Package pygeodesy :: Module sphericalNvector |
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# Module sphericalNvector

N-vector-based spherical geodetic (lat-/longitude) classes LatLon and Nvector and functions areaOf, intersection, meanOf, nearestOn2, triangulate and trilaterate.

Pure Python implementation of n-vector-based spherical geodetic (lat-/longitude) methods, transcribed from JavaScript originals by (C) Chris Veness 2011-2016, published under the same MIT Licence**. See Vector-based geodesy and Module latlon-nvector-spherical.

Tools for working with points and paths on (a spherical model of) the earth’s surface using using n-vectors rather than the more common spherical trigonometry. N-vectors make many calculations much simpler, and easier to follow, compared with the trigonometric equivalents.

Based on Kenneth Gade’s ‘Non-singular Horizontal Position Representation’, The Journal of Navigation (2010), vol 63, nr 3, pp 395-417.

Note that the formulations below take x => 0°N,0°E, y => 0°N,90°E and z => 90°N while Gade uses x => 90°N, y => 0°N,90°E, z => 0°N,0°E.

Also note that on a spherical earth model, an n-vector is equivalent to a normalised version of an (ECEF) cartesian coordinate.

Version: 19.04.05

Classes
| LatLon New n-vector based point on spherical earth model.
| Nvector An n-vector is a position representation using a (unit) vector normal to the earth's surface.

Functions
ispolar(points, wrap=False) Check whether a polygon encloses a pole. |
areaOf(points, radius=6371008.77141) Calculate the area of a (spherical) polygon (with great circle arcs joining consecutive points). |
intersection(start1, end1, start2, end2, height=None, LatLon=<class 'pygeodesy.sphericalNvector.LatLon'>) Locate the intersection of two paths each defined by two points or by a start point and an initial bearing. |
meanOf(points, height=None, LatLon=<class 'pygeodesy.sphericalNvector.LatLon'>) Compute the geographic mean of the supplied points. |
nearestOn2(point, points, closed=False, radius=6371008.77141, height=None) Locate the point on a polygon (with great circle arcs joining consecutive points) closest to an other point. |
triangulate(point1, bearing1, point2, bearing2, height=None, LatLon=<class 'pygeodesy.sphericalNvector.LatLon'>) Locate a point given two known points and initial bearings from those points. |
trilaterate(point1, distance1, point2, distance2, point3, distance3, radius=6371008.77141, height=None, LatLon=<class 'pygeodesy.sphericalNvector.LatLon'>) Locate a point at given distances from three other points. |

Function Details

### ispolar(points, wrap=False) |
Check whether a polygon encloses a pole. Parameters: `points` - The polygon points (`LatLon`[]). `wrap` - Wrap and unroll longitudes (`bool`). Returns: `True` if a pole is enclosed by the polygon, `False` otherwise. Raises: `ValueError` - Insufficient number of points. `TypeError` - Some points are not `LatLon` or don't have `bearingTo2`, `initialBearingTo` and `finalBearingTo` methods.

### areaOf(points, radius=6371008.77141) |
Calculate the area of a (spherical) polygon (with great circle arcs joining consecutive points). Parameters: `points` - The polygon points (LatLon[]). `radius` - Optional, mean earth radius (`meter`). Returns: Polygon area (`meter`, same units as radius, squared). Raises: `TypeError` - Some points are not LatLon. `ValueError` - Insufficient number of points. See Also: pygeodesy.areaOf, sphericalTrigonometry.areaOf and ellipsoidalKarney.areaOf. Example:
    >>> b = LatLon(45, 1), LatLon(45, 2), LatLon(46, 2), LatLon(46, 1)
    >>> areaOf(b)  # 8666058750.718977

