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 C{(lon1, lon2)}, both in C{degrees180} or
C{None} if the great circle doesn't reach B{C{lat}}.
'''
self.others(other)
a1, b1 = self.to2ab()
a2, b2 = other.to2ab()
a = radians(lat)
db, b2 = unrollPI(b1, b2, wrap=wrap)
sa, ca, sa1, ca1, \
sa2, ca2, sdb, cdb = sincos2(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 maxLat(self, bearing):
'''Return the maximum latitude reached when travelling
on a great circle on given bearing from this point
based on Clairaut's formula.
The maximum latitude is independent of longitude
and the same for all points on a given latitude.
Negate the result for the minimum latitude (on the
Southern hemisphere).
@param bearing: Initial bearing (compass C{degrees360}).
@return: Maximum latitude (C{degrees90}).
@JSname: I{maxLatitude}.
'''
a, _ = self.to2ab()
m = acos1(abs(sin(radians(bearing)) * cos(a)))
return degrees90(m)
def maxLat(self, bearing):
'''Return the maximum latitude reached when travelling
on a great circle on given bearing from this point
based on Clairaut's formula.
The maximum latitude is independent of longitude
and the same for all points on a given latitude.
Negate the result for the minimum latitude (on the
Southern hemisphere).
@arg bearing: Initial bearing (compass C{degrees360}).
@return: Maximum latitude (C{degrees90}).
@raise ValueError: Invalid B{C{bearing}}.
@JSname: I{maxLatitude}.
'''
m = acos1(abs(sin(Bearing_(bearing)) * cos(self.phi)))
return degrees90(m)
def alongTrackDistanceTo(self, start, end, radius=R_M, wrap=False):
'''Compute the (signed) distance from the start to the closest
point on the great circle path defined by a start and an
end point.
That is, if a perpendicular is drawn from this point to the
great circle path, the along-track distance is the distance
from the start point to the point where the perpendicular
crosses the path.
@arg start: Start point of great circle path (L{LatLon}).
@arg end: End point of great circle path (L{LatLon}).
@kwarg radius: Mean earth radius (C{meter}).
@kwarg wrap: Wrap and unroll longitudes (C{bool}).
@return: Distance along the great circle path (C{meter},
same units as B{C{radius}}), positive if after the
B{C{start}} toward the B{C{end}} point of the path or
negative if before the B{C{start}} point.
@raise TypeError: The B{C{start}} or B{C{end}} point is not L{LatLon}.
@raise ValueError: Invalid B{C{radius}}.
@example:
>>> p = LatLon(53.2611, -0.7972)
>>> s = LatLon(53.3206, -1.7297)
>>> e = LatLon(53.1887, 0.1334)
>>> d = p.alongTrackDistanceTo(s, e) # 62331.58
'''
r, x, b = self._trackDistanceTo3(start, end, radius, wrap)
cx = cos(x)
if abs(cx) > EPS:
return copysign(acos1(cos(r) / cx), cos(b)) * radius
else:
return 0.0
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 to return the intersection
point (L{LatLon}) or C{None}.
@return: The intersection point (B{C{LatLon}}) or a
L{LatLon3Tuple}C{(lat, lon, height)} if B{C{LatLon}}
is C{None}. An alternate intersection point might
be the L{antipode} to the returned result.
@raise TypeError: A B{C{start}} or B{C{end}} point 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')
hs = [start1.height, start2. height]
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 <https://www.EdWilliams.org/avform.htm#Intersection>
elif isscalar(end1) and isscalar(end2): # both bearings
sa1, ca1, sa2, ca2, sr12, cr12 = sincos2(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
# <https://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, cx1, sx2, cx2 = sincos2(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)))
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', hs)
x2, d2 = _x3d2(start2, end2, wrap, '2', hs)
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 = fmean(hs) if height is None else height
r = LatLon3Tuple(a, b, h) if LatLon is None else \
LatLon(a, b, height=h)
return _xnamed(r, intersection.__name__)
def intersections2(
center1,
rad1,
center2,
rad2,
radius=R_M, # MCCABE 13
height=None,
wrap=False,
LatLon=LatLon,
**LatLon_kwds):
'''Compute the intersection points of two circles each defined
by a center point and radius.
@arg center1: Center of the first circle (L{LatLon}).
@arg rad1: Radius of the second circle (C{meter} or C{radians},
see B{C{radius}}).
@arg center2: Center of the second circle (L{LatLon}).
@arg rad2: Radius of the second circle (C{meter} or C{radians},
see B{C{radius}}).
@kwarg radius: Mean earth radius (C{meter} or C{None} if both
B{C{rad1}} and B{C{rad2}} are given in C{radians}).
@kwarg height: Optional height for the intersection point,
overriding the mean height (C{meter}).
@kwarg wrap: Wrap and unroll longitudes (C{bool}).
@kwarg LatLon: Optional class to return the intersection
points (L{LatLon}) or C{None}.
@kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword
arguments, ignored if B{C{LatLon=None}}.
@return: 2-Tuple of the intersection points, each a B{C{LatLon}}
instance or L{LatLon3Tuple}C{(lat, lon, height)} if
B{C{LatLon}} is C{None}. The intersection points are
the same instance for abutting circles.
@raise IntersectionError: Concentric, antipodal, invalid or
non-intersecting circles.
@raise TypeError: If B{C{center1}} or B{C{center2}} not L{LatLon}.
@raise ValueError: Invalid B{C{rad1}}, B{C{rad2}}, B{C{radius}} or
B{C{height}}.
@note: Courtesy U{Samuel Čavoj<https://GitHub.com/mrJean1/PyGeodesy/issues/41>}.
