# Coverage for pygeodesy/sphericalTrigonometry.py : 96%

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-*- coding: utf-8 -*- u'''Trigonometric classes geodetic (lat-/longitude) L{LatLon} and geocentric (ECEF) L{Cartesian} and functions L{areaOf}, L{intersection}, L{isPoleEnclosedBy}, L{meanOf}, L{nearestOn2} and L{perimeterOf}, I{all spherical}. Pure Python implementation of geodetic (lat-/longitude) methods using spherical trigonometry, transcribed from JavaScript originals by I{(C) Chris Veness 2011-2016} published under the same MIT Licence**, see U{Latitude/Longitude<https://www.Movable-Type.co.UK/scripts/latlong.html>}. @newfield example: Example, Examples ''' from pygeodesy.basics import EPS, PI, PI2, PI_2, PI_4, R_M, \ isscalar, map1, _xkwds from pygeodesy.errors import _AssertionError, CrossError, crosserrors, \ IntersectionError, _ValueError, _xkwds_get from pygeodesy.fmath import favg, fdot, fmean, fsum, fsum_ from pygeodesy.formy import antipode_, bearing_, _radical2, vincentys_ from pygeodesy.interns import _1_, _2_, _center1_, _center2_, _coincident_, \ _colinear_, _end_, _fraction_, _invalid_, \ _item_sq, _near_concentric_, _not_convex_, \ _points_, _start_, _start1_, _start2_, \ _too_distant_fmt_ from pygeodesy.lazily import _ALL_LAZY, _ALL_OTHER from pygeodesy.named import LatLon2Tuple, LatLon3Tuple, NearestOn3Tuple, \ _xnamed from pygeodesy.nvectorBase import NvectorBase as _Nvector from pygeodesy.points import _imdex2, ispolar, nearestOn5 as _nearestOn5 from pygeodesy.sphericalBase import _angular, CartesianSphericalBase, \ LatLonSphericalBase, _rads3 from pygeodesy.units import Bearing_, Height, Radius, Radius_, Scalar from pygeodesy.utily import acos1, asin1, degrees90, degrees180, degrees2m, \ iterNumpy2, radiansPI2, sincos2, tan_2, \ unrollPI, wrapPI from pygeodesy.vector3d import sumOf, Vector3d from math import asin, atan2, copysign, cos, degrees, hypot, radians, sin __all__ = _ALL_LAZY.sphericalTrigonometry __version__ = '20.08.09' _EPS_I2 = 4.0 * EPS _PI_EPS_I2 = PI - _EPS_I2 if _PI_EPS_I2 >= PI: raise _AssertionError(_PI_EPS_I2=_PI_EPS_I2) def _destination2(a, b, r, t): '''(INTERNAL) Destination phi- and longitude in C{radians}. @arg a: Latitude (C{radians}). @arg b: Longitude (C{radians}). @arg r: Angular distance (C{radians}). @arg t: Bearing (compass C{radians}). @return: 2-Tuple (phi, lam) of (C{radians}, C{radiansPI}). ''' # see <https://www.EdWilliams.org/avform.htm#LL> sa, ca, sr, cr, st, ct = sincos2(a, r, t) a = asin1(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 class Cartesian(CartesianSphericalBase): '''Extended to convert geocentric, L{Cartesian} points to spherical, geodetic L{LatLon}. ''' def toLatLon(self, **LatLon_datum_kwds): # PYCHOK LatLon=LatLon '''Convert this cartesian point to an C{Nvector}-based geodetic point. @kwarg LatLon_datum_kwds: Optional L{LatLon}, B{C{datum}} and other keyword arguments, ignored if B{C{LatLon=None}}. Use B{C{LatLon=...}} to override this L{LatLon} class or specify B{C{LatLon=None}}. @return: The geodetic point (L{LatLon}) or if B{C{LatLon}} is C{None}, an L{Ecef9Tuple}C{(x, y, z, lat, lon, height, C, M, datum)} with C{C} and C{M} if available. @raise TypeError: Invalid B{C{LatLon_datum_kwds}}. ''' kwds = _xkwds(LatLon_datum_kwds, LatLon=LatLon, datum=self.datum) return CartesianSphericalBase.toLatLon(self, **kwds) class LatLon(LatLonSphericalBase): '''New point on spherical model earth model. @example: >>> p = LatLon(52.205, 0.119) # height=0 ''' def _trackDistanceTo3(self, start, end, radius, wrap): '''(INTERNAL) Helper