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 initialBearingTo(self, other, wrap=False):
'''Compute the initial bearing (aka forward azimuth) from
this to an other point.
@param other: The other point (L{LatLon}).
@keyword wrap: Wrap and unroll longitudes (bool).
@return: Initial bearing (compass degrees).
@raise TypeError: The I{other} point is not L{LatLon}.
@example:
>>> p1 = LatLon(52.205, 0.119)
>>> p2 = LatLon(48.857, 2.351)
>>> b = p1.initialBearingTo(p2) # 156.2
@JSname: I{bearingTo}.
'''
self.others(other)
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)
# see <http://mathforum.org/library/drmath/view/55417.html>
x = ca1 * sa2 - sa1 * ca2 * cdb
y = sdb * ca2
return degrees360(atan2(y, x))
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 _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 greatCircle(self, bearing):
'''Compute the vector normal to great circle obtained by heading
on the given initial bearing from this point.
Direction of vector is such that initial bearing vector
b = c × n, where n is an n-vector representing this point.
@param bearing: Bearing from this point (compass C{degrees360}).
@return: Vector representing great circle (L{Vector3d}).
@example:
>>> p = LatLon(53.3206, -1.7297)
>>> g = p.greatCircle(96.0)
>>> g.toStr() # (-0.794, 0.129, 0.594)
'''
a, b = self.to2ab()
t = radians(bearing)
ca, cb, ct = map1(cos, a, b, t)
sa, sb, st = map1(sin, a, b, t)
return Vector3d(sb * ct - cb * sa * st, -cb * ct - sb * sa * st,
ca * st) # XXX .unit()?
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 cross(self, other, raiser=None):
'''Compute the cross product of this and an other vector.
@param other: The other vector (L{Vector3d}).
@keyword raiser: Optional, L{CrossError} label if raised (C{str}).
@return: Cross product (L{Vector3d}).
@raise CrossError: Zero or near-zero cross product and both
I{raiser} and L{crosserrors} set.
@raise TypeError: Incompatible I{other} C{type}.
@raise ValueError: Coincident or colinear to I{other}.
'''
self.others(other)
x = self.y * other.z - self.z * other.y
y = self.z * other.x - self.x * other.z
z = self.x * other.y - self.y * other.x
if raiser and self.crosserrors and max(map1(abs, x, y, z)) < EPS:
t = 'coincident' if self.isequalTo(other) else 'colinear'
r = getattr(other, '_fromll', None) or other
raise CrossError('%s %s: %r' % (t, raiser, r))
return self.classof(x, y, z)
def __init__(self, x, y, radius=R_MA, name=''):
'''New Web Mercator (WM) coordinate.
@param x: Easting from central meridian (C{meter}).
@param y: Northing from equator (C{meter}).
@keyword radius: Optional earth radius (C{meter}).
@keyword name: Optional name (C{str}).
@raise ValueError: Invalid I{x}, I{y} or I{radius}.
@example:
>>> import pygeodesy
>>> w = pygeodesy.Wm(448251, 5411932)
'''
if name:
self.name = name
try:
self._x, self._y, r = map1(float, x, y, radius)
except (TypeError, ValueError):
raise ValueError('%s invalid: %r' % (Wm.__name__, (x, y, radius)))
if r < EPS: # check radius
t = '%s.%s' % (classname(self), 'radius')
raise ValueError('%s invalid: %r' % (t, r))
self._radius = r
def _xyhs3(atype, m, knots, off=True):
# convert knot C{LatLon}s to tuples or C{NumPy} arrays and C{SciPy} sphericals
xs, ys, hs = zip(*_xyhs(knots, off=off)) # PYCHOK unzip
n = len(hs)
if n < m:
raise HeightError('insufficient %s: %s, need %s' % ('knots', n, m))
return map1(atype, xs, ys, hs)
def toOsgr(latlon, lon=None, datum=Datums.WGS84, Osgr=Osgr):
'''Convert a lat-/longitude point to an OSGR coordinate.
@param latlon: Latitude (degrees) or an (ellipsoidal)
geodetic I{LatLon} point.
@keyword lon: Optional longitude in degrees (scalar or None).
