orthodrome

exception Slow[source]
class Loc(lat, lon)[source]

Simple location representation.

Attrib lat:

Latitude in [deg].

Attrib lon:

Longitude in [deg].

clip(x, mi, ma)[source]

Clip values of an array.

Parameters:
Returns:

Clipped data.

Return type:

numpy.ndarray

wrap(x, mi, ma)[source]

Wrapping continuous data to fundamental phase values.

x_{\mathrm{wrapped}} = x_{\mathrm{cont},i} - \frac{ x_{\mathrm{cont},i} - r_{\mathrm{min}} } { r_{\mathrm{max}} - r_{\mathrm{min}}} \cdot ( r_{\mathrm{max}} - r_{\mathrm{min}}),\quad x_{\mathrm{wrapped}}\; \in \;[ r_{\mathrm{min}},\, r_{\mathrm{max}}].

Parameters:

  • x (numpy.ndarray) – Continunous data to be wrapped.

  • mi (float) – Minimum value of wrapped data.

  • ma (float) – Maximum value of wrapped data.

Returns:

Wrapped data.

Return type:

numpy.ndarray

cosdelta(*args)[source]

Cosine of the angular distance between two points a and b on a sphere.

This function (find implementation below) returns the cosine of the distance angle ‘delta’ between two points a and b, coordinates of which are expected to be given in geographical coordinates and in degrees. For numerical stability a maximum of 1.0 is enforced.

A_{\mathrm{lat'}} = \frac{ \pi}{180} \cdot A_{lat}, \quad A_{\mathrm{lon'}} = \frac{ \pi}{180} \cdot A_{lon}, \quad B_{\mathrm{lat'}} = \frac{ \pi}{180} \cdot B_{lat}, \quad B_{\mathrm{lon'}} = \frac{ \pi}{180} \cdot B_{lon}\\[0.5cm] \cos(\Delta) = \min( 1.0, \quad \sin( A_{\mathrm{lat'}}) \sin( B_{\mathrm{lat'}} ) + \cos(A_{\mathrm{lat'}}) \cos( B_{\mathrm{lat'}} ) \cos( B_{\mathrm{lon'}} - A_{\mathrm{lon'}} )

Parameters:
Returns:

Cosdelta.

Return type:

float

cosdelta_numpy(a_lats, a_lons, b_lats, b_lons)[source]

Cosine of the angular distance between two points a and b on a sphere.

This function returns the cosines of the distance angles delta between two points a and b given as numpy.ndarray. The coordinates are expected to be given in geographical coordinates and in degrees. For numerical stability a maximum of 1.0 is enforced.

Please find the details of the implementation in the documentation of the function pyrocko.orthodrome.cosdelta() above.

Parameters:
Returns:

Cosdelta.

azimuth(*args)[source]

Azimuth calculation.

This function (find implementation below) returns azimuth … between points a and b, coordinates of which are expected to be given in geographical coordinates and in degrees.

A_{\mathrm{lat'}} = \frac{ \pi}{180} \cdot A_{lat}, \quad A_{\mathrm{lon'}} = \frac{ \pi}{180} \cdot A_{lon}, \quad B_{\mathrm{lat'}} = \frac{ \pi}{180} \cdot B_{lat}, \quad B_{\mathrm{lon'}} = \frac{ \pi}{180} \cdot B_{lon}\\ \varphi_{\mathrm{azi},AB} = \frac{180}{\pi} \arctan \left[ \frac{ \cos( A_{\mathrm{lat'}}) \cos( B_{\mathrm{lat'}} ) \sin(B_{\mathrm{lon'}} - A_{\mathrm{lon'}} )} {\sin ( B_{\mathrm{lat'}} ) - \sin( A_{\mathrm{lat'}} cosdelta) } \right]

Parameters:
Returns:

Azimuth in degree

azimuth_numpy(a_lats, a_lons, b_lats, b_lons, _cosdelta=None)[source]

Calculation of the azimuth (track angle) from a location A towards B.

This function returns azimuths (track angles) from locations A towards B given in numpy.ndarray. Coordinates are expected to be given in geographical coordinates and in degrees.

Please find the details of the implementation in the documentation of the function pyrocko.orthodrome.azimuth().

Parameters:
Returns:

Azimuths in [deg].

Return type:

numpy.ndarray, (N)

azibazi(*args, **kwargs)[source]

Azimuth and backazimuth from location A towards B and back.