for along-/crossTrackDistanceTo. ''' self.others(start, name=_start_) self.others(end, name=_end_) r = Radius_(radius) r = start.distanceTo(self, r, wrap=wrap) / r b = radians(start.initialBearingTo(self, wrap=wrap)) e = radians(start.initialBearingTo(end, wrap=wrap)) x = asin(sin(r) * sin(b - e)) return r, x, (e - b) 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 bearingTo(self, other, wrap=False, raiser=False): # PYCHOK no cover '''DEPRECATED, use method C{initialBearingTo}. ''' return self.initialBearingTo(other, wrap=wrap, raiser=raiser) 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. @arg other: The other point defining great circle (L{LatLon}). @arg lat: Latitude at the crossing (C{degrees}). @kwarg 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.philam a2, b2 = other.philam 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 crossTrackDistanceTo(self, start, end, radius=R_M, wrap=False): '''Compute the (signed) distance from this point to the great circle defined by a start and an end point. @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 to great circle (negative if to the left or positive if to the right of the path). @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.crossTrackDistanceTo(s, e) # -307.5 ''' _, x, _ = self._trackDistanceTo3(start, end, radius, wrap) return x * radius def destination(self, distance, bearing, radius=R_M, height=None): '''Locate the destination from this point after having travelled the given distance on the given initial bearing. @arg distance: Distance travelled (C{meter}, same units as B{C{radius}}). @arg bearing: Bearing from this point (compass C{degrees360}). @kwarg radius: Mean earth radius (C{meter}). @kwarg height: Optional height at destination (C{meter}, same units a B{C{radius}}). @return: Destination point (L{LatLon}). @raise ValueError: Invalid B{C{distance}}, B{C{bearing}}, B{C{radius}} or B{C{height}}. @example: >>> p1 = LatLon(51.4778, -0.0015) >>> p2 = p1.destination(7794, 300.7) >>> p2.toStr() # '51.5135°N, 000.0983°W' @JSname: I{destinationPoint}. ''' a, b = self.philam r, t = _angular(distance, radius), Bearing_(bearing) a, b = _destination2(a, b, r, t) h = self.height if height is None else Height(height) return self.classof(degrees90(a), degrees180(b), height=h) def distanceTo(self, other, radius=R_M, wrap=False): '''Compute the distance from this to an other point. @arg other: The other point (L{LatLon}). @kwarg radius: Mean earth radius (C{meter}). @kwarg wrap: Wrap and unroll longitudes (C{bool}). @return: Distance between this and the B{C{other}} point (C{meter}, same units as B{C{radius}}). @raise TypeError: The B{C{other}} point is not L{LatLon}. @raise ValueError: Invalid B{C{radius}}. @example: >>> p1 = LatLon(52.205, 0.119) >>> p2 = LatLon(48.857, 2.351); >>> d = p1.distanceTo(p2) # 404300 ''' self.others(other) a1, b1 = self.philam a2, b2 = other.philam db, _ = unrollPI(b1, b2, wrap=wrap) r = vincentys_(a2, a1, db) return r * Radius(radius) 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. @arg bearing: Bearing from this point (compass C{degrees360}). @return: Vector representing great circle (L{Vector3d}). @raise ValueError: Invalid B{C{bearing}}. @example: >>> p = LatLon(53.3206, -1.7297) >>> g = p.greatCircle(96.0) >>> g.toStr() # (-0.794, 0.129, 0.594) ''' a, b = self.philam t = Bearing_(bearing) sa, ca, sb, cb, st, ct = sincos2(a, b, t) return Vector3d(sb * ct - cb * sa * st, -cb * ct - sb * sa * st, ca * st) # XXX .unit()? def initialBearingTo(self, other, wrap=False, raiser=False): '''Compute