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#
# This is a rather literal translation of the GeographicLib::Geodesic
# class to python. See the documentation for the C++ class for more
# information at
#
# http://geographiclib.sourceforge.net/html/annotated.html
#
# The algorithms are derived in
#
# Charles F. F. Karney,
# Geodesics on an ellipsoid of revolution, Feb. 2011,
# http://arxiv.org/abs/1102.1215
# errata: http://geographiclib.sourceforge.net/geod-errata.html
#
# Charles F. F. Karney,
# Algorithms for geodesics, Sept. 2011,
# http://arxiv.org/abs/1109.4448
#
# Copyright (c) Charles Karney (2011, 2012) <charles@karney.com> and
# licensed under the MIT/X11 License. For more information, see
# http://geographiclib.sourceforge.net/
#
# $Id: 95edebf1ac240458e1ba8a63577c589f330f849d $
######################################################################
import math
from geographiclib.geomath import Math
from geographiclib.constants import Constants
from geographiclib.geodesiccapability import GeodesicCapability
class Geodesic(object):
"""
Solve geodesic problems. The following illustrates its use
import sys
sys.path.append("/usr/local/lib/python/site-packages");
from geographiclib.geodesic import Geodesic
# The geodesic inverse problem
Geodesic.WGS84.Inverse(-41.32, 174.81, 40.96, -5.50)
# The geodesic direct problem
Geodesic.WGS84.Direct(40.6, -73.8, 45, 10000e3)
# How to obtain several points along a geodesic
line = Geodesic.WGS84.Line(40.6, -73.8, 45)
line.Position( 5000e3)
line.Position(10000e3)
# Computing the area of a geodesic polygon
def p(lat,lon): return {'lat': lat, 'lon': lon}
Geodesic.WGS84.Area([p(0, 0), p(0, 90), p(90, 0)])
Documentation on these routines is available via
help(Geodesic.__init__)
help(Geodesic.Inverse)
help(Geodesic.Direct)
help(Geodesic.Line)
help(line.Position)
help(Geodesic.Area)
"""
GEOD_ORD = 6
nA1_ = GEOD_ORD
nC1_ = GEOD_ORD
nC1p_ = GEOD_ORD
nA2_ = GEOD_ORD
nC2_ = GEOD_ORD
nA3_ = GEOD_ORD
nA3x_ = nA3_
nC3_ = GEOD_ORD
nC3x_ = (nC3_ * (nC3_ - 1)) / 2
nC4_ = GEOD_ORD
nC4x_ = (nC4_ * (nC4_ + 1)) / 2
maxit_ = 50
tiny_ = math.sqrt(Math.minval)
tol0_ = Math.epsilon
tol1_ = 200 * tol0_
tol2_ = math.sqrt(Math.epsilon)
xthresh_ = 1000 * tol2_
CAP_NONE = GeodesicCapability.CAP_NONE
CAP_C1 = GeodesicCapability.CAP_C1
CAP_C1p = 1<<1
CAP_C2 = 1<<2
CAP_C3 = 1<<3
CAP_C4 = 1<<4
CAP_ALL = 0x1F
OUT_ALL = 0x7F80
NONE = 0
LATITUDE = 1<<7 | CAP_NONE
LONGITUDE = 1<<8 | CAP_C3
AZIMUTH = 1<<9 | CAP_NONE
DISTANCE = 1<<10 | CAP_C1
DISTANCE_IN = 1<<11 | CAP_C1 | CAP_C1p
REDUCEDLENGTH = 1<<12 | CAP_C1 | CAP_C2
GEODESICSCALE = 1<<13 | CAP_C1 | CAP_C2
AREA = 1<<14 | CAP_C4
ALL = OUT_ALL| CAP_ALL
def SinCosSeries(sinp, sinx, cosx, c, n):
# Evaluate
# y = sinp ? sum(c[i] * sin( 2*i * x), i, 1, n) :
# sum(c[i] * cos((2*i+1) * x), i, 0, n-1) :
# using Clenshaw summation. N.B. c[0] is unused for sin series
# Approx operation count = (n + 5) mult and (2 * n + 2) add
k = (n + sinp) # Point to one beyond last element
ar = 2 * (cosx - sinx) * (cosx + sinx) # 2 * cos(2 * x)
y1 = 0 # accumulators for sum
if n & 1:
k -= 1; y0 = c[k]
else:
y0 = 0
# Now n is even
n //= 2
while n: # while n--:
n -= 1
# Unroll loop x 2, so accumulators return to their original role
k -= 1; y1 = ar * y0 - y1 + c[k]
k -= 1; y0 = ar * y1 - y0 + c[k]
return ( 2 * sinx * cosx * y0 if sinp # sin(2 * x) * y0
else cosx * (y0 - y1) ) # cos(x) * (y0 - y1)
SinCosSeries = staticmethod(SinCosSeries)
def AngNormalize(x):
# Place angle in [-180, 180). Assumes x is in [-540, 540).
return (x - 360 if x >= 180 else
(x + 360 if x < -180 else x))
AngNormalize = staticmethod(AngNormalize)
def AngRound(x):
# The makes the smallest gap in x = 1/16 - nextafter(1/16, 0) = 1/2^57
# for reals = 0.7 pm on the earth if x is an angle in degrees. (This
# is about 1000 times more resolution than we get with angles around 90
# degrees.) We use this to avoid having to deal with near singular
# cases when x is non-zero but tiny (e.g., 1.0e-200).
z = 0.0625 # 1/16
y = abs(x)
# The compiler mustn't "simplify" z - (z - y) to y
y = z - (z - y) if y < z else y
return -y if x < 0 else y
AngRound = staticmethod(AngRound)
def SinCosNorm(sinx, cosx):
r = math.hypot(sinx, cosx)
return sinx/r, cosx/r
SinCosNorm = staticmethod(SinCosNorm)
def Astroid(x, y):