Returns:

Azimuth in [deg] from A to B, back azimuth in [deg] from B to A.

Return type:

tuple[float, float]

azibazi_numpy(a_lats, a_lons, b_lats, b_lons, implementation='c')[source]

Azimuth and backazimuth from location A towards B and back.

Arguments are given as numpy.ndarray.

Parameters:
Returns:

Azimuth(s) in [deg] from A to B, back azimuth(s) in [deg] from B to A.

Return type:

numpy.ndarray, numpy.ndarray

azidist_numpy(*args)[source]

Calculation of the azimuth (track angle) and the distance from locations A towards B on a sphere.

The assisting functions used are pyrocko.orthodrome.cosdelta() and pyrocko.orthodrome.azimuth()

Parameters:
Returns:

Azimuths in [deg], distances in [deg].

Return type:

numpy.ndarray, (2xN)

distance_accurate50m(*args, **kwargs)[source]

Accurate distance calculation based on a spheroid of rotation.

Function returns distance in meter between points A and B, coordinates of which must be given in geographical coordinates and in degrees. The returned distance should be accurate to 50 m using WGS84. Values for the Earth’s equator radius and the Earth’s oblateness (f_oblate) are defined in the pyrocko configuration file pyrocko.config.

From wikipedia (http://de.wikipedia.org/wiki/Orthodrome), based on:

Meeus, J.: Astronomical Algorithms, S 85, Willmann-Bell, Richmond 2000 (2nd ed., 2nd printing), ISBN 0-943396-61-1

F = \frac{\pi}{180} \frac{(A_{lat} + B_{lat})}{2}, \quad G = \frac{\pi}{180} \frac{(A_{lat} - B_{lat})}{2}, \quad l = \frac{\pi}{180} \frac{(A_{lon} - B_{lon})}{2} \quad \\[0.5cm] S = \sin^2(G) \cdot \cos^2(l) + \cos^2(F) \cdot \sin^2(l), \quad \quad C = \cos^2(G) \cdot \cos^2(l) + \sin^2(F) \cdot \sin^2(l)

w = \arctan \left( \sqrt{ \frac{S}{C}} \right) , \quad r = \sqrt{\frac{S}{C} }

The spherical-earth distance D between A and B, can be given with:

D_{sphere} = 2w \cdot R_{equator}

The oblateness of the Earth requires some correction with correction factors h1 and h2:

h_1 = \frac{3r - 1}{2C}, \quad h_2 = \frac{3r +1 }{2S}\\[0.5cm] D = D_{\mathrm{sphere}} \cdot [ 1 + h_1 \,f_{\mathrm{oblate}} \cdot \sin^2(F) \cos^2(G) - h_2\, f_{\mathrm{oblate}} \cdot \cos^2(F) \sin^2(G)]

Parameters:
Returns:

Distance in [m].

Return type:

float

distance_accurate50m_numpy(a_lats, a_lons, b_lats, b_lons, implementation='c')[source]

Accurate distance calculation based on a spheroid of rotation.

Function returns distance in meter between points a and b, coordinates of which must be given in geographical coordinates and in degrees. The returned distance should be accurate to 50 m using WGS84. Values for the Earth’s equator radius and the Earth’s oblateness (f_oblate) are defined in the pyrocko configuration file pyrocko.config.

From wikipedia (http://de.wikipedia.org/wiki/Orthodrome), based on:

Meeus, J.: Astronomical Algorithms, S 85, Willmann-Bell, Richmond 2000 (2nd ed., 2nd printing), ISBN 0-943396-61-1

F_i = \frac{\pi}{180} \frac{(a_{lat,i} + a_{lat,i})}{2}, \quad G_i = \frac{\pi}{180} \frac{(a_{lat,i} - b_{lat,i})}{2}, \quad l_i= \frac{\pi}{180} \frac{(a_{lon,i} - b_{lon,i})}{2} \\[0.5cm] S_i = \sin^2(G_i) \cdot \cos^2(l_i) + \cos^2(F_i) \cdot \sin^2(l_i), \quad \quad C_i = \cos^2(G_i) \cdot \cos^2(l_i) + \sin^2(F_i) \cdot \sin^2(l_i)

w_i = \arctan \left( \sqrt{\frac{S_i}{C_i}} \right), \quad r_i = \sqrt{\frac{S_i}{C_i} }

The spherical-earth distance D between a and b, can be given with:

D_{\mathrm{sphere},i} = 2w_i \cdot R_{\mathrm{equator}}

The oblateness of the Earth requires some correction with correction factors h1 and h2:

h_{1.i} = \frac{3r - 1}{2C_i}, \quad h_{2,i} = \frac{3r +1 }{2S_i}\\[0.5cm] D_{AB,i} = D_{\mathrm{sphere},i} \cdot [1 + h_{1,i} \,f_{\mathrm{oblate}} \cdot \sin^2(F_i) \cos^2(G_i) - h_{2,i}\, f_{\mathrm{oblate}} \cdot \cos^2(F_i) \sin^2(G_i)]

Parameters:
Returns:

Distances in [m].