the initial bearing (forward azimuth) from this to an other point. @arg other: The other point (spherical L{LatLon}). @kwarg wrap: Wrap and unroll longitudes (C{bool}). @kwarg raiser: Optionally, raise L{CrossError} (C{bool}), use B{C{raiser}}=C{True} for behavior like C{sphericalNvector.LatLon.initialBearingTo}. @return: Initial bearing (compass C{degrees360}). @raise CrossError: If this and the B{C{other}} point coincide, provided B{C{raiser}} is C{True} and L{crosserrors} is C{True}. @raise TypeError: The B{C{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.philam a2, b2 = other.philam # XXX behavior like sphericalNvector.LatLon.initialBearingTo if raiser and crosserrors() and max(abs(a2 - a1), abs(b2 - b1)) < EPS: raise CrossError(_points_, self, txt=_coincident_) return degrees(bearing_(a1, b1, a2, b2, final=False, wrap=wrap)) def intermediateTo(self, other, fraction, height=None, wrap=False): '''Locate the point at given fraction between this and an other point. @arg other: The other point (L{LatLon}). @arg fraction: Fraction between both points (float, between 0.0 for this and 1.0 for the other point). @kwarg height: Optional height, overriding the fractional height (C{meter}). @kwarg wrap: Wrap and unroll longitudes (C{bool}). @return: Intermediate point (L{LatLon}). @raise TypeError: The B{C{other}} point is not L{LatLon}. @raise ValueError: Invalid B{C{fraction}} or B{C{height}}. @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) f = Scalar(fraction, name=_fraction_) a1, b1 = self.philam a2, b2 = other.philam db, b2 = unrollPI(b1, b2, wrap=wrap) r = vincentys_(a2, a1, db) sr = sin(r) if abs(sr) > EPS: sa1, ca1, sa2, ca2, \ sb1, cb1, sb2, cb2 = sincos2(a1, a2, b1, b2) A = sin((1 - f) * r) / sr B = sin( f * 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=f) b = favg(b1, b2, f=f) if height is None: h = self._havg(other, f=f) else: h = Height(height) return self.classof(degrees90(a), degrees180(b), height=h) def intersection(self, end1, other, end2, height=None, wrap=False): '''Locate the intersection point of two paths both defined by two points or a start point and bearing from North. @arg end1: End point of this path (L{LatLon}) or the initial bearing at this point (compass C{degrees360}). @arg other: Start point of the other path (L{LatLon}). @arg end2: End point of the other path (L{LatLon}) or the initial bearing at the other start point (compass C{degrees360}). @kwarg height: Optional height for intersection point, overriding the mean height (C{meter}). @kwarg wrap: Wrap and unroll longitudes (C{bool}). @return: The intersection point (L{LatLon}). 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: If B{C{end1}}, B{C{other}} or B{C{end2}} is not L{LatLon}. @raise ValueError: Invalid B{C{height}}. @example: >>> p = LatLon(51.8853, 0.2545) >>> s = LatLon(49.0034, 2.5735) >>> i = p.intersection(108.547, s, 32.435) # '50.9078°N, 004.5084°E' ''' return intersection(self, end1, other, end2, height=height, wrap=wrap, LatLon=self.classof) def intersections2(self, rad1, other, rad2, radius=R_M, height=None, wrap=True): '''Compute the intersection points of two circles each defined by a center point and radius. @arg rad1: Radius of the this circle (C{meter} or C{radians}, see B{C{radius}}). @arg other: Center of the other circle (L{LatLon}). @arg rad2: Radius of the other 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 points, overriding the "radical height" at the "radical line" between both centers (C{meter}) or C{None}. @kwarg wrap: Wrap and unroll longitudes (C{bool}). @return: 2-Tuple of the intersection points, each a L{LatLon} instance. For abutting circles, the intersection points are the same instance. @raise IntersectionError: Concentric, antipodal, invalid or non-intersecting circles. @raise TypeError: If B{C{other}} is not L{LatLon}. @raise ValueError: Invalid B{C{rad1}}, B{C{rad2}}, B{C{radius}} or B{C{height}}. ''' c2 = self.others(other) return intersections2(self, rad1, c2, rad2, radius=radius, height=height, wrap=wrap, LatLon=self.classof) def isenclosedBy(self, points): '''Check whether a (convex) polygon encloses this point. @arg points: The polygon points (L{LatLon}[]). @return: C{True} if the polygon encloses this point, C{False} otherwise. @raise PointsError: Insufficient number of B{C{points}}. @raise TypeError: Some B{C{points}} are not L{LatLon}. @raise ValueError: Invalid B{C{points}}, non-convex polygon. @example: >>> b = LatLon(45,1), LatLon(45,2), LatLon(46,2), LatLon(46,1) >>> p = LatLon(45,1, 1.1) >>> inside = p.isEnclosedBy(b) # True ''' n, points = self.points2(points, closed=True) n0 = self._N_vector if iterNumpy2(points): v1 = points[n-1]._N_vector v2 = points[n-2]._N_vector gc1 = v2.cross(v1) t0 = gc1.angleTo(n0) > PI_2 for i in range(n): v2 = points[i]._N_vector gc = v1.cross(v2) v1 = v2 ti = gc.angleTo(n0) > PI_2 if ti != t0: return False # outside if gc1.angleTo(gc, vSign=n0) < 0: raise _ValueError(_item_sq(points=i), points[i], txt=_not_convex_) gc1 = gc else: # get great-circle vector for each edge gc, v1 = [], points[n-1]._N_vector for i in range(n): v2 = points[i]._N_vector gc.append(v1.cross(v2)) v1 = v2 # check whether this point on same side of all # polygon edges (to the left or right depending # on anti-/clockwise polygon direction) t0 = gc[0].angleTo(n0) > PI_2 # True if on the right for i in range(1, n): ti = gc[i].angleTo(n0) > PI_2 if ti != t0: # different sides of edge i return False # outside # check for convex polygon (otherwise # the test above is not reliable) gc1 = gc[n-1] for i, gc2 in enumerate(gc): # angle between gc vectors, signed by direction of n0 if gc1.angleTo(gc2, vSign=n0) < 0: raise _ValueError(_item_sq(points=i), points[i], txt=_not_convex_) gc1 = gc2 return True # inside def isEnclosedBy(self, points): # PYCHOK no cover '''DEPRECATED, use method C{isenclosedBy}. ''' return self.isenclosedBy(points) def midpointTo(self, other, height=None, wrap=False): '''Find the midpoint between this and an other point. @arg other: The other point (L{LatLon}). @kwarg height: Optional height for midpoint, overriding the mean height (C{meter}). @kwarg wrap: Wrap and unroll longitudes (C{bool}). @return: Midpoint (L{LatLon}). @raise TypeError: The B{C{other}} point is not L{LatLon}. @raise ValueError: Invalid B{C{height}}. @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 <https://MathForum.org/library/drmath/view/51822.html> a1, b1 = self.philam a2, b2 = other.philam db, b2 = unrollPI(b1, b2, wrap=wrap) sa1, ca1, sa2, ca2, sdb, cdb = sincos2(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(height) return self.classof(degrees90(a), degrees180(b), height=h) def nearestOn(self, point1, point2, radius=R_M, **options): '''Locate the point between two points closest and this point. Distances are approximated by function L{equirectangular_}, subject to the supplied B{C{options}}. @arg point1: Start point (L{LatLon}). @arg point2: End point (L{LatLon}). @kwarg radius: Mean earth radius (C{meter}). @kwarg options: Optional keyword arguments for function L{equirectangular_}. @return: Closest point on the arc (L{LatLon}). @raise LimitError: Lat- and/or longitudinal delta exceeds B{C{limit}}, see function L{equirectangular_}. @raise TypeError: If B{C{point1}} or B{C{point2}} is not L{LatLon}. @raise ValueError: Invalid B{C{radius}}. @see: Functions L{equirectangular_} and L{nearestOn5} and method L{sphericalTrigonometry.LatLon.nearestOn3}. ''' return self.nearestOn3([point1, point2], closed=False, radius=radius, **options)[0] def nearestOn2(self, points, closed=False, radius=R_M, **options): # PYCHOK no cover '''DEPRECATED, use method L{sphericalTrigonometry.LatLon.nearestOn3}. @return: ... 2-Tuple C{(closest, distance)} of the closest point (L{LatLon}) on the polygon and the distance to that point from this point in C{meter}, same units of B{C{radius}}. ''' r = self.nearestOn3(points, closed=closed, radius=radius, **options) return tuple(r[:2]) def nearestOn3(self, points, closed=False, radius=R_M, **options): '''Locate the point on a polygon closest to this point. Distances are approximated by function L{equirectangular_}, subject to the supplied B{C{options}}. @arg points: The polygon points (L{LatLon}[]). @kwarg closed: Optionally, close the polygon (C{bool}). @kwarg radius: Mean earth radius (C{meter}). @kwarg options: Optional keyword arguments for function L{equirectangular_}. @return: A L{NearestOn3Tuple}C{(closest, distance, angle)} of the C{closest} point (L{LatLon}), the L{equirectangular_} C{distance} between this and the C{closest} point in C{meter}, same units as B{C{radius}}. The C{angle} from this to the C{closest} point is in compass C{degrees360}, like function L{compassAngle}. @raise LimitError: Lat- and/or longitudinal delta exceeds B{C{limit}}, see function L{equirectangular_}. @raise PointsError: Insufficient number of B{C{points}}. @raise TypeError: Some B{C{points}} are not C{LatLon}. @raise ValueError: Invalid B{C{radius}}. @see: Functions L{compassAngle}, L{equirectangular_} and L{nearestOn5}. ''' lat, lon, d, c, h = _nearestOn5(self, points, closed=closed, **options) return NearestOn3Tuple(self.classof(lat, lon, height=h), degrees2m(d, radius=radius), c) def toCartesian(self, **Cartesian_datum_kwds): # PYCHOK Cartesian=Cartesian, datum=None '''Convert this point to C{Karney}-based cartesian (ECEF) coordinates. @kwarg Cartesian_datum_kwds: Optional L{Cartesian}, B{C{datum}} and other keyword arguments, ignored if B{C{Cartesian=None}}. Use B{C{Cartesian=...}} to override this L{Cartesian} class or specify B{C{Cartesian=None}}. @return: The cartesian point (L{Cartesian}) or if B{C{Cartesian}} is C{None}, an L{Ecef9Tuple}C{(x, y, z, lat, lon, height, C, M, datum)} with C{C} and C{M} if available. @raise TypeError: Invalid B{C{Cartesian_datum_kwds}}. ''' kwds = _xkwds(Cartesian_datum_kwds, Cartesian=Cartesian, datum=self.datum) return LatLonSphericalBase.toCartesian(self, **kwds) _T00 = LatLon(0, 0) #: (INTERNAL) Reference instance (L{LatLon}). def areaOf(points, radius=R_M, wrap=True): '''Calculate the area of a (spherical) polygon (with great circle arcs joining the points). @arg points: The polygon points (L{LatLon}[]). @kwarg radius: Mean earth radius (C{meter}). @kwarg wrap: Wrap and unroll longitudes (C{bool}). @return: Polygon area (C{meter}, same units as B{C{radius}}, squared). @raise PointsError: Insufficient number of B{C{points}}. @raise TypeError: Some B{C{points}} are not L{LatLon}. @raise ValueError: Invalid B{C{radius}}. @note: The area is based on I{Karney}'s U{'Area of a spherical polygon' <https://OSGeo-org.1560.x6.nabble.com/ Area-of-a-spherical-polygon-td3841625.html>}. @see: L{pygeodesy.areaOf}, L{sphericalNvector.areaOf} and L{ellipsoidalKarney.areaOf}. @example: >>> b = LatLon(45, 1), LatLon(45, 2), LatLon(46, 2), LatLon(46, 1) >>> areaOf(b) # 8666058750.718977 >>> c = LatLon(0, 0), LatLon(1, 0), LatLon(0, 1) >>> areaOf(c) # 6.18e9 ''' n, points = _T00.points2(points, closed=True) # Area method due to Karney: for each edge of the polygon, # # tan(Δλ/2) · (tan(φ1/2) + tan(φ2/2)) # tan(E/2) = ------------------------------------ # 1 + tan(φ1/2) · tan(φ2/2) # # where E is the spherical excess of the trapezium obtained by # extending the edge to the equator-circle vector for each edge def _exs(n, points): # iterate over spherical edge excess a1, b1 = points[n-1].philam ta1 = tan_2(a1) for i in range(n): a2, b2 = points[i].philam db, b2 = unrollPI(b1, b2, wrap=wrap) ta2, tdb = map1(tan_2, a2, db) yield atan2(tdb * (ta1 + ta2), 1 + ta1 * ta2) ta1, b1 = ta2, b2 s = fsum(_exs(n, points)) * 2 if isPoleEnclosedBy(points): s = abs(s) - PI2 return abs(s * Radius(radius)**2) def _xb(a1, b1, end, a, b, wrap): # difference between the bearing to (a, b) and the given # bearing is negative if both are in opposite directions r = bearing_(a1, b1, a, b, wrap=wrap) return PI_2 - abs(wrapPI(r - radians(end))) def _xdot(d, a1, b1, a, b, wrap): # compute dot product d . (-b + b1, a - a1) db, _ = unrollPI(b1, b, wrap=wrap) return fdot(d, db, a - a1) def _x3d2(start, end, wrap, n, hs): # see <https://www.EdWilliams.org/intersect.htm> (5) ff a1, b1 = start.philam if isscalar(end): # bearing, make a point a2, b2 = _destination2(a1, b1, PI_4, radians(end)) else: # must be a point _T00.others(end, name=_end_ + n) hs.append(end.height) a2, b2 = end.philam db, b2 = unrollPI(b1, b2, wrap=wrap) if max(abs(db), abs(a2 - a1)) < EPS: raise IntersectionError(start=start, end=end, txt='null path' + n) # note, in EdWilliams.org/avform.htm W is + and E is - b21, b12 = db * 0.5, -(b1 + b2) * 0.5 sb21, cb21, sb12, cb12, \ sa21, _, sa12, _ = sincos2(b21, b12, a1 - a2, a1 + a2) x = _Nvector(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 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' ''' _T00.others(start1, name=_start1_) _T00.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) def intersections2(center1, rad1, center2, rad2, radius=R_M, height=None, wrap=True, LatLon=LatLon, **LatLon_kwds): '''Compute the intersection points of two circles each defined by a center point and a radius. @arg center1: Center of the first circle (L{LatLon}). @arg rad1: Radius of the first 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 points, overriding the "radical height" at the "radical line" between both centers (C{meter}) or C{None}. @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}. For abutting circles, both intersection points are the same instance. @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>}. ''' c1 = _T00.others(center1, name=_center1_) c2 = _T00.others(center2, name=_center2_) try: return _intersects2(c1, rad1, c2, rad2, radius=radius, height=height, wrap=wrap, LatLon=LatLon, **LatLon_kwds) except (TypeError, ValueError) as x: raise IntersectionError(center1=center1, rad1=rad1, center2=center2, rad2=rad2, txt=str(x)) def _intersects2(c1, rad1, c2, rad2, radius=R_M, # in .ellipsoidalBase._intersects2 height=None, wrap=True, too_d=None, LatLon=LatLon, **LatLon_kwds): # (INTERNAL) Intersect two spherical circles, see L{intersections2} # above, separated to allow callers to embellish any exceptions def _dest3(bearing, h): a, b = _destination2(a1, b1, r1, bearing) return _latlon3(degrees90(a), degrees180(b), h, intersections2, LatLon, **LatLon_kwds) r1, r2, f = _rads3(rad1, rad2, radius) if f: # swapped c1, c2 = c2, c1 # PYCHOK swap a1, b1 = c1.philam a2, b2 = c2.philam db, b2 = unrollPI(b1, b2, wrap=wrap) d = vincentys_(a2, a1, db) # radians if d < max(r1 - r2, EPS): raise ValueError(_near_concentric_) x = fsum_(r1, r2, -d) # overlap if x > EPS: sd, cd, sr1, cr1, _, cr2 = sincos2(d, r1, r2) x = sd * sr1 if abs(x) < EPS: raise ValueError(_invalid_) x = acos1((cr2 - cd * cr1) / x) # 0 <= x <= PI elif x < 0: t = (d * radius) if too_d is None else too_d raise ValueError(_too_distant_fmt_ % (t,)) if height is None: # "radical height" f = _radical2(d, r1, r2).ratio h = Height(favg(c1.height, c2.height, f=f)) else: h = Height(height) b = bearing_(a1, b1, a2, b2, final=False, wrap=wrap) if x < _EPS_I2: # externally ... r = _dest3(b, h) elif x > _PI_EPS_I2: # internally ... r = _dest3(b + PI, h) else: return _dest3(b + x, h), _dest3(b - x, h) return r, r # ... abutting circles def isPoleEnclosedBy(points, wrap=False): # PYCHOK no cover '''DEPRECATED, use function L{ispolar}. ''' return ispolar(points, wrap=wrap) def _latlon3(lat, lon, height, func, LatLon, **LatLon_kwds): '''(INTERNAL) Helper for L{intersection}, L{intersections2} and L{meanof}. ''' if LatLon is None: r = LatLon3Tuple(lat, lon, height) else: kwds = _xkwds(LatLon_kwds, height=height) r = LatLon(lat, lon, **kwds) return _xnamed(r, func.__name__) def meanOf(points, height=None, LatLon=LatLon, **LatLon_kwds): '''Compute the geographic mean of several points. @arg points: Points to be averaged (L{LatLon}[]). @kwarg height: Optional height at mean point, overriding the mean height (C{meter}). @kwarg LatLon: Optional class to return the mean point (L{LatLon}) or C{None}. @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword arguments, ignored if B{C{LatLon=None}}. @return: Point at geographic mean and height (B{C{LatLon}}) or a L{LatLon3Tuple}C{(lat, lon, height)} if B{C{LatLon}} is C{None}. @raise TypeError: Some B{C{points}} are not L{LatLon}. @raise ValueError: No B{C{points}} or invalid B{C{height}}. ''' # geographic mean n, points = _T00.points2(points, closed=False) m = sumOf(points[i]._N_vector for i in range(n)) lat, lon = m._N_vector.latlon if height is None: h = fmean(points[i].height for i in range(n)) else: h = Height(height) return _latlon3(lat, lon, h, meanOf, LatLon, **LatLon_kwds) def nearestOn2(point, points, **closed_radius_LatLon_options): # PYCHOK no cover '''DEPRECATED, use function L{sphericalTrigonometry.nearestOn3}. @return: ... 2-tuple C{(closest, distance)} of the C{closest} point (L{LatLon}) on the polygon and the C{distance} between the C{closest} and the given B{C{point}}. The C{closest} is a B{C{LatLon}} or a L{LatLon2Tuple}C{(lat, lon)} if B{C{LatLon}} is C{None} ... ''' ll, d, _ = nearestOn3(point, points, **closed_radius_LatLon_options) if _xkwds_get(closed_radius_LatLon_options, LatLon=LatLon) is None: ll = LatLon2Tuple(ll.lat, ll.lon) return ll, d def nearestOn3(point, points, closed=False, radius=R_M, LatLon=LatLon, **options): '''Locate the point on a polygon closest to an other, reference point. Distances are approximated by function L{equirectangular_}, subject to the supplied B{C{options}}. @arg point: The other, reference point (L{LatLon}). @arg points: The polygon points (L{LatLon}[]). @kwarg closed: Optionally, close the polygon (C{bool}). @kwarg radius: Mean earth radius (C{meter}). @kwarg LatLon: Optional class to return the closest point (L{LatLon}) or C{None}. @kwarg options: Optional keyword arguments for function L{equirectangular_}. @return: A L{NearestOn3Tuple}C{(closest, distance, angle)} with the C{closest} point as B{L{LatLon}} or L{LatLon3Tuple}C{(lat, lon, height)} if B{C{LatLon}} is C{None}. The C{distance} is the L{equirectangular_} distance between the C{closest} and the given B{C{point}} in C{meter}, same units as B{C{radius}}. The C{angle} from the given B{C{point}} to the C{closest} is in compass C{degrees360}, like function L{compassAngle}. The C{height} is the (interpolated) height at the C{closest} point. @raise LimitError: Lat- and/or longitudinal delta exceeds the B{C{limit}}, see function L{equirectangular_}. @raise PointsError: Insufficient number of B{C{points}}. @raise TypeError: Some B{C{points}} are not C{LatLon}. @raise ValueError: Invalid B{C{radius}}. @see: Functions L{equirectangular_} and L{nearestOn5}. ''' lat, lon, d, c, h = _nearestOn5(point, points, closed=closed, LatLon=None, **options) r = LatLon3Tuple(lat, lon, h) if LatLon is None else \ LatLon(lat, lon, height=h) r = NearestOn3Tuple(r, degrees2m(d, radius=radius), c) return _xnamed(r, nearestOn3.__name__) def perimeterOf(points, closed=False, radius=R_M, wrap=True): '''Compute the perimeter of a (spherical) polygon (with great circle arcs joining the points). @arg points: The polygon points (L{LatLon}[]). @kwarg closed: Optionally, close the polygon (C{bool}). @kwarg radius: Mean earth radius (C{meter}). @kwarg wrap: Wrap and unroll longitudes (C{bool}). @return: Polygon perimeter (C{meter}, same units as B{C{radius}}). @raise PointsError: Insufficient number of B{C{points}}. @raise TypeError: Some B{C{points}} are not L{LatLon}. @raise ValueError: Invalid B{C{radius}}. @note: This perimeter is based on the L{haversine} formula. @see: L{pygeodesy.perimeterOf}, L{sphericalNvector.perimeterOf} and L{ellipsoidalKarney.perimeterOf}. ''' n, points = _T00.points2(points, closed=closed) def _rads(n, points, closed): # angular edge lengths in radians i, m = _imdex2(closed, n) a1, b1 = points[i].philam for i in range(m, n): a2, b2 = points[i].philam db, b2 = unrollPI(b1, b2, wrap=wrap) yield vincentys_(a2, a1, db) a1, b1 = a2, b2 r = fsum(_rads(n, points, closed)) return r * Radius(radius) __all__ += _ALL_OTHER(Cartesian, LatLon, # classes areaOf, # functions intersection, intersections2, ispolar, isPoleEnclosedBy, # DEPRECATED, use ispolar meanOf, nearestOn2, nearestOn3, perimeterOf, sumOf) # == vector3d.sumOf # **) MIT License # # Copyright (C) 2016-2020 -- mrJean1 at Gmail -- All Rights Reserved. # # Permission is hereby granted, free of charge, to any person obtaining a # copy of this software and associated documentation files (the "Software"), # to deal in the Software without restriction, including without limitation # the rights to use, copy, modify, merge, publish, distribute, sublicense, # and/or sell copies of the Software, and to permit persons to whom the # Software is furnished to do so, subject to the following conditions: # # The above copyright notice and this permission notice shall be included # in all copies or substantial portions of the Software. # # THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS # OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, # FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL # THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR # OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, # ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR # OTHER DEALINGS IN THE SOFTWARE.