula.
@see: L{pygeodesy.perimeterOf}, L{sphericalNvector.perimeterOf}
and L{ellipsoidalKarney.perimeterOf}.
'''
n, points = _Trll.points2(points, closed=closed)
def _rads(n, points, closed): # angular edge lengths in radians
i, m = _imdex2(closed, n)
a1, b1 = points[i].to2ab()
for i in range(m, n):
a2, b2 = points[i].to2ab()
db, b2 = unrollPI(b1, b2, wrap=wrap)
yield haversine_(a2, a1, db)
a1, b1 = a2, b2
r = fsum(_rads(n, points, closed))
return r * float(radius)
def perimeterOf(points, closed=False, radius=R_M):
'''Compute the perimeter of a (spherical) polygon (with great circle
arcs joining consecutive points).
@param points: The polygon points (L{LatLon}[]).
@keyword closed: Optionally, close the polygon (C{bool}).
@keyword radius: Optional, mean earth radius (C{meter}).
@return: Polygon perimeter (C{meter}, same units as B{C{radius}}).
@raise TypeError: Some B{C{points}} are not L{LatLon}.
@raise ValueError: Insufficient number of B{C{points}}.
@see: L{pygeodesy.perimeterOf}, L{sphericalTrigonometry.perimeterOf}
and L{ellipsoidalKarney.perimeterOf}.
'''
n, points = _Nvll.points2(points, closed=closed)
def _rads(n, points, closed): # angular edge lengths in radians
i, m = _imdex2(closed, n)
v1 = points[i].toNvector()
for i in range(m, n):
v2 = points[i].toNvector()
yield v1.angleTo(v2)
v1 = v2
r = fsum(_rads(n, points, closed))
return r * float(radius)
def sumOf(nvectors, Vector=None, h=None, **Vector_kwds):
'''Return the vectorial sum of two or more n-vectors.
@arg nvectors: Vectors to be added (C{Nvector}[]).
@kwarg Vector: Optional class for the vectorial sum (C{Nvector})
or C{None}.
@kwarg h: Optional height, overriding the mean height (C{meter}).
@kwarg Vector_kwds: Optional, additional B{C{Vector}} keyword
arguments, ignored if C{B{Vector}=None}.
@return: Vectorial sum (B{C{Vector}}) or a L{Vector4Tuple}C{(x, y,
z, h)} if B{C{Vector}} is C{None}.
@raise VectorError: No B{C{nvectors}}.
'''
n, nvectors = len2(nvectors)
if n < 1:
raise VectorError(nvectors=n, txt=MISSING)
if h is None:
h = fsum(v.h for v in nvectors) / float(n)
if Vector is None:
r = _sumOf(nvectors, Vector=Vector3Tuple).to4Tuple(h)
else:
r = _sumOf(nvectors, Vector=Vector, h=h, **Vector_kwds)
return r
def sumOf(vectors, Vector=Vector3d, **kwds):
'''Compute the vectorial sum of several vectors.
@param vectors: Vectors to be added (L{Vector3d}[]).
@keyword Vector: Optional class for the vectorial sum (L{Vector3d}).
@keyword kwds: Optional, additional B{C{Vector}} keyword arguments.
@return: Vectorial sum (B{C{Vector}}).
@raise VectorError: No B{C{vectors}}.
'''
n, vectors = len2(vectors)
if n < 1:
raise VectorError('no vectors: %r' & (n, ))
return Vector(fsum(v.x for v in vectors), fsum(v.y for v in vectors),
fsum(v.z for v in vectors), **kwds)
def areaOf(points, radius=R_M, wrap=True):
'''Calculate the area of a (spherical) polygon (with great circle
arcs joining the points).
@arg points: The polygon points (L{LatLon}[]).
@kwarg radius: Mean earth radius (C{meter}).
@kwarg wrap: Wrap and unroll longitudes (C{bool}).
@return: Polygon area (C{meter}, same units as B{C{radius}}, squared).
@raise PointsError: Insufficient number of B{C{points}}.