@see: This U{Answer<https://StackOverflow.com/questions/53324667/
find-intersection-coordinates-of-two-circles-on-earth/53331953>}.
'''
def _destination1(bearing):
a, b = _destination2(a1, b1, r1, bearing)
return _latlon3(degrees90(a), degrees180(b), h, intersections2, LatLon,
**LatLon_kwds)
_Trll.others(center1, name='center1')
_Trll.others(center2, name='center2')
a1, b1 = center1.philam
a2, b2 = center2.philam
r1, r2, x = _rads3(rad1, rad2, radius)
if x:
a1, b1, a2, b2 = a2, b2, a1, b1
db, _ = unrollPI(b1, b2, wrap=wrap)
d = vincentys_(a2, a1, db) # radians
if d < max(r1 - r2, EPS):
raise IntersectionError(center1=center1,
rad1=rad1,
center2=center2,
rad2=rad2,
txt=_near_concentric_)
x = fsum_(r1, r2, -d)
if x > EPS:
try:
sd, cd, s1, c1, _, c2 = sincos2(d, r1, r2)
x = sd * s1
if abs(x) < EPS:
raise ValueError
x = acos1((c2 - cd * c1) / x)
except ValueError:
raise IntersectionError(center1=center1,
rad1=rad1,
center2=center2,
rad2=rad2)
elif x < 0:
raise IntersectionError(center1=center1,
rad1=rad1,
center2=center2,
rad2=rad2,
txt=_too_distant_)
b = bearing_(a1, b1, a2, b2, final=False, wrap=wrap)
if height is None:
h = fmean((center1.height, center2.height))
else:
Height(height)
if abs(x) > EPS:
return _destination1(b + x), _destination1(b - x)
else: # abutting circles
x = _destination1(b)
return x, x
def intersection(start1,
end1,
start2,
end2,
height=None,
wrap=False,
LatLon=LatLon,
**LatLon_kwds):
'''Compute the intersection point of two paths both defined
by two points or a start point and bearing from North.
@arg start1: Start point of the first path (L{LatLon}).
@arg end1: End point ofthe first path (L{LatLon}) or
the initial bearing at the first start point
(compass C{degrees360}).
@arg start2: Start point of the second path (L{LatLon}).
@arg end2: End point of the second path (L{LatLon}) or
the initial bearing at the second start point
(compass C{degrees360}).
@kwarg height: Optional height for the intersection point,
overriding the mean height (C{meter}).
@kwarg wrap: Wrap and unroll longitudes (C{bool}).
@kwarg LatLon: Optional class to return the intersection
point (L{LatLon}) or C{None}.
@kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword
arguments, ignored if B{C{LatLon=None}}.
@return: The intersection point (B{C{LatLon}}) or a
L{LatLon3Tuple}C{(lat, lon, height)} if B{C{LatLon}}
is C{None}. An alternate intersection point might
be the L{antipode} to the returned result.
@raise IntersectionError: Intersection is ambiguous or infinite
or the paths are coincident, colinear
or parallel.
@raise TypeError: A B{C{start}} or B{C{end}} point not L{LatLon}.
@raise ValueError: Invalid B{C{height}}.
@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_)
hs = [start1.height, start2.height]
a1, b1 = start1.philam
a2, b2 = start2.philam
db, b2 = unrollPI(b1, b2, wrap=wrap)
r12 = vincentys_(a2, a1, db)
if abs(r12) < EPS: # [nearly] coincident points
a, b = favg(a1, a2), favg(b1, b2)
# see <https://www.EdWilliams.org/avform.htm#Intersection>
elif isscalar(end1) and isscalar(end2): # both bearings
sa1, ca1, sa2, ca2, sr12, cr12 = sincos2(a1, a2, r12)
x1, x2 = (sr12 * ca1), (sr12 * ca2)
if abs(x1) < EPS or abs(x2) < EPS:
raise IntersectionError(start1=start1,
end1=end1,
start2=start2,
end2=end2,
txt='parallel')
# handle domain error for equivalent longitudes,
# see also functions asin_safe and acos_safe at
# <https://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, cx1, sx2, cx2 = sincos2(x1, x2)
if sx1 == 0 and sx2 == 0: # max(abs(sx1), abs(sx2)) < EPS
raise IntersectionError(start1=start1,
end1=end1,
start2=start2,
end2=end2,
txt='infinite')
sx3 = sx1 * sx2
# if sx3 < 0:
# raise IntersectionError(start1=start1, end1=end1,
# start2=start2, end2=end2, txt=_ambiguous_)
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) # PYCHOK PhiLam2Tuple
else: # end point(s) or bearing(s)
x1, d1 = _x3d2(start1, end1, wrap, _1_, hs)
x2, d2 = _x3d2(start2, end2, wrap, _2_, hs)
x = x1.cross(x2)
if x.length < EPS: # [nearly] colinear or parallel paths
raise IntersectionError(start1=start1,
end1=end1,
start2=start2,
end2=end2,
txt=_colinear_)
a, b = x.philam
# choose intersection similar to sphericalNvector
d1 = _xdot(d1, a1, b1, a, b, wrap)
if d1:
d2 = _xdot(d2, a2, b2, a, b, wrap)
if (d2 < 0 and d1 > 0) or (d2 > 0 and d1 < 0):
a, b = antipode_(a, b) # PYCHOK PhiLam2Tuple
h = fmean(hs) if height is None else Height(height)
return _latlon3(degrees90(a), degrees180(b), h, intersection, LatLon,
**LatLon_kwds)