@keyword datum: Optional datum to convert (I{Datum}).
@keyword Osgr: Optional Osgr class to use for the
OSGR coordinate (L{Osgr}).
@return: The OSGR coordinate (L{Osgr}).
@raise TypeError: If I{latlon} is not ellipsoidal or if
I{datum} conversion failed.
@raise ValueError: Invalid I{latlon} or I{lon}.
@example:
>>> p = LatLon(52.65798, 1.71605)
>>> r = toOsgr(p) # TG 51409 13177
>>> # for conversion of (historical) OSGB36 lat-/longitude:
>>> r = toOsgr(52.65757, 1.71791, datum=Datums.OSGB36)
'''
if not isinstance(latlon, _eLLb):
# XXX fix failing _eLLb.convertDatum()
latlon = _eLLb(*parseDMS2(latlon, lon), datum=datum)
elif lon is not None:
raise ValueError('%s not %s: %r' % ('lon', None, lon))
E = _OSGB36.ellipsoid
ll = _ll2datum(latlon, _OSGB36, 'latlon')
a, b = map1(radians, ll.lat, ll.lon)
ca, sa, ta = cos(a), sin(a), tan(a)
s = E.e2s2(sa)
v = E.a * _F0 / sqrt(s) # nu
r = s / E.e12 # nu / rho == v / (v * E.e12 / s)
x2 = r - 1 # η2
ca3, ca5 = fpowers(ca, 5, 3) # PYCHOK false!
ta2, ta4 = fpowers(ta, 4, 2) # PYCHOK false!
vsa = v * sa
I4 = (E.b * _F0 * _M(E.Mabcd, a) + _N0, (vsa / 2) * ca,
(vsa / 24) * ca3 * fsum_(5, -ta2, 9 * x2),
(vsa / 720) * ca5 * fsum_(61, ta4, -58 * ta2))
V4 = (_E0, (v * ca), (v / 6) * ca3 * (r - ta2), (v / 120) * ca5 * fdot(
(-18, 1, 14, -58), ta2, 5 + ta4, x2, ta2 * x2))
d, d2, d3, d4, d5, d6 = fpowers(b - _B0, 6) # PYCHOK false!
n = fdot(I4, 1, d2, d4, d6)
e = fdot(V4, 1, d, d3, d5)
return Osgr(e, n)
def to2ab(self):
'''Return this point's lat- and longitude in radians.
@return: 2-Tuple (lat, lon) in (C{radiansPI_2}, C{radiansPI}).
'''
if not self._ab:
self._ab = map1(radians, self.lat, self.lon)
return self._ab
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 to2ab(self):
'''Return this point's lat- and longitude in C{radians}.
@return: A L{PhiLam2Tuple}C{(phi, lambda)}.
'''
if not self._ab:
a, b = map1(radians, self.lat, self.lon)
self._ab = self._xnamed(PhiLam2Tuple(a, b))
return self._ab
def _destination2_(a, b, r, t):
'''(INTERNAL) Computes destination lat- and longitude.
@param a: Latitude (C{radians}).
@param b: Longitude (C{radians}).
@param r: Angular distance (C{radians}).
@param t: Bearing (compass C{radians}).
@return: 2-Tuple (lat, lon) of (radians, radians).
'''
# see <http://www.EdWilliams.org/avform.htm#LL>
ca, cr, ct = map1(cos, a, r, t)
sa, sr, st = map1(sin, a, r, t)
a = asin(ct * sr * ca + cr * sa)
d = atan2(st * sr * ca, cr - sa * sin(a))
# note, in EdWilliams.org/avform.htm W is + and E is -
return a, b + d
def _destination2(a, b, r, t):
'''(INTERNAL) Computes destination lat-/longitude.
@param a: Latitude (C{radians}).
@param b: Longitude (C{radians}).
@param r: Angular distance (C{radians}).
@param t: Bearing (compass C{radians}).
@return: 2-Tuple (lat, lon) of (C{degrees90}, C{degrees180}).