@raise TypeError: Some B{C{points}} are not L{LatLon}.
@raise ValueError: Invalid B{C{radius}}.
@note: The area is based on Karney's U{'Area of a spherical polygon'
<https://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
def _exs(n, points): # iterate over spherical edge excess
a1, b1 = points[n - 1].philam
ta1 = tan_2(a1)
for i in range(n):
a2, b2 = points[i].philam
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
if isPoleEnclosedBy(points):
s = abs(s) - PI2
return abs(s * Radius(radius)**2)
def sumOf(vectors, Vector=Vector3d, **Vector_kwds):
'''Compute the vectorial sum of several vectors.
@arg vectors: Vectors to be added (L{Vector3d}[]).
@kwarg Vector: Optional class for the vectorial sum (L{Vector3d}).
@kwarg Vector_kwds: Optional B{C{Vector}} keyword arguments,
ignored if C{B{Vector}=None}.
@return: Vectorial sum as B{C{Vector}} or if B{C{Vector}} is
C{None}, a L{Vector3Tuple}C{(x, y, z)}.
@raise VectorError: No B{C{vectors}}.
'''
n, vectors = len2(vectors)
if n < 1:
raise VectorError(vectors=n, txt=MISSING)
v = Vector3Tuple(fsum(v.x for v in vectors),
fsum(v.y for v in vectors),
fsum(v.z for v in vectors))
return _V_n(v, sumOf.__name__, Vector, Vector_kwds)
def sumOf(vectors, Vector=Vector3d, **kwds):
'''Compute the vectorial sum of several vectors.
@param vectors: Vectors to be added (L{Vector3d}[]).
@keyword Vector: Optional class for the vectorial sum (L{Vector3d}).
@keyword kwds: Optional, additional B{C{Vector}} keyword arguments,
ignored if C{B{Vector}=None}.
@return: Vectorial sum (B{C{Vector}}).
@raise VectorError: No B{C{vectors}}.
'''
n, vectors = len2(vectors)
if n < 1:
raise VectorError('no vectors: %r' & (n, ))
r = Vector3Tuple(fsum(v.x for v in vectors), fsum(v.y for v in vectors),
fsum(v.z for v in vectors))
if Vector is not None:
r = Vector(r.x, r.y, r.z, **kwds) # PYCHOK x, y, z
return r
def sumOf(vectors, Vector=Vector3d, **Vector_kwds):
'''Compute the vectorial sum of several vectors.
@arg vectors: Vectors to be added (L{Vector3d}[]).
@kwarg Vector: Optional class for the vectorial sum (L{Vector3d}).
@kwarg Vector_kwds: Optional B{C{Vector}} keyword arguments,
ignored if B{C{Vector=None}}.
@return: Vectorial sum (B{C{Vector}}).
@raise VectorError: No B{C{vectors}}.
'''
n, vectors = len2(vectors)
if n < 1:
raise VectorError(vectors=n, txt=_Missing)
r = Vector3Tuple(fsum(v.x for v in vectors), fsum(v.y for v in vectors),
fsum(v.z for v in vectors))
if Vector is not None:
r = Vector(r.x, r.y, r.z, **Vector_kwds) # PYCHOK x, y, z
return r
def sumOf(nvectors, Vector=Nvector, h=None, **kwds):
'''Return the vectorial sum of two or more n-vectors.
@param nvectors: Vectors to be added (L{Nvector}[]).
@keyword Vector: Optional class for the vectorial sum (L{Nvector}).
@keyword kwds: Optional, additional B{C{Vector}} keyword arguments.
@keyword h: Optional height, overriding the mean height (C{meter}).
@return: Vectorial sum (B{C{Vector}}).
@raise VectorError: No B{C{nvectors}}.
'''
n, nvectors = len2(nvectors)
if n < 1:
raise VectorError('no nvectors: %r' & (n, ))
if h is None:
h = fsum(v.h for v in nvectors) / float(n)
return _sumOf(nvectors, Vector=Vector, h=h, **kwds)
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
