def bearing_(a1, b1, a2, b2, final=False, wrap=False):
'''Compute the initial or final bearing (forward or reverse
azimuth) between a (spherical) start and end point.
@param a1: Start latitude (C{radians}).
@param b1: Start longitude (C{radians}).
@param a2: End latitude (C{radians}).
@param b2: End longitude (C{radians}).
@keyword final: Return final bearing if C{True}, initial
otherwise (C{bool}).
@keyword wrap: Wrap and L{unrollPI} longitudes (C{bool}).
@return: Initial or final bearing (compass C{radiansPI2}) or
zero if start and end point coincide.
'''
if final:
a1, b1, a2, b2 = a2, b2, a1, b1
r = PI + PI2
else:
r = PI2
db, _ = unrollPI(b1, b2, wrap=wrap)
ca1, ca2, cdb = map1(cos, a1, a2, db)
sa1, sa2, sdb = map1(sin, a1, a2, db)
# see <http://MathForum.org/library/drmath/view/55417.html>
x = ca1 * sa2 - sa1 * ca2 * cdb
y = sdb * ca2
return (atan2(y, x) + r) % PI2 # wrapPI2
def crossingParallels(self, other, lat, wrap=False):
'''Return the pair of meridians at which a great circle defined
by this and an other point crosses the given latitude.
@param other: The other point defining great circle (L{LatLon}).
@param lat: Latitude at the crossing (C{degrees}).
@keyword wrap: Wrap and unroll longitudes (C{bool}).
@return: 2-Tuple (lon1, lon2) in (C{degrees180}) or C{None}
if the great circle doesn't reach the given I{lat}.
'''
self.others(other)
a1, b1 = self.to2ab()
a2, b2 = other.to2ab()
a = radians(lat)
db, b2 = unrollPI(b1, b2, wrap=wrap)
ca, ca1, ca2, cdb = map1(cos, a, a1, a2, db)
sa, sa1, sa2, sdb = map1(sin, a, a1, a2, db)
x = sa1 * ca2 * ca * sdb
y = sa1 * ca2 * ca * cdb - ca1 * sa2 * ca
z = ca1 * ca2 * sa * sdb
h = hypot(x, y)
if h < EPS or abs(z) > h:
return None # great circle doesn't reach latitude
m = atan2(-y, x) + b1 # longitude at max latitude
d = acos1(z / h) # delta longitude to intersections
return degrees180(m - d), degrees180(m + d)
def _x3d2(start, end, wrap, n):
# see <http://www.EdWilliams.org/intersect.htm> (5) ff
a1, b1 = start.to2ab()
if isscalar(end): # bearing, make a point
a2, b2 = _destination2_(a1, b1, PI_4, radians(end))
else: # must be a point
_Trll.others(end, name='end' + n)
a2, b2 = end.to2ab()
db, b2 = unrollPI(b1, b2, wrap=wrap)
if max(abs(db), abs(a2 - a1)) < EPS:
raise ValueError('intersection %s%s null: %r' % ('path', n,
(start, end)))
# note, in EdWilliams.org/avform.htm W is + and E is -
b21, b12 = db * 0.5, -(b1 + b2) * 0.5
cb21, cb12 = map1(cos, b21, b12)
sb21, sb12 = map1(sin, b21, b12)
sa21, sa12 = map1(sin, a1 - a2, a1 + a2)
x = Vector3d(sa21 * sb12 * cb21 - sa12 * cb12 * sb21,
sa21 * cb12 * cb21 + sa12 * sb12 * sb21,
cos(a1) * cos(a2) * sin(db),
ll=start)
return x.unit(), (db, (a2 - a1)) # negated d
def distanceTo(self, other, radius=R_M, wrap=False):
'''Compute the distance from this to an other point.
@param other: The other point (L{LatLon}).
@keyword radius: Optional, mean earth radius (C{meter}).
@keyword wrap: Wrap and unroll longitudes (C{bool}).
@return: Distance between this and the I{other} point
(C{meter}, same units as I{radius}).