### intersection(start1, end1, start2, end2, height=None, LatLon=<class 'pygeodesy.sphericalNvector.LatLon'>) |
Locate the intersection of two paths each defined by two points or by a start point and an initial bearing. Parameters: `start1` - Start point of the first path (LatLon). `end1` - End point of the first path (LatLon) or the initial bearing at the first start point (compass `degrees360`). `start2` - Start point of the second path (LatLon). `end2` - End point of the second path (LatLon) or the initial bearing at the second start point (compass `degrees360`). `height` - Optional height at the intersection point, overriding the mean height (`meter`). `LatLon` - Optional (sub-)class for the intersection point (LatLon). Returns: The intersection point (LatLon) or 3-tuple (`degrees90`, `degrees180`, height) if LatLon is `None` or `None` if no unique intersection exists. Raises: `TypeError` - Start or end point(s) not LatLon. `ValueError` - Intersection is ambiguous or infinite or the paths are parallel, coincident or null. Example:
    >>> p = LatLon(51.8853, 0.2545)
    >>> q = LatLon(49.0034, 2.5735)
    >>> i = intersection(p, 108.55, q, 32.44)  # 50.9076°N, 004.5086°E

### meanOf(points, height=None, LatLon=<class 'pygeodesy.sphericalNvector.LatLon'>) |
Compute the geographic mean of the supplied points. Parameters: `points` - Array of points to be averaged (LatLon[]). `height` - Optional height, overriding the mean height (`meter`). `LatLon` - Optional (sub-)class for the mean point (LatLon). Returns: Point at geographic mean and mean height (LatLon). Raises: `ValueError` - Insufficient number of points.

### nearestOn2(point, points, closed=False, radius=6371008.77141, height=None) |
Locate the point on a polygon (with great circle arcs joining consecutive points) closest to an other point. If the given point is within the extent of any great circle arc, the closest point is on that arc. Otherwise, the closest is the nearest of the arc's end points. Parameters: `point` - The other, reference point (LatLon). `points` - The polygon points (LatLon[]). `closed` - Optionally, close the polygon (`bool`). `radius` - Optional, mean earth radius (`meter`). `height` - Optional height, overriding the mean height for a point within the arc (`meter`). Returns: 2-Tuple (closest, distance) of the closest point (LatLon) on the polygon and the distance from the closest to the other point in `meter`, same units as radius. Raises: `TypeError` - Some points or the point not `LatLon`. `ValueError` - No points.

### triangulate(point1, bearing1, point2, bearing2, height=None, LatLon=<class 'pygeodesy.sphericalNvector.LatLon'>) |
Locate a point given two known points and initial bearings from those points. Parameters: `point1` - First reference point (LatLon). `bearing1` - Bearing at the first point (compass `degrees360`). `point2` - Second reference point (LatLon). `bearing2` - Bearing at the second point (compass `degrees360`). `height` - Optional height at the triangulated point, overriding the mean height (`meter`). `LatLon` - Optional (sub-)class for the triangulated point (LatLon). Returns: Triangulated point (LatLon). Raises: `TypeError` - One of the points is not LatLon. `Valuerror` - Points coincide. Example:
    >>> p = LatLon("47°18.228'N","002°34.326'W")  # Basse Castouillet
    >>> q = LatLon("47°18.664'N","002°31.717'W")  # Basse Hergo
    >>> t = triangulate(p, 7, q, 295)  # 47.323667°N, 002.568501°W'

### trilaterate(point1, distance1, point2, distance2, point3, distance3, radius=6371008.77141, height=None, LatLon=<class 'pygeodesy.sphericalNvector.LatLon'>) |
Locate a point at given distances from three other points. See also Trilateration. Parameters: `point1` - First point (LatLon). `distance1` - Distance to the first point (`meter`, same units as radius). `point2` - Second point (LatLon). `distance2` - Distance to the second point (`meter`, same units as radius). `point3` - Third point (LatLon). `distance3` - Distance to the third point (`meter`, same units as radius). `radius` - Optional, mean earth radius (`meter`). `height` - Optional height at the trilaterated point, overriding the mean height (`meter`, same units as radius). `LatLon` - Optional (sub-)class for the trilaterated point (LatLon). Returns: Trilaterated point (LatLon). Raises: `TypeError` - One of the points is not LatLon. `ValueError` - Invalid radius, some distances exceed trilateration or some points coincide.