# Solve k^4+2*k^3-(x^2+y^2-1)*k^2-2*y^2*k-y^2 = 0 for positive root k.
# This solution is adapted from Geocentric::Reverse.
p = Math.sq(x)
q = Math.sq(y)
r = (p + q - 1) / 6
if not(q == 0 and r <= 0):
# Avoid possible division by zero when r = 0 by multiplying equations
# for s and t by r^3 and r, resp.
S = p * q / 4 # S = r^3 * s
r2 = Math.sq(r)
r3 = r * r2
# The discrimant of the quadratic equation for T3. This is zero on
# the evolute curve p^(1/3)+q^(1/3) = 1
disc = S * (S + 2 * r3)
u = r
if (disc >= 0):
T3 = S + r3
# Pick the sign on the sqrt to maximize abs(T3). This minimizes loss
# of precision due to cancellation. The result is unchanged because
# of the way the T is used in definition of u.
T3 += -math.sqrt(disc) if T3 < 0 else math.sqrt(disc) # T3 = (r * t)^3
# N.B. cbrt always returns the real root. cbrt(-8) = -2.
T = Math.cbrt(T3) # T = r * t
# T can be zero; but then r2 / T -> 0.
u += T + (r2 / T if T != 0 else 0)
else:
# T is complex, but the way u is defined the result is real.
ang = math.atan2(math.sqrt(-disc), -(S + r3))
# There are three possible cube roots. We choose the root which
# avoids cancellation. Note that disc < 0 implies that r < 0.
u += 2 * r * math.cos(ang / 3)
v = math.sqrt(Math.sq(u) + q) # guaranteed positive
# Avoid loss of accuracy when u < 0.
uv = q / (v - u) if u < 0 else u + v # u+v, guaranteed positive
w = (uv - q) / (2 * v) # positive?
# Rearrange expression for k to avoid loss of accuracy due to
# subtraction. Division by 0 not possible because uv > 0, w >= 0.
k = uv / (math.sqrt(uv + Math.sq(w)) + w) # guaranteed positive
else: # q == 0 && r <= 0
# y = 0 with |x| <= 1. Handle this case directly.
# for y small, positive root is k = abs(y)/sqrt(1-x^2)
k = 0
return k
Astroid = staticmethod(Astroid)
def A1m1f(eps):
eps2 = Math.sq(eps)
t = eps2*(eps2*(eps2+4)+64)/256
return (t + eps) / (1 - eps)
A1m1f = staticmethod(A1m1f)
def C1f(eps, c):
eps2 = Math.sq(eps)
d = eps
c[1] = d*((6-eps2)*eps2-16)/32
d *= eps
c[2] = d*((64-9*eps2)*eps2-128)/2048
d *= eps
c[3] = d*(9*eps2-16)/768
d *= eps
c[4] = d*(3*eps2-5)/512
d *= eps
c[5] = -7*d/1280
d *= eps
c[6] = -7*d/2048
C1f = staticmethod(C1f)
def C1pf(eps, c):
eps2 = Math.sq(eps)
d = eps
c[1] = d*(eps2*(205*eps2-432)+768)/1536
d *= eps
c[2] = d*(eps2*(4005*eps2-4736)+3840)/12288
d *= eps
c[3] = d*(116-225*eps2)/384
d *= eps
c[4] = d*(2695-7173*eps2)/7680
d *= eps
c[5] = 3467*d/7680
d *= eps
c[6] = 38081*d/61440
C1pf = staticmethod(C1pf)
def A2m1f(eps):
eps2 = Math.sq(eps)
t = eps2*(eps2*(25*eps2+36)+64)/256
return t * (1 - eps) - eps
A2m1f = staticmethod(A2m1f)
def C2f(eps, c):
eps2 = Math.sq(eps)
d = eps
c[1] = d*(eps2*(eps2+2)+16)/32
d *= eps
c[2] = d*(eps2*(35*eps2+64)+384)/2048
d *= eps
c[3] = d*(15*eps2+80)/768
d *= eps
c[4] = d*(7*eps2+35)/512
d *= eps
c[5] = 63*d/1280
d *= eps
c[6] = 77*d/2048
C2f = staticmethod(C2f)
def __init__(self, a, f):
"""
Construct a Geodesic object for ellipsoid with major radius a and
flattening f.