Return type:

numpy.ndarray, (N)

ne_to_latlon(lat0, lon0, north_m, east_m)[source]

Transform local cartesian coordinates to latitude and longitude.

From east and north coordinates (x and y coordinate numpy.ndarray) relative to a reference differences in longitude and latitude are calculated, which are effectively changes in azimuth and distance, respectively:

\text{distance change:}\; \Delta {\bf{a}} &= \sqrt{{\bf{y}}^2 + {\bf{x}}^2 }/ \mathrm{R_E}, \text{azimuth change:}\; \Delta \bf{\gamma} &= \arctan( \bf{x} / \bf{y}).

The projection used preserves the azimuths of the input points.

Parameters:

  • lat0 (float) – Latitude origin of the cartesian coordinate system in [deg].

  • lon0 (float) – Longitude origin of the cartesian coordinate system in [deg].

  • north_m (numpy.ndarray, (N)) – Northing distances from origin in [m].

  • east_m (numpy.ndarray, (N)) – Easting distances from origin in [m].

Returns:

Array with latitudes and longitudes in [deg].

Return type:

numpy.ndarray, (2xN)

azidist_to_latlon(lat0, lon0, azimuth_deg, distance_deg)[source]

Absolute latitudes and longitudes are calculated from relative changes.

Convenience wrapper to azidist_to_latlon_rad() with azimuth and distance given in degrees.

Parameters:

  • lat0 (float) – Latitude origin of the cartesian coordinate system in [deg].

  • lon0 (float) – Longitude origin of the cartesian coordinate system in [deg].

  • azimuth_deg (numpy.ndarray, (N)) – Azimuth from origin in [deg].

  • distance_deg (numpy.ndarray, (N)) – Distances from origin in [deg].

Returns:

Array with latitudes and longitudes in [deg].

Return type:

numpy.ndarray, (2xN)

azidist_to_latlon_rad(lat0, lon0, azimuth_rad, distance_rad)[source]

Absolute latitudes and longitudes are calculated from relative changes.

For numerical stability a range between of -1.0 and 1.0 is enforced for c and alpha.

\Delta {\bf a}_i \; \text{and} \; \Delta \gamma_i \; \text{are relative distances and azimuths from lat0 and lon0 for \textit{i} source points of a finite source.}

\mathrm{b} &= \frac{\pi}{2} -\frac{\pi}{180}\;\mathrm{lat_0}\\ {\bf c}_i &=\arccos[\; \cos(\Delta {\bf{a}}_i) \cos(\mathrm{b}) + |\Delta \gamma_i| \, \sin(\Delta {\bf a}_i) \sin(\mathrm{b})\; ] \\ \mathrm{lat}_i &= \frac{180}{\pi} \left(\frac{\pi}{2} - {\bf c}_i \right)

\alpha_i &= \arcsin \left[ \; \frac{ \sin(\Delta {\bf a}_i ) \sin(|\Delta \gamma_i|)}{\sin({\bf c}_i)}\; \right] \\ \alpha_i &= \begin{cases} \alpha_i, &\text{if} \; \cos(\Delta {\bf a}_i) - \cos(\mathrm{b}) \cos({\bf{c}}_i) > 0, \; \text{else} \\ \pi - \alpha_i, & \text{if} \; \alpha_i > 0,\; \text{else}\\ -\pi - \alpha_i, & \text{if} \; \alpha_i < 0. \end{cases} \\ \mathrm{lon}_i &= \mathrm{lon_0} + \frac{180}{\pi} \, \frac{\Delta \gamma_i }{|\Delta \gamma_i|} \cdot \alpha_i \text{, with $\alpha_i \in [-\pi,\pi]$}

Parameters:

  • lat0 (float) – Latitude origin of the cartesian coordinate system in [deg].

  • lon0 (float) – Longitude origin of the cartesian coordinate system in [deg].