'''
# see <http://www.EdWilliams.org/avform.htm#LL>
ca, cr, ct = map1(cos, a, r, t)
sa, sr, st = map1(sin, a, r, t)
a = asin(ct * sr * ca + cr * sa)
d = atan2(st * sr * ca, cr - sa * sin(a))
# note, use addition, not subtraction of d (since
# in EdWilliams.org/avform.htm W is + and E is -)
return degrees90(a), degrees180(b + d)
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 (meter).
@keyword wrap: Wrap and unroll longitudes (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
if height is None:
h = self._havg(other)
else:
h = height
return self.classof(degrees90(a), degrees180(b), height=h)
def bearing(lat1, lon1, lat2, lon2, final=False, wrap=False):
'''Compute the initial or final bearing (forward or reverse
azimuth) between a (spherical) start and end point.
@param lat1: Start latitude (C{degrees}).
@param lon1: Start longitude (C{degrees}).
@param lat2: End latitude (C{degrees}).
@param lon2: End longitude (C{degrees}).
@keyword final: Return final or initial bearing (C{bool}).
@keyword wrap: Wrap and L{unroll180} longitudes (C{bool}).
@return: Initial or final bearing (compass C{degrees360}) or
zero if start and end point coincide.
'''
a1, b1, a2, b2 = map1(radians, lat1, lon1, lat2, lon2)
return degrees(bearing_(a1, b1, a2, b2, final=final, wrap=wrap))
def bearing(lat1, lon1, lat2, lon2, **options):
'''Compute the initial or final bearing (forward or reverse
azimuth) between a (spherical) start and end point.
@param lat1: Start latitude (C{degrees}).
@param lon1: Start longitude (C{degrees}).
@param lat2: End latitude (C{degrees}).
@param lon2: End longitude (C{degrees}).
@keyword options: Optional keyword arguments for function
L{bearing_}.
@return: Initial or final bearing (compass C{degrees360}) or
zero if start and end point coincide.
'''
ab4 = map1(radians, lat1, lon1, lat2, lon2)
return degrees(bearing_(*ab4, **options))
def __init__(self, x, y, radius=R_MA):
'''New Web Mercator (WM) coordinate.
@param x: Easting from central meridian (meter).
@param y: Northing from equator (meter).
@keyword radius: Optional earth radius (meter).
@raise ValueError: Invalid I{x}, I{y} or I{radius}.
@example:
>>> import pygeodesy
>>> w = pygeodesy.Wm(448251, 5411932)
'''
try:
self._x, self._y, self._radius = map1(float, x, y, radius)
except (TypeError, ValueError):
raise ValueError('%s invalid: %r' % (Wm.__name__, (x, y, radius)))
_ = self.radius # PYCHOK check radius
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 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 trilaterate(point1,
distance1,
point2,
distance2,
point3,
distance3,
radius=R_M,
height=None,
LatLon=LatLon,
useZ=False):
'''Locate a point at given distances from three other points.
See also U{Trilateration<https://WikiPedia.org/wiki/Trilateration>}.
@param point1: First point (L{LatLon}).
@param distance1: Distance to the first point (C{meter}, same units
as B{C{radius}}).
@param point2: Second point (L{LatLon}).
@param distance2: Distance to the second point (C{meter}, same units
as B{C{radius}}).
@param point3: Third point (L{LatLon}).
@param distance3: Distance to the third point (C{meter}, same units
as B{C{radius}}).
@keyword radius: Optional, mean earth radius (C{meter}).
@keyword height: Optional height at the trilaterated point, overriding
the IDW height (C{meter}, same units as B{C{radius}}).
@keyword LatLon: Optional (sub-)class to return the trilaterated
point (L{LatLon}).
@keyword useZ: Include Z component iff non-NaN, non-zero (C{bool}).
@return: Trilaterated point (B{C{LatLon}}).
@raise TypeError: If B{C{point1}}, B{C{point2}} or B{C{point3}}
is not L{LatLon}.
@raise ValueError: Invalid B{C{radius}}, some B{C{distances}} exceed
trilateration or some B{C{points}} coincide.
'''
def _nd2(p, d, name, *qs):
# return Nvector and radial distance squared
_Nvll.others(p, name=name)
for q in qs:
if p.isequalTo(q, EPS):
raise ValueError('%s %s: %r' % ('coincident', 'points', p))
return p.toNvector(), (float(d) / radius)**2
if float(radius or 0) < EPS:
raise ValueError('%s %s: %r' % ('radius', 'invalid', radius))
n1, d12 = _nd2(point1, distance1, 'point1')
n2, d22 = _nd2(point2, distance2, 'point2', point1)
n3, d32 = _nd2(point3, distance3, 'point3', point1, point2)
# the following uses x,y coordinate system with origin at n1, x axis n1->n2
y = n3.minus(n1)
x = n2.minus(n1)
d = x.length # distance n1->n2
if d > EPS_2: # and y.length > EPS_2:
X = x.unit() # unit vector in x direction n1->n2
i = X.dot(y) # signed magnitude of x component of n1->n3
Y = y.minus(X.times(i)).unit() # unit vector in y direction
j = Y.dot(y) # signed magnitude of y component of n1->n3
if abs(j) > EPS_2:
# courtesy Carlos Freitas <https://GitHub.com/mrJean1/PyGeodesy/issues/33>
x = fsum_(d12, -d22, d**2) / (2 * d) # n1->intersection x- and ...
y = fsum_(d12, -d32, i**2, j**2) / (2 * j) - (x * i / j
) # ... y-component
n = n1.plus(X.times(x)).plus(Y.times(y)) # .plus(Z.times(z))
if useZ: # include non-NaN, non-zero Z component
z = fsum_(d12, -(x**2), -(y**2))
if z > EPS:
Z = X.cross(Y) # unit vector perpendicular to plane
n = n.plus(Z.times(sqrt(z)))
if height is None:
h = fidw((point1.height, point2.height, point3.height),
map1(fabs, distance1, distance2, distance3))
else:
h = height
return n.toLatLon(
height=h,
LatLon=LatLon) # Nvector(n.x, n.y, n.z).toLatLon(...)
# no intersection, d < EPS_2 or j < EPS_2
raise ValueError('no %s for %r, %r, %r at %r, %r, %r' %
('trilaterate', point1, point2, point3, distance1,
distance2, distance3))
def toOsgr(latlon, lon=None, datum=Datums.WGS84, Osgr=Osgr, name=''):
'''Convert a lat-/longitude point to an OSGR coordinate.
@param latlon: Latitude (C{degrees}) or an (ellipsoidal)
geodetic C{LatLon} point.
@keyword lon: Optional longitude in degrees (scalar or C{None}).
@keyword datum: Optional datum to convert (C{Datum}).
@keyword Osgr: Optional (sub-)class to return the OSGR
coordinate (L{Osgr}) or C{None}.
@keyword name: Optional I{Osgr} name (C{str}).
@return: The OSGR coordinate (L{Osgr}) or 2-tuple (easting,
northing) if I{Osgr} is C{None}.
@raise TypeError: Non-ellipsoidal I{latlon} or I{datum}
conversion failed.
@raise ValueError: Invalid I{latlon} or I{lon}.
@example:
>>> p = LatLon(52.65798, 1.71605)
>>> r = toOsgr(p) # TG 51409 13177
>>> # for conversion of (historical) OSGB36 lat-/longitude:
>>> r = toOsgr(52.65757, 1.71791, datum=Datums.OSGB36)
'''
if not isinstance(latlon, _LLEB):
# XXX fix failing _LLEB.convertDatum()
latlon = _LLEB(*parseDMS2(latlon, lon), datum=datum)
elif lon is not None:
raise ValueError('%s not %s: %r' % ('lon', None, lon))
elif not name: # use latlon.name
name = _nameof(latlon) or name # PYCHOK no effect
E = _OSGB36.ellipsoid
ll = _ll2datum(latlon, _OSGB36, 'latlon')
a, b = map1(radians, ll.lat, ll.lon)
sa, ca = sincos2(a)
s = E.e2s2(sa)
v = E.a * _F0 / sqrt(s) # nu
r = s / E.e12 # nu / rho == v / (v * E.e12 / s)
x2 = r - 1 # η2
ta = tan(a)
ca3, ca5 = fpowers(ca, 5, 3) # PYCHOK false!
ta2, ta4 = fpowers(ta, 4, 2) # PYCHOK false!
vsa = v * sa
I4 = (E.b * _F0 * _M(E.Mabcd, a) + _N0, (vsa / 2) * ca,
(vsa / 24) * ca3 * fsum_(5, -ta2, 9 * x2),
(vsa / 720) * ca5 * fsum_(61, ta4, -58 * ta2))
V4 = (_E0, (v * ca), (v / 6) * ca3 * (r - ta2), (v / 120) * ca5 * fdot(
(-18, 1, 14, -58), ta2, 5 + ta4, x2, ta2 * x2))
d, d2, d3, d4, d5, d6 = fpowers(b - _B0, 6) # PYCHOK false!
n = fdot(I4, 1, d2, d4, d6)
e = fdot(V4, 1, d, d3, d5)
return (e, n) if Osgr is None else _xnamed(Osgr(e, n), name)
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 isenclosedBy(point, points, wrap=False): # MCCABE 15
'''Determine whether a point is enclosed by a polygon.
@param point: The point (C{LatLon} or 2-tuple (lat, lon)).
@param points: The polygon points (C{LatLon}[]).
@keyword wrap: Wrap lat-, wrap and unroll longitudes (C{bool}).
@return: C{True} if I{point} is inside the polygon, C{False}
otherwise.
@raise TypeError: Some I{points} are not C{LatLon}.
@raise ValueError: Insufficient number of I{points} or invalid
I{point}.
@see: L{sphericalNvector.LatLon.isEnclosedBy},
L{sphericalTrigonometry.LatLon.isEnclosedBy} and
U{MultiDop GeogContainPt<http://GitHub.com/NASA/MultiDop>}
(U{Shapiro et al. 2009, JTECH
<http://journals.AMetSoc.org/doi/abs/10.1175/2009JTECHA1256.1>}
and U{Potvin et al. 2012, JTECH
<http://journals.AMetSoc.org/doi/abs/10.1175/JTECH-D-11-00019.1>}).
'''
try:
y0, x0 = point.lat, point.lon
except AttributeError:
try:
y0, x0 = map1(float, *point[:2])
except (IndexError, TypeError, ValueError):
raise ValueError('%s invalid: %r' % ('point', point))
pts = LatLon2psxy(points, closed=True, radius=None, wrap=wrap)
n = len(pts)
if wrap:
x0, y0 = wrap180(x0), wrap90(y0)
def _dxy(x1, i):
x2, y2, _ = pts[i]
dx, x2 = unroll180(x1, x2, wrap=i < (n - 1))
return dx, x2, y2
else:
x0 = fmod(x0, 360.0) # not x0 % 360
def _dxy(x1, i): # PYCHOK expected
x, y, _ = pts[i]
x %= 360.0
if x < (x0 - 180):
x += 360
elif x >= (x0 + 180):
x -= 360
return (x - x1), x, y
e = m = False
s = Fsum()
_, x1, y1 = _dxy(x0, n - 1)
for i in range(n):
dx, x2, y2 = _dxy(x1, i)
# ignore duplicate and near-duplicate pts
if max(abs(dx), abs(y2 - y1)) > EPS:
# determine if polygon edge (x1, y1)..(x2, y2) straddles
# point (lat, lon) or is on boundary, but do not count
# edges on boundary as more than one crossing
if abs(dx) < 180 and (x1 < x0 <= x2 or x2 < x0 <= x1):
m = not m
dy = (x0 - x1) * (y2 - y1) - (y0 - y1) * dx
if (dy > 0 and dx >= 0) or (dy < 0 and dx <= 0):
e = not e
s.fadd(sin(radians(y2)))
x1, y1 = x2, y2
# An odd number of meridian crossings means, the polygon
# contains a pole. Assume it is the pole on the hemisphere
# containing the polygon mean point and if the polygon does
# contain the North Pole, flip the result.
if m and s.fsum() > 0:
e = not e
return e
def _T(*fs): # tuple of single floats
return map1(_F, *fs)