"""
self._a = float(a)
self._f = float(f) if f <= 1 else 1.0/f
self._f1 = 1 - self._f
self._e2 = self._f * (2 - self._f)
self._ep2 = self._e2 / Math.sq(self._f1) # e2 / (1 - e2)
self._n = self._f / ( 2 - self._f)
self._b = self._a * self._f1
# authalic radius squared
self._c2 = (Math.sq(self._a) + Math.sq(self._b) *
(1 if self._e2 == 0 else
(Math.atanh(math.sqrt(self._e2)) if self._e2 > 0 else
math.atan(math.sqrt(-self._e2))) /
math.sqrt(abs(self._e2))))/2
# The sig12 threshold for "really short"
self._etol2 = 10 * Geodesic.tol2_ / max(0.1, math.sqrt(abs(self._e2)))
if not(Math.isfinite(self._a) and self._a > 0):
raise ValueError("Major radius is not positive")
if not(Math.isfinite(self._b) and self._b > 0):
raise ValueError("Minor radius is not positive")
self._A3x = range(Geodesic.nA3x_)
self._C3x = range(Geodesic.nC3x_)
self._C4x = range(Geodesic.nC4x_)
self.A3coeff()
self.C3coeff()
self.C4coeff()
def A3coeff(self):
_n = self._n
self._A3x[0] = 1
self._A3x[1] = (_n-1)/2
self._A3x[2] = (_n*(3*_n-1)-2)/8
self._A3x[3] = ((-_n-3)*_n-1)/16
self._A3x[4] = (-2*_n-3)/64
self._A3x[5] = -3/128.0
def C3coeff(self):
_n = self._n
self._C3x[0] = (1-_n)/4
self._C3x[1] = (1-_n*_n)/8
self._C3x[2] = ((3-_n)*_n+3)/64
self._C3x[3] = (2*_n+5)/128
self._C3x[4] = 3/128.0
self._C3x[5] = ((_n-3)*_n+2)/32
self._C3x[6] = ((-3*_n-2)*_n+3)/64
self._C3x[7] = (_n+3)/128
self._C3x[8] = 5/256.0
self._C3x[9] = (_n*(5*_n-9)+5)/192
self._C3x[10] = (9-10*_n)/384
self._C3x[11] = 7/512.0
self._C3x[12] = (7-14*_n)/512
self._C3x[13] = 7/512.0
self._C3x[14] = 21/2560.0
def C4coeff(self):
_ep2 = self._ep2
self._C4x[0] = (_ep2*(_ep2*(_ep2*((832-640*_ep2)*_ep2-1144)+1716)-3003)+
30030)/45045
self._C4x[1] = (_ep2*(_ep2*((832-640*_ep2)*_ep2-1144)+1716)-3003)/60060
self._C4x[2] = (_ep2*((208-160*_ep2)*_ep2-286)+429)/18018
self._C4x[3] = ((104-80*_ep2)*_ep2-143)/10296
self._C4x[4] = (13-10*_ep2)/1430
self._C4x[5] = -1/156.0
self._C4x[6] = (_ep2*(_ep2*(_ep2*(640*_ep2-832)+1144)-1716)+3003)/540540
self._C4x[7] = (_ep2*(_ep2*(160*_ep2-208)+286)-429)/108108
self._C4x[8] = (_ep2*(80*_ep2-104)+143)/51480
self._C4x[9] = (10*_ep2-13)/6435
self._C4x[10] = 5/3276.0
self._C4x[11] = (_ep2*((208-160*_ep2)*_ep2-286)+429)/900900
self._C4x[12] = ((104-80*_ep2)*_ep2-143)/257400
self._C4x[13] = (13-10*_ep2)/25025
self._C4x[14] = -1/2184.0
self._C4x[15] = (_ep2*(80*_ep2-104)+143)/2522520
self._C4x[16] = (10*_ep2-13)/140140
self._C4x[17] = 5/45864.0
self._C4x[18] = (13-10*_ep2)/1621620
self._C4x[19] = -1/58968.0
self._C4x[20] = 1/792792.0
def A3f(self, eps):
# Evaluation sum(_A3c[k] * eps^k, k, 0, nA3x_-1) by Horner's method
v = 0
for i in range(Geodesic.nA3x_-1, -1, -1):
v = eps * v + self._A3x[i]
return v
def C3f(self, eps, c):
# Evaluation C3 coeffs by Horner's method
# Elements c[1] thru c[nC3_ - 1] are set
j = Geodesic.nC3x_; k = Geodesic.nC3_ - 1
while k:
t = 0
for i in range(Geodesic.nC3_ - k):
j -= 1
t = eps * t + self._C3x[j]
c[k] = t
k -= 1
mult = 1
for k in range(1, Geodesic.nC3_):
mult *= eps
c[k] *= mult
def C4f(self, k2, c):
# Evaluation C4 coeffs by Horner's method
# Elements c[0] thru c[nC4_ - 1] are set
j = Geodesic.nC4x_; k = Geodesic.nC4_
while k:
t = 0
for i in range(Geodesic.nC4_ - k + 1):
j -= 1
t = k2 * t + self._C4x[j]
k -= 1
c[k] = t
mult = 1
for k in range(1, Geodesic.nC4_):
mult *= k2
c[k] *= mult
# return s12b, m12a, m0, M12, M21
def Lengths(self, eps, sig12,
ssig1, csig1, ssig2, csig2, cbet1, cbet2, scalep,
# Scratch areas of the right size
C1a, C2a):
# Return m12a = (reduced length)/_a; also calculate s12b = distance/_b,
# and m0 = coefficient of secular term in expression for reduced length.
Geodesic.C1f(eps, C1a)
Geodesic.C2f(eps, C2a)
A1m1 = Geodesic.A1m1f(eps)
AB1 = (1 + A1m1) * (
Geodesic.SinCosSeries(True, ssig2, csig2, C1a, Geodesic.nC1_) -
Geodesic.SinCosSeries(True, ssig1, csig1, C1a, Geodesic.nC1_))
A2m1 = Geodesic.A2m1f(eps)
AB2 = (1 + A2m1) * (
Geodesic.SinCosSeries(True, ssig2, csig2, C2a, Geodesic.nC2_) -
Geodesic.SinCosSeries(True, ssig1, csig1, C2a, Geodesic.nC2_))
cbet1sq = Math.sq(cbet1)
cbet2sq = Math.sq(cbet2)
w1 = math.sqrt(1 - self._e2 * cbet1sq)
w2 = math.sqrt(1 - self._e2 * cbet2sq)
# Make sure it's OK to have repeated dummy arguments
m0x = A1m1 - A2m1
J12 = m0x * sig12 + (AB1 - AB2)