@raise TypeError: The I{other} point is not L{LatLon}.
@example:
>>> p1 = LatLon(52.205, 0.119)
>>> p2 = LatLon(48.857, 2.351);
>>> d = p1.distanceTo(p2) # 404300
'''
self.others(other)
a1, b1 = self.to2ab()
a2, b2 = other.to2ab()
db, b2 = unrollPI(b1, b2, wrap=wrap)
r = haversine_(a2, a1, db)
return r * float(radius)
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
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
def intermediateTo(self, other, fraction, height=None, wrap=False):
'''Locate the point at given fraction between this and an
other point.
@param other: The other point (L{LatLon}).
@param fraction: Fraction between both points (float, 0.0 =
this point, 1.0 = the other point).
@keyword height: Optional height, overriding the fractional
height (C{meter}).
@keyword wrap: Wrap and unroll longitudes (C{bool}).
@return: Intermediate point (L{LatLon}).
@raise TypeError: The I{other} point is not L{LatLon}.
@example:
>>> p1 = LatLon(52.205, 0.119)
>>> p2 = LatLon(48.857, 2.351)
>>> p = p1.intermediateTo(p2, 0.25) # 51.3721°N, 000.7073°E
@JSname: I{intermediatePointTo}.
'''
self.others(other)
a1, b1 = self.to2ab()
a2, b2 = other.to2ab()
db, b2 = unrollPI(b1, b2, wrap=wrap)
r = haversine_(a2, a1, db)
sr = sin(r)
if abs(sr) > EPS:
cb1, cb2, ca1, ca2 = map1(cos, b1, b2, a1, a2)
sb1, sb2, sa1, sa2 = map1(sin, b1, b2, a1, a2)
A = sin((1 - fraction) * r) / sr
B = sin(fraction * r) / sr
x = A * ca1 * cb1 + B * ca2 * cb2
y = A * ca1 * sb1 + B * ca2 * sb2
z = A * sa1 + B * sa2
a = atan2(z, hypot(x, y))
b = atan2(y, x)
else: # points too close
a = favg(a1, a2, f=fraction)
b = favg(b1, b2, f=fraction)
if height is None:
h = self._havg(other, f=fraction)
else:
h = height
return self.classof(degrees90(a), degrees180(b), height=h)
def _rads2(n, pts): # trapezoidal areas in rads**2
x1, y1, _ = pts[n - 1]
for i in range(n):
x2, y2, _ = pts[i]
w, x2 = unrollPI(x1, x2, wrap=wrap if i < (n - 1) else False)
# approximate trapezoid by a rectangle, adjusting
# the top width by the cosine of the latitudinal
# average and bottom width by some fudge factor
h = (y2 + y1) * 0.5
if adjust:
w *= (cos(h) + 1.2876) * 0.5
yield h * w # signed trapezoidal area
x1, y1 = x2, y2
def centroidOf(points, wrap=True, LatLon=None):
'''Determine the centroid of a polygon.
@param points: The polygon points (C{LatLon}[]).
@keyword wrap: Wrap lat-, wrap and unroll longitudes (C{bool}).
@keyword LatLon: Optional (sub-)class to return the centroid
(L{LatLon}) or C{None}.
@return: Centroid location (I{LatLon}) or as 2-tuple (C{lat, lon})
in C{degrees} if I{LatLon} is C{None}.
@raise TypeError: Some I{points} are not C{LatLon}.
@raise ValueError: Insufficient number of I{points} or I{points}
enclose a pole or zero area.
@see: U{Centroid<http://WikiPedia.org/wiki/Centroid#Of_a_polygon>}
and U{Calculating The Area And Centroid Of A Polygon
<http://www.Seas.UPenn.edu/~sys502/extra_materials/
Polygon%20Area%20and%20Centroid.pdf>}.