  • distance_rad (numpy.ndarray, (N)) – Distances from origin in [rad].

  • azimuth_rad (numpy.ndarray, (N)) – Azimuth from origin in [rad].

Returns:

Array with latitudes and longitudes in [deg].

Return type:

numpy.ndarray, (2xN)

ne_to_latlon_alternative_method(lat0, lon0, north_m, east_m)[source]

Transform local cartesian coordinates to latitude and longitude.

Like pyrocko.orthodrome.ne_to_latlon(), but this method (implementation below), although it should be numerically more stable, suffers problems at points which are across the pole as seen from the cartesian origin.

\text{distance change:}\; \Delta {{\bf a}_i} &= \sqrt{{\bf{y}}^2_i + {\bf{x}}^2_i }/ \mathrm{R_E},\\ \text{azimuth change:}\; \Delta {\bf \gamma}_i &= \arctan( {\bf x}_i {\bf y}_i). \\ \mathrm{b} &= \frac{\pi}{2} -\frac{\pi}{180} \;\mathrm{lat_0}\\

{{\bf z}_1}_i &= \cos{\left( \frac{\Delta {\bf a}_i - \mathrm{b}}{2} \right)} \cos {\left( \frac{|\gamma_i|}{2} \right) }\\ {{\bf n}_1}_i &= \cos{\left( \frac{\Delta {\bf a}_i + \mathrm{b}}{2} \right)} \sin {\left( \frac{|\gamma_i|}{2} \right) }\\ {{\bf z}_2}_i &= \sin{\left( \frac{\Delta {\bf a}_i - \mathrm{b}}{2} \right)} \cos {\left( \frac{|\gamma_i|}{2} \right) }\\ {{\bf n}_2}_i &= \sin{\left( \frac{\Delta {\bf a}_i + \mathrm{b}}{2} \right)} \sin {\left( \frac{|\gamma_i|}{2} \right) }\\ {{\bf t}_1}_i &= \arctan{\left( \frac{{{\bf z}_1}_i} {{{\bf n}_1}_i} \right) }\\ {{\bf t}_2}_i &= \arctan{\left( \frac{{{\bf z}_2}_i} {{{\bf n}_2}_i} \right) } \\[0.5cm] c &= \begin{cases} 2 \cdot \arccos \left( {{\bf z}_1}_i / \sin({{\bf t}_1}_i) \right),\; \text{if } |\sin({{\bf t}_1}_i)| > |\sin({{\bf t}_2}_i)|,\; \text{else} \\ 2 \cdot \arcsin{\left( {{\bf z}_2}_i / \sin({{\bf t}_2}_i) \right)}. \end{cases}\\

{\bf {lat}}_i &= \frac{180}{ \pi } \left( \frac{\pi}{2} - {\bf {c}}_i \right) \\ {\bf {lon}}_i &= {\bf {lon}}_0 + \frac{180}{ \pi } \frac{\gamma_i}{|\gamma_i|}, \text{ with}\; \gamma \in [-\pi,\pi]

Parameters:

  • lat0 (float) – Latitude origin of the cartesian coordinate system in [deg].

  • lon0 (float) – Longitude origin of the cartesian coordinate system in [deg].

  • north_m (numpy.ndarray, (N)) – Northing distances from origin in [m].

  • east_m (numpy.ndarray, (N)) – Easting distances from origin in [m].

Returns:

Array with latitudes and longitudes in [deg].

Return type:

numpy.ndarray, (2xN)

latlon_to_ne(*args)[source]

Relative cartesian coordinates with respect to a reference location.

For two locations, a reference location A and another location B, given in geographical coordinates in degrees, the corresponding cartesian coordinates are calculated. Assisting functions are pyrocko.orthodrome.azimuth() and pyrocko.orthodrome.distance_accurate50m().

D_{AB} &= \mathrm{distance\_accurate50m(}A, B \mathrm{)}, \quad \varphi_{\mathrm{azi},AB} = \mathrm{azimuth(}A,B \mathrm{)}\\[0.3cm] n &= D_{AB} \cdot \cos( \frac{\pi }{180} \varphi_{\mathrm{azi},AB} )\\ e &= D_{AB} \cdot \sin( \frac{\pi }{180} \varphi_{\mathrm{azi},AB})

Parameters:
Returns:

Northing and easting from refloc to location in [m].

Return type:

tuple[float, float]

latlon_to_ne_numpy