m0 = m0x
# Missing a factor of _a.
# Add parens around (csig1 * ssig2) and (ssig1 * csig2) to ensure accurate
# cancellation in the case of coincident points.
m12a = ((w2 * (csig1 * ssig2) - w1 * (ssig1 * csig2))
- self._f1 * csig1 * csig2 * J12)
# Missing a factor of _b
s12b = (1 + A1m1) * sig12 + AB1
if scalep:
csig12 = csig1 * csig2 + ssig1 * ssig2
J12 *= self._f1
M12 = csig12 + (self._e2 * (cbet1sq - cbet2sq) * ssig2 / (w1 + w2)
- csig2 * J12) * ssig1 / w1
M21 = csig12 - (self._e2 * (cbet1sq - cbet2sq) * ssig1 / (w1 + w2)
- csig1 * J12) * ssig2 / w2
else:
M12 = M21 = Math.nan
return s12b, m12a, m0, M12, M21
# return sig12, salp1, calp1, salp2, calp2
def InverseStart(self, sbet1, cbet1, sbet2, cbet2, lam12,
# Scratch areas of the right size
C1a, C2a):
# Return a starting point for Newton's method in salp1 and calp1 (function
# value is -1). If Newton's method doesn't need to be used, return also
# salp2 and calp2 and function value is sig12.
sig12 = -1; salp2 = calp2 = Math.nan # Return values
# bet12 = bet2 - bet1 in [0, pi); bet12a = bet2 + bet1 in (-pi, 0]
sbet12 = sbet2 * cbet1 - cbet2 * sbet1
cbet12 = cbet2 * cbet1 + sbet2 * sbet1
# Volatile declaration needed to fix inverse cases
# 88.202499451857 0 -88.202499451857 179.981022032992859592
# 89.262080389218 0 -89.262080389218 179.992207982775375662
# 89.333123580033 0 -89.333123580032997687 179.99295812360148422
# which otherwise fail with g++ 4.4.4 x86 -O3
sbet12a = sbet2 * cbet1
sbet12a += cbet2 * sbet1
shortline = cbet12 >= 0 and sbet12 < 0.5 and lam12 <= math.pi / 6
omg12 = (lam12 if not shortline else
lam12 / math.sqrt(1 - self._e2 * Math.sq((cbet1 + cbet2) / 2)))
somg12 = math.sin(omg12); comg12 = math.cos(omg12)
salp1 = cbet2 * somg12
calp1 = (
sbet12 + cbet2 * sbet1 * Math.sq(somg12) / (1 + comg12) if comg12 >= 0
else sbet12a - cbet2 * sbet1 * Math.sq(somg12) / (1 - comg12))
ssig12 = math.hypot(salp1, calp1)
csig12 = sbet1 * sbet2 + cbet1 * cbet2 * comg12
if shortline and ssig12 < self._etol2:
# really short lines
salp2 = cbet1 * somg12
calp2 = sbet12 - cbet1 * sbet2 * Math.sq(somg12) / (1 + comg12)
salp2, calp2 = Geodesic.SinCosNorm(salp2, calp2)
# Set return value
sig12 = math.atan2(ssig12, csig12)
elif csig12 >= 0 or ssig12 >= 3 * abs(self._f) * math.pi * Math.sq(cbet1):
# Nothing to do, zeroth order spherical approximation is OK
pass
else:
# Scale lam12 and bet2 to x, y coordinate system where antipodal point
# is at origin and singular point is at y = 0, x = -1.
# real y, lamscale, betscale
# Volatile declaration needed to fix inverse case
# 56.320923501171 0 -56.320923501171 179.664747671772880215
# which otherwise fails with g++ 4.4.4 x86 -O3
# volatile real x
if self._f >= 0: # In fact f == 0 does not get here
# x = dlong, y = dlat
k2 = Math.sq(sbet1) * self._ep2
eps = k2 / (2 * (1 + math.sqrt(1 + k2)) + k2)
lamscale = self._f * cbet1 * self.A3f(eps) * math.pi
betscale = lamscale * cbet1
x = (lam12 - math.pi) / lamscale
y = sbet12a / betscale
else: # _f < 0
# x = dlat, y = dlong
cbet12a = cbet2 * cbet1 - sbet2 * sbet1
bet12a = math.atan2(sbet12a, cbet12a)
# real m12a, m0, dummy
# In the case of lon12 = 180, this repeats a calculation made in
# Inverse.
dummy, m12a, m0, dummy, dummy = self.Lengths(
self._n, math.pi + bet12a, sbet1, -cbet1, sbet2, cbet2,
cbet1, cbet2, dummy, False, C1a, C2a)
x = -1 + m12a/(self._f1 * cbet1 * cbet2 * m0 * math.pi)
betscale = (sbet12a / x if x < -real(0.01)
else -self._f * Math.sq(cbet1) * math.pi)
lamscale = betscale / cbet1
y = (lam12 - math.pi) / lamscale
if y > -Geodesic.tol1_ and x > -1 - Geodesic.xthresh_:
# strip near cut
if self._f >= 0:
salp1 = min(1.0, -x); calp1 = - math.sqrt(1 - Math.sq(salp1))
else:
calp1 = max((0.0 if x > -Geodesic.tol1_ else -1.0), x)
salp1 = math.sqrt(1 - Math.sq(calp1))
else:
# Estimate alp1, by solving the astroid problem.
#
# Could estimate alpha1 = theta + pi/2, directly, i.e.,
# calp1 = y/k; salp1 = -x/(1+k); for _f >= 0
# calp1 = x/(1+k); salp1 = -y/k; for _f < 0 (need to check)
#
# However, it's better to estimate omg12 from astroid and use
# spherical formula to compute alp1. This reduces the mean number of
# Newton iterations for astroid cases from 2.24 (min 0, max 6) to 2.12
# (min 0 max 5). The changes in the number of iterations are as
# follows:
#
# change percent
# 1 5
# 0 78
# -1 16
# -2 0.6
# -3 0.04
# -4 0.002
#
# The histogram of iterations is (m = number of iterations estimating
# alp1 directly, n = number of iterations estimating via omg12, total
# number of trials = 148605):
#
# iter m n
# 0 148 186
# 1 13046 13845
# 2 93315 102225
# 3 36189 32341
# 4 5396 7
# 5 455 1
# 6 56 0
#
# Because omg12 is near pi, estimate work with omg12a = pi - omg12
k = Geodesic.Astroid(x, y)
omg12a = lamscale * ( -x * k/(1 + k) if self._f >= 0
else -y * (1 + k)/k )
somg12 = math.sin(omg12a); comg12 = -math.cos(omg12a)
# Update spherical estimate of alp1 using omg12 instead of lam12
salp1 = cbet2 * somg12
calp1 = sbet12a - cbet2 * sbet1 * Math.sq(somg12) / (1 - comg12)
salp1, calp1 = Geodesic.SinCosNorm(salp1, calp1)
return sig12, salp1, calp1, salp2, calp2
# return lam12, salp2, calp2, sig12, ssig1, csig1, ssig2, csig2, eps,
# domg12, dlam12
def Lambda12(self, sbet1, cbet1, sbet2, cbet2, salp1, calp1, diffp,
# Scratch areas of the right size
C1a, C2a, C3a):
if sbet1 == 0 and calp1 == 0:
# Break degeneracy of equatorial line. This case has already been
# handled.
calp1 = -Geodesic.tiny_
# sin(alp1) * cos(bet1) = sin(alp0)
salp0 = salp1 * cbet1
calp0 = math.hypot(calp1, salp1 * sbet1) # calp0 > 0
# real somg1, comg1, somg2, comg2, omg12, lam12
# tan(bet1) = tan(sig1) * cos(alp1)
# tan(omg1) = sin(alp0) * tan(sig1) = tan(omg1)=tan(alp1)*sin(bet1)
ssig1 = sbet1; somg1 = salp0 * sbet1
csig1 = comg1 = calp1 * cbet1
ssig1, csig1 = Geodesic.SinCosNorm(ssig1, csig1)
# SinCosNorm(somg1, comg1); -- don't need to normalize!
# Enforce symmetries in the case abs(bet2) = -bet1. Need to be careful
# about this case, since this can yield singularities in the Newton
# iteration.
# sin(alp2) * cos(bet2) = sin(alp0)
salp2 = salp0 / cbet2 if cbet2 != cbet1 else salp1
# calp2 = sqrt(1 - sq(salp2))
# = sqrt(sq(calp0) - sq(sbet2)) / cbet2
# and subst for calp0 and rearrange to give (choose positive sqrt
# to give alp2 in [0, pi/2]).
calp2 = (math.sqrt(Math.sq(calp1 * cbet1) +
((cbet2 - cbet1) * (cbet1 + cbet2) if cbet1 < -sbet1
else (sbet1 - sbet2) * (sbet1 + sbet2))) / cbet2
if cbet2 != cbet1 or abs(sbet2) != -sbet1 else abs(calp1))
# tan(bet2) = tan(sig2) * cos(alp2)
# tan(omg2) = sin(alp0) * tan(sig2).
ssig2 = sbet2; somg2 = salp0 * sbet2
csig2 = comg2 = calp2 * cbet2
ssig2, csig2 = Geodesic.SinCosNorm(ssig2, csig2)
# SinCosNorm(somg2, comg2); -- don't need to normalize!
# sig12 = sig2 - sig1, limit to [0, pi]
sig12 = math.atan2(max(csig1 * ssig2 - ssig1 * csig2, 0.0),
csig1 * csig2 + ssig1 * ssig2)
# omg12 = omg2 - omg1, limit to [0, pi]
omg12 = math.atan2(max(comg1 * somg2 - somg1 * comg2, 0.0),
comg1 * comg2 + somg1 * somg2)
# real B312, h0
k2 = Math.sq(calp0) * self._ep2
eps = k2 / (2 * (1 + math.sqrt(1 + k2)) + k2)
self.C3f(eps, C3a)
B312 = (Geodesic.SinCosSeries(True, ssig2, csig2, C3a, Geodesic.nC3_-1) -
Geodesic.SinCosSeries(True, ssig1, csig1, C3a, Geodesic.nC3_-1))
h0 = -self._f * self.A3f(eps)
domg12 = salp0 * h0 * (sig12 + B312)
lam12 = omg12 + domg12
if diffp:
if calp2 == 0:
dlam12 = - 2 * math.sqrt(1 - self._e2 * Math.sq(cbet1)) / sbet1
else:
dummy, dlam12, dummy, dummy, dummy = self.Lengths(
eps, sig12, ssig1, csig1, ssig2, csig2, cbet1, cbet2, False, C1a, C2a)
dlam12 /= calp2 * cbet2
else:
dlam12 = Math.nan
return (lam12, salp2, calp2, sig12, ssig1, csig1, ssig2, csig2, eps,
domg12, dlam12)
# return a12, s12, azi1, azi2, m12, M12, M21, S12
def GenInverse(self, lat1, lon1, lat2, lon2, outmask):
a12 = s12 = azi1 = azi2 = m12 = M12 = M21 = S12 = Math.nan # return vals
outmask &= Geodesic.OUT_ALL
lon1 = Geodesic.AngNormalize(lon1)
lon12 = Geodesic.AngNormalize(Geodesic.AngNormalize(lon2) - lon1)
# If very close to being on the same meridian, then make it so.
# Not sure this is necessary...
lon12 = Geodesic.AngRound(lon12)
# Make longitude difference positive.
lonsign = 1 if lon12 >= 0 else -1
lon12 *= lonsign
if lon12 == 180:
lonsign = 1
# If really close to the equator, treat as on equator.
lat1 = Geodesic.AngRound(lat1)
lat2 = Geodesic.AngRound(lat2)
# Swap points so that point with higher (abs) latitude is point 1
swapp = 1 if abs(lat1) >= abs(lat2) else -1
if swapp < 0:
lonsign *= -1
lat2, lat1 = lat1, lat2
# Make lat1 <= 0
latsign = 1 if lat1 < 0 else -1
lat1 *= latsign
lat2 *= latsign
# Now we have
#
# 0 <= lon12 <= 180
# -90 <= lat1 <= 0
# lat1 <= lat2 <= -lat1
#
# longsign, swapp, latsign register the transformation to bring the
# coordinates to this canonical form. In all cases, 1 means no change was
# made. We make these transformations so that there are few cases to
# check, e.g., on verifying quadrants in atan2. In addition, this
# enforces some symmetries in the results returned.
# real phi, sbet1, cbet1, sbet2, cbet2, s12x, m12x
phi = lat1 * Math.degree
# Ensure cbet1 = +epsilon at poles
sbet1 = self._f1 * math.sin(phi)
cbet1 = Geodesic.tiny_ if lat1 == -90 else math.cos(phi)
sbet1, cbet1 = Geodesic.SinCosNorm(sbet1, cbet1)
phi = lat2 * Math.degree
# Ensure cbet2 = +epsilon at poles
sbet2 = self._f1 * math.sin(phi)
cbet2 = Geodesic.tiny_ if abs(lat2) == 90 else math.cos(phi)
sbet2, cbet2 = Geodesic.SinCosNorm(sbet2, cbet2)
# If cbet1 < -sbet1, then cbet2 - cbet1 is a sensitive measure of the
# |bet1| - |bet2|. Alternatively (cbet1 >= -sbet1), abs(sbet2) + sbet1 is
# a better measure. This logic is used in assigning calp2 in Lambda12.
# Sometimes these quantities vanish and in that case we force bet2 = +/-
# bet1 exactly. An example where is is necessary is the inverse problem
# 48.522876735459 0 -48.52287673545898293 179.599720456223079643
# which failed with Visual Studio 10 (Release and Debug)
if cbet1 < -sbet1:
if cbet2 == cbet1:
sbet2 = sbet1 if sbet2 < 0 else -sbet1
else:
if abs(sbet2) == -sbet1:
cbet2 = cbet1
lam12 = lon12 * Math.degree
slam12 = 0 if lon12 == 180 else math.sin(lam12)
clam12 = math.cos(lam12) # lon12 == 90 isn't interesting
# real a12, sig12, calp1, salp1, calp2, salp2
# index zero elements of these arrays are unused
C1a = range(Geodesic.nC1_ + 1)
C2a = range(Geodesic.nC2_ + 1)
C3a = range(Geodesic.nC3_)
meridian = lat1 == -90 or slam12 == 0
if meridian:
# Endpoints are on a single full meridian, so the geodesic might lie on
# a meridian.
calp1 = clam12; salp1 = slam12 # Head to the target longitude
calp2 = 1; salp2 = 0 # At the target we're heading north
# tan(bet) = tan(sig) * cos(alp)
ssig1 = sbet1; csig1 = calp1 * cbet1
ssig2 = sbet2; csig2 = calp2 * cbet2
# sig12 = sig2 - sig1
sig12 = math.atan2(max(csig1 * ssig2 - ssig1 * csig2, 0.0),
csig1 * csig2 + ssig1 * ssig2)
s12x, m12x, dummy, M12, M21 = self.Lengths(
self._n, sig12, ssig1, csig1, ssig2, csig2, cbet1, cbet2,
(outmask & Geodesic.GEODESICSCALE) != 0, C1a, C2a)
# Add the check for sig12 since zero length geodesics might yield m12 <
# 0. Test case was
#
# echo 20.001 0 20.001 0 | Geod -i
#
# In fact, we will have sig12 > pi/2 for meridional geodesic which is
# not a shortest path.
if sig12 < 1 or m12x >= 0:
m12x *= self._a
s12x *= self._b
a12 = sig12 / Math.degree
else:
# m12 < 0, i.e., prolate and too close to anti-podal
meridian = False
# end if meridian:
#real omg12
if (not meridian and
sbet1 == 0 and # and sbet2 == 0
# Mimic the way Lambda12 works with calp1 = 0
(self._f <= 0 or lam12 <= math.pi - self._f * math.pi)):
# Geodesic runs along equator
calp1 = calp2 = 0; salp1 = salp2 = 1
s12x = self._a * lam12
m12x = self._b * math.sin(lam12 / self._f1)
if outmask & Geodesic.GEODESICSCALE:
M12 = M21 = math.cos(lam12 / self._f1)
a12 = lon12 / self._f1
sig12 = omg12 = lam12 / self._f1
elif not meridian:
# Now point1 and point2 belong within a hemisphere bounded by a
# meridian and geodesic is neither meridional or equatorial.
# Figure a starting point for Newton's method
sig12, salp1, calp1, salp2, calp2 = self.InverseStart(
sbet1, cbet1, sbet2, cbet2, lam12, C1a, C2a)
if sig12 >= 0:
# Short lines (InverseStart sets salp2, calp2)
wm = math.sqrt(1 - self._e2 * Math.sq((cbet1 + cbet2) / 2))
s12x = sig12 * self._a * wm
m12x = (Math.sq(wm) * self._a / self._f1 *
math.sin(sig12 * self._f1 / wm))
if outmask & Geodesic.GEODESICSCALE:
M12 = M21 = math.cos(sig12 * self._f1 / wm)
a12 = sig12 / Math.degree
omg12 = lam12 / wm
else:
# Newton's method
# real ssig1, csig1, ssig2, csig2, eps
ov = numit = trip = 0
while numit < Geodesic.maxit_:
(nlam12, salp2, calp2, sig12, ssig1, csig1, ssig2, csig2,
eps, omg12, dv) = self.Lambda12(
sbet1, cbet1, sbet2, cbet2, salp1, calp1, trip < 1, C1a, C2a, C3a)
v = nlam12 - lam12
if not(abs(v) > Geodesic.tiny_) or not(trip < 1):
if not(abs(v) <= max(Geodesic.tol1_, ov)):
numit = Geodesic.maxit_
break
dalp1 = -v/dv
sdalp1 = math.sin(dalp1); cdalp1 = math.cos(dalp1)
nsalp1 = salp1 * cdalp1 + calp1 * sdalp1
calp1 = calp1 * cdalp1 - salp1 * sdalp1
salp1 = max(0.0, nsalp1)
salp1, calp1 = Geodesic.SinCosNorm(salp1, calp1)