'''
# setting radius=1 converts degrees to radians
pts = LatLon2psxy(points, closed=True, radius=1, wrap=wrap)
n = len(pts)
A, X, Y = Fsum(), Fsum(), Fsum()
x1, y1, _ = pts[n-1]
for i in range(n):
x2, y2, _ = pts[i]
if wrap and i < (n - 1):
_, x2 = unrollPI(x1, x2, wrap=True)
t = x1 * y2 - x2 * y1
A += t
X += t * (x1 + x2)
Y += t * (y1 + y2)
# XXX more elaborately:
# t1, t2 = x1 * y2, -(x2 * y1)
# A.fadd_(t1, t2)
# X.fadd_(t1 * x1, t1 * x2, t2 * x1, t2 * x2)
# Y.fadd_(t1 * y1, t1 * y2, t2 * y1, t2 * y2)
x1, y1 = x2, y2
A = A.fsum() * 3.0 # 6.0 / 2.0
if abs(A) < EPS:
raise ValueError('polar or zero area: %r' % (pts,))
Y, X = degrees90(Y.fsum() / A), degrees180(X.fsum() / A)
return (Y, X) if LatLon is None else LatLon(Y, X)
def distance3To(self, other, radius=R_M, wrap=False):
'''Compute the great-circle distance between this and an other
geohash using the U{Haversine
<http://www.Movable-Type.co.UK/scripts/latlong.html>} formula.
@param other: The other geohash (L{Geohash}).
@keyword radius: Optional, mean earth radius (C{meter}).
@keyword wrap: Wrap and unroll longitudes (C{bool}).
@return: Great-circle distance (C{meter}, same units as I{radius}).
@raise TypeError: The I{other} is not a L{Geohash}, C{LatLon}
or C{str}.
'''
other = _2Geohash(other)
a1, b1 = self.ab
a2, b2 = other.ab
db, b2 = unrollPI(b1, b2, wrap=wrap)
return haversine_(a2, a1, db) * float(radius)
def midpointTo(self, other, height=None, wrap=False):
'''Find the midpoint between this and an other point.
@param other: The other point (L{LatLon}).
@keyword height: Optional height for midpoint, overriding
the mean height (C{meter}).
@keyword wrap: Wrap and unroll longitudes (C{bool}).
@return: Midpoint (L{LatLon}).
@raise TypeError: The I{other} point is not L{LatLon}.
@example:
>>> p1 = LatLon(52.205, 0.119)
>>> p2 = LatLon(48.857, 2.351)
>>> m = p1.midpointTo(p2) # '50.5363°N, 001.2746°E'
'''
self.others(other)
# see <http://MathForum.org/library/drmath/view/51822.html>
a1, b1 = self.to2ab()
a2, b2 = other.to2ab()
db, b2 = unrollPI(b1, b2, wrap=wrap)
ca1, ca2, cdb = map1(cos, a1, a2, db)
sa1, sa2, sdb = map1(sin, a1, a2, db)
x = ca2 * cdb + ca1
y = ca2 * sdb
a = atan2(sa1 + sa2, hypot(x, y))
b = atan2(y, x) + b1
h = self._havg(other) if height is None else height
return self.classof(degrees90(a), degrees180(b), height=h)
def intersection(start1, bearing1, start2, bearing2,
height=None, wrap=False, LatLon=LatLon):
'''Compute the intersection point of two paths each defined
by a start point and an initial bearing.
@param start1: Start point of first path (L{LatLon}).
@param bearing1: Initial bearing from start1 (compass C{degrees360}).
@param start2: Start point of second path (L{LatLon}).
@param bearing2: Initial bearing from start2 (compass C{degrees360}).
@keyword height: Optional height for the intersection point,
overriding the mean height (C{meter}).
@keyword wrap: Wrap and unroll longitudes (C{bool}).
@keyword LatLon: Optional (sub-)class for the intersection point
(L{LatLon}) or C{None}.
@return: Intersection point (L{LatLon}) or 3-tuple (C{degrees90},
C{degrees180}, height) if C{LatLon} is C{None}.
@raise TypeError: Point I{start1} or I{start2} is not L{LatLon}.
@raise ValueError: Intersection is ambiguous or infinite or
the paths are parallel or coincident.
@example:
>>> p = LatLon(51.8853, 0.2545)
>>> s = LatLon(49.0034, 2.5735)
>>> i = intersection(p, 108.547, s, 32.435) # '50.9078°N, 004.5084°E'
'''
_Trll.others(start1, name='start1')
_Trll.others(start2, name='start2')
# see <http://www.EdWilliams.org/avform.htm#Intersection>
a1, b1 = start1.to2ab()
a2, b2 = start2.to2ab()
db, b2 = unrollPI(b1, b2, wrap=wrap)
r12 = haversine_(a2, a1, db)
if abs(r12) < EPS: # [nearly] coincident points
a, b = map1(degrees, favg(a1, a2), favg(b1, b2))
else:
ca1, ca2, cr12 = map1(cos, a1, a2, r12)
sa1, sa2, sr12 = map1(sin, a1, a2, r12)
x1, x2 = (sr12 * ca1), (sr12 * ca2)
if abs(x1) < EPS or abs(x2) < EPS:
raise ValueError('intersection %s: %r vs %r' % ('parallel', start1, start2))
# handle domain error for equivalent longitudes,
# see also functions asin_safe and acos_safe at
# <http://www.EdWilliams.org/avform.htm#Math>
t1, t2 = map1(acos1, (sa2 - sa1 * cr12) / x1,
(sa1 - sa2 * cr12) / x2)
if sin(db) > 0:
t12, t21 = t1, PI2 - t2
else:
t12, t21 = PI2 - t1, t2
t13, t23 = map1(radiansPI2, bearing1, bearing2)
x1, x2 = map1(wrapPI, t13 - t12, # angle 2-1-3
t21 - t23) # angle 1-2-3
sx1, sx2 = map1(sin, x1, x2)
if sx1 == 0 and sx2 == 0:
raise ValueError('intersection %s: %r vs %r' % ('infinite', start1, start2))
sx3 = sx1 * sx2
if sx3 < 0:
raise ValueError('intersection %s: %r vs %r' % ('ambiguous', start1, start2))
cx1, cx2 = map1(cos, x1, x2)
x3 = acos1(cr12 * sx3 - cx2 * cx1)
r13 = atan2(sr12 * sx3, cx2 + cx1 * cos(x3))
a, b = _destination2(a1, b1, r13, t13)
h = start1._havg(start2) if height is None else height
return (a, b, h) if LatLon is None else LatLon(a, b, height=h)
def intersection(start1,
end1,
start2,
end2,
height=None,
wrap=False,
LatLon=LatLon):
'''Compute the intersection point of two paths both defined
by two points or a start point and bearing from North.
@param start1: Start point of the first path (L{LatLon}).
@param end1: End point ofthe first path (L{LatLon}) or
the initial bearing at the first start point
(compass C{degrees360}).
@param start2: Start point of the second path (L{LatLon}).
@param end2: End point of the second path (L{LatLon}) or
the initial bearing at the second start point
(compass C{degrees360}).
@keyword height: Optional height for the intersection point,
overriding the mean height (C{meter}).
@keyword wrap: Wrap and unroll longitudes (C{bool}).
@keyword LatLon: Optional (sub-)class for the intersection point
(L{LatLon}) or C{None}.
@return: The intersection point (I{LatLon}) or 3-tuple
(C{degrees90}, C{degrees180}, height) if I{LatLon}
is C{None}. An alternate intersection point might
be the L{antipode} to the returned result.
@raise TypeError: Start or end point(s) not L{LatLon}.
@raise ValueError: Intersection is ambiguous or infinite or
the paths are parallel, coincident or null.
@example:
>>> p = LatLon(51.8853, 0.2545)
>>> s = LatLon(49.0034, 2.5735)
>>> i = intersection(p, 108.547, s, 32.435) # '50.9078°N, 004.5084°E'
'''
_Trll.others(start1, name='start1')
_Trll.others(start2, name='start2')
a1, b1 = start1.to2ab()
a2, b2 = start2.to2ab()
db, b2 = unrollPI(b1, b2, wrap=wrap)
r12 = haversine_(a2, a1, db)
if abs(r12) < EPS: # [nearly] coincident points
a, b = map1(degrees, favg(a1, a2), favg(b1, b2))
# see <http://www.EdWilliams.org/avform.htm#Intersection>
elif isscalar(end1) and isscalar(end2): # both bearings
ca1, ca2, cr12 = map1(cos, a1, a2, r12)
sa1, sa2, sr12 = map1(sin, a1, a2, r12)
x1, x2 = (sr12 * ca1), (sr12 * ca2)
if abs(x1) < EPS or abs(x2) < EPS:
raise ValueError('intersection %s: %r vs %r' % ('parallel',
(start1, end1),
(start2, end2)))
# handle domain error for equivalent longitudes,
# see also functions asin_safe and acos_safe at
# <http://www.EdWilliams.org/avform.htm#Math>
t1, t2 = map1(acos1, (sa2 - sa1 * cr12) / x1, (sa1 - sa2 * cr12) / x2)
if sin(db) > 0:
t12, t21 = t1, PI2 - t2
else:
t12, t21 = PI2 - t1, t2
t13, t23 = map1(radiansPI2, end1, end2)
x1, x2 = map1(
wrapPI,
t13 - t12, # angle 2-1-3
t21 - t23) # angle 1-2-3
sx1, sx2 = map1(sin, x1, x2)
if sx1 == 0 and sx2 == 0: # max(abs(sx1), abs(sx2)) < EPS
raise ValueError('intersection %s: %r vs %r' % ('infinite',
(start1, end1),
(start2, end2)))
sx3 = sx1 * sx2
# if sx3 < 0:
# raise ValueError('intersection %s: %r vs %r' % ('ambiguous',
# (start1, end1), (start2, end2)))
cx1, cx2 = map1(cos, x1, x2)
x3 = acos1(cr12 * sx3 - cx2 * cx1)
r13 = atan2(sr12 * sx3, cx2 + cx1 * cos(x3))
a, b = _destination2(a1, b1, r13, t13)
# choose antipode for opposing bearings
if _xb(a1, b1, end1, a, b, wrap) < 0 or \
_xb(a2, b2, end2, a, b, wrap) < 0:
a, b = antipode(a, b)
else: # end point(s) or bearing(s)
x1, d1 = _x3d2(start1, end1, wrap, '1')
x2, d2 = _x3d2(start2, end2, wrap, '2')
x = x1.cross(x2)
if x.length < EPS: # [nearly] colinear or parallel paths
raise ValueError('intersection %s: %r vs %r' % ('colinear',
(start1, end1),
(start2, end2)))
a, b = x.to2ll()
# choose intersection similar to sphericalNvector
d1 = _xdot(d1, a1, b1, a, b, wrap)
d2 = _xdot(d2, a2, b2, a, b, wrap)
if (d1 < 0 and d2 > 0) or (d1 > 0 and d2 < 0):
a, b = antipode(a, b)
h = start1._havg(start2) if height is None else height
return (a, b, h) if LatLon is None else LatLon(a, b, height=h)
def _xdot(d, a1, b1, a, b, wrap):
# compute dot product d . (-b + b1, a - a1)
db, _ = unrollPI(b1, radians(b), wrap=wrap)
return fdot(d, db, radians(a) - a1)
def _distances(self, x, y): # (x, y) radians
for xk, yk in zip(self._xs, self._ys):
dx, _ = unrollPI(xk, x, wrap=self._wrap)
if self._adjust:
dx *= _scaler(yk, y)
yield dx**2 + (y - yk)**2 # like equirectangular_ distance2
def _distances(self, x, y): # (x, y) radians
for xk, yk in zip(self._xs, self._ys):
d, _ = unrollPI(xk, x, wrap=self._wrap)
yield haversine_(y, yk, d)
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)