# In some regimes we don't get quadratic convergence because slope
# -> 0. So use convergence conditions based on epsilon instead of
# sqrt(epsilon). The first criterion is a test on abs(v) against
# 100 * epsilon. The second takes credit for an anticipated
# reduction in abs(v) by v/ov (due to the latest update in alp1) and
# checks this against epsilon.
if not(abs(v) >= Geodesic.tol1_ and
Math.sq(v) >= ov * Geodesic.tol0_):
trip += 1
ov = abs(v)
numit += 1
if numit >= Geodesic.maxit_:
# Signal failure.
return a12, s12, azi1, azi2, m12, M12, M21, S12
s12x, m12x, dummy, M12, M21 = self.Lengths(
eps, sig12, ssig1, csig1, ssig2, csig2, cbet1, cbet2,
(outmask & Geodesic.GEODESICSCALE) != 0, C1a, C2a)
m12x *= self._a
s12x *= self._b
a12 = sig12 / Math.degree
omg12 = lam12 - omg12
# end elif not meridian
if outmask & Geodesic.DISTANCE:
s12 = 0 + s12x # Convert -0 to 0
if outmask & Geodesic.REDUCEDLENGTH:
m12 = 0 + m12x # Convert -0 to 0
if outmask & Geodesic.AREA:
# From Lambda12: sin(alp1) * cos(bet1) = sin(alp0)
salp0 = salp1 * cbet1
calp0 = math.hypot(calp1, salp1 * sbet1) # calp0 > 0
# real alp12
if calp0 != 0 and salp0 != 0:
# From Lambda12: tan(bet) = tan(sig) * cos(alp)
ssig1 = sbet1; csig1 = calp1 * cbet1
ssig2 = sbet2; csig2 = calp2 * cbet2
k2 = Math.sq(calp0) * self._ep2
# Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0).
A4 = Math.sq(self._a) * calp0 * salp0 * self._e2
ssig1, csig1 = Geodesic.SinCosNorm(ssig1, csig1)
ssig2, csig2 = Geodesic.SinCosNorm(ssig2, csig2)
C4a = range(Geodesic.nC4_)
self.C4f(k2, C4a)
B41 = Geodesic.SinCosSeries(False, ssig1, csig1, C4a, Geodesic.nC4_)
B42 = Geodesic.SinCosSeries(False, ssig2, csig2, C4a, Geodesic.nC4_)
S12 = A4 * (B42 - B41)
else:
# Avoid problems with indeterminate sig1, sig2 on equator
S12 = 0
if (not meridian and
omg12 < 0.75 * math.pi and # Long difference too big
sbet2 - sbet1 < 1.75): # Lat difference too big
# Use tan(Gamma/2) = tan(omg12/2)
# * (tan(bet1/2)+tan(bet2/2))/(1+tan(bet1/2)*tan(bet2/2))
# with tan(x/2) = sin(x)/(1+cos(x))
somg12 = math.sin(omg12); domg12 = 1 + math.cos(omg12)
dbet1 = 1 + cbet1; dbet2 = 1 + cbet2
alp12 = 2 * math.atan2( somg12 * ( sbet1 * dbet2 + sbet2 * dbet1 ),
domg12 * ( sbet1 * sbet2 + dbet1 * dbet2 ) )
else:
# alp12 = alp2 - alp1, used in atan2 so no need to normalize
salp12 = salp2 * calp1 - calp2 * salp1
calp12 = calp2 * calp1 + salp2 * salp1
# The right thing appears to happen if alp1 = +/-180 and alp2 = 0, viz
# salp12 = -0 and alp12 = -180. However this depends on the sign
# being attached to 0 correctly. The following ensures the correct
# behavior.
if salp12 == 0 and calp12 < 0:
salp12 = Geodesic.tiny_ * calp1
calp12 = -1
alp12 = math.atan2(salp12, calp12)
S12 += self._c2 * alp12
S12 *= swapp * lonsign * latsign
# Convert -0 to 0
S12 += 0
# Convert calp, salp to azimuth accounting for lonsign, swapp, latsign.
if swapp < 0:
salp2, salp1 = salp1, salp2
calp2, calp1 = calp1, calp2
if outmask & Geodesic.GEODESICSCALE:
M21, M12 = M12, M21
salp1 *= swapp * lonsign; calp1 *= swapp * latsign
salp2 *= swapp * lonsign; calp2 *= swapp * latsign
if outmask & Geodesic.AZIMUTH:
# minus signs give range [-180, 180). 0- converts -0 to +0.
azi1 = 0 - math.atan2(-salp1, calp1) / Math.degree
azi2 = 0 - math.atan2(-salp2, calp2) / Math.degree
# Returned value in [0, 180]
return a12, s12, azi1, azi2, m12, M12, M21, S12
def CheckPosition(lat, lon):
if not (abs(lat) <= 90):
raise ValueError("latitude " + str(lat) + " not in [-90, 90]")
if not (lon >= -180 and lon <= 360):
raise ValueError("longitude " + str(lon) + " not in [-180, 360]")
return Geodesic.AngNormalize(lon)
CheckPosition = staticmethod(CheckPosition)
def CheckAzimuth(azi):
if not (azi >= -180 and azi <= 360):
raise ValueError("azimuth " + str(azi) + " not in [-180, 360]")
return Geodesic.AngNormalize(azi)
CheckAzimuth = staticmethod(CheckAzimuth)
def CheckDistance(s):
if not (Math.isfinite(s)):
raise ValueError("distance " + str(s) + " not a finite number")
CheckDistance = staticmethod(CheckDistance)
def Inverse(self, lat1, lon1, lat2, lon2, outmask = DISTANCE | AZIMUTH):
"""
Solve the inverse geodesic problem. Compute geodesic between
(lat1, lon1) and (lat2, lon2). Return a dictionary with (some) of
the following entries:
lat1 latitude of point 1
lon1 longitude of point 1
azi1 azimuth of line at point 1
lat2 latitude of point 2
lon2 longitude of point 2
azi2 azimuth of line at point 2
s12 distance from 1 to 2
a12 arc length on auxiliary sphere from 1 to 2
m12 reduced length of geodesic
M12 geodesic scale 2 relative to 1
M21 geodesic scale 1 relative to 2
S12 area between geodesic and equator
outmask determines which fields get included and if outmask is
omitted, then only the basic geodesic fields are computed. The mask
is an or'ed combination of the following values
Geodesic.LATITUDE
Geodesic.LONGITUDE
Geodesic.AZIMUTH
Geodesic.DISTANCE
Geodesic.REDUCEDLENGTH
Geodesic.GEODESICSCALE
Geodesic.AREA
Geodesic.ALL
"""
lon1 = Geodesic.CheckPosition(lat1, lon1)
lon2 = Geodesic.CheckPosition(lat2, lon2)
result = {'lat1': lat1, 'lon1': lon1, 'lat2': lat2, 'lon2': lon2}
a12, s12, azi1, azi2, m12, M12, M21, S12 = self.GenInverse(
lat1, lon1, lat2, lon2, outmask)
outmask &= Geodesic.OUT_ALL
result['a12'] = a12
if outmask & Geodesic.DISTANCE: result['s12'] = s12
if outmask & Geodesic.AZIMUTH:
result['azi1'] = azi1; result['azi2'] = azi2
if outmask & Geodesic.REDUCEDLENGTH: result['m12'] = m12
if outmask & Geodesic.GEODESICSCALE:
result['M12'] = M12; result['M21'] = M21
if outmask & Geodesic.AREA: result['S12'] = S12
return result
# return a12, lat2, lon2, azi2, s12, m12, M12, M21, S12
def GenDirect(self, lat1, lon1, azi1, arcmode, s12_a12, outmask):
from geographiclib.geodesicline import GeodesicLine
line = GeodesicLine(
self, lat1, lon1, azi1,
# Automatically supply DISTANCE_IN if necessary
outmask | ( Geodesic.NONE if arcmode else Geodesic.DISTANCE_IN))
return line.GenPosition(arcmode, s12_a12, outmask)
def Direct(self, lat1, lon1, azi1, s12,
outmask = LATITUDE | LONGITUDE | AZIMUTH):
"""
Solve the direct geodesic problem. Compute geodesic starting at
(lat1, lon1) with azimuth azi1 and length s12. Return a dictionary
with (some) of the following entries:
lat1 latitude of point 1
lon1 longitude of point 1
azi1 azimuth of line at point 1
lat2 latitude of point 2
lon2 longitude of point 2
azi2 azimuth of line at point 2
s12 distance from 1 to 2
a12 arc length on auxiliary sphere from 1 to 2
m12 reduced length of geodesic
M12 geodesic scale 2 relative to 1
M21 geodesic scale 1 relative to 2
S12 area between geodesic and equator
outmask determines which fields get included and if outmask is
omitted, then only the basic geodesic fields are computed. The mask
is an or'ed combination of the following values
Geodesic.LATITUDE
Geodesic.LONGITUDE
Geodesic.AZIMUTH
Geodesic.DISTANCE
Geodesic.REDUCEDLENGTH
Geodesic.GEODESICSCALE
Geodesic.AREA
Geodesic.ALL
"""
lon1 = Geodesic.CheckPosition(lat1, lon1)
azi1 = Geodesic.CheckAzimuth(azi1)
Geodesic.CheckDistance(s12)
result = {'lat1': lat1, 'lon1': lon1, 'azi1': azi1, 's12': s12}
a12, lat2, lon2, azi2, s12, m12, M12, M21, S12 = self.GenDirect(
lat1, lon1, azi1, False, s12, outmask)
outmask &= Geodesic.OUT_ALL
result['a12'] = a12
if outmask & Geodesic.LATITUDE: result['lat2'] = lat2
if outmask & Geodesic.LONGITUDE: result['lon2'] = lon2
if outmask & Geodesic.AZIMUTH: result['azi2'] = azi2
if outmask & Geodesic.REDUCEDLENGTH: result['m12'] = m12
if outmask & Geodesic.GEODESICSCALE:
result['M12'] = M12; result['M21'] = M21
if outmask & Geodesic.AREA: result['S12'] = S12
return result
def Line(self, lat1, lon1, azi1, caps = ALL):
"""
Return a GeodesicLine object to compute points along a geodesic
starting at lat1, lon1, with azimuth azi1. caps is an or'ed
combination of bit the following values indicating the capabilities
of the return object
Geodesic.LATITUDE
Geodesic.LONGITUDE
Geodesic.AZIMUTH
Geodesic.DISTANCE
Geodesic.REDUCEDLENGTH
Geodesic.GEODESICSCALE
Geodesic.AREA
Geodesic.DISTANCE_IN
Geodesic.ALL
"""
from geographiclib.geodesicline import GeodesicLine
lon1 = Geodesic.CheckPosition(lat1, lon1)
azi1 = Geodesic.CheckAzimuth(azi1)
return GeodesicLine(
self, lat1, lon1, azi1,
# Automatically supply DISTANCE_IN
caps | Geodesic.DISTANCE_IN)
def Area(self, points, polyline = False):
"""
Compute the area of a geodesic polygon given by points, an array of
dictionaries with entries lat and lon. Return a dictionary with
entries
number the number of verices
perimeter the perimeter
area the area (counter-clockwise traversal positive)
There is no need to "close" the polygon. If polyline is set to
True, then the points define a polyline instead of a polygon, the
length is returned as the perimeter, and the area is not returned.
"""
from geographiclib.polygonarea import PolygonArea
for p in points:
Geodesic.CheckPosition(p['lat'], p['lon'])
num, perimeter, area = PolygonArea.Area(self, points, polyline)
result = {'number': num, 'perimeter': perimeter}
if not polyline: result['area'] = area
return result
Geodesic.WGS84 = Geodesic(Constants.WGS84_a, Constants.WGS84_f)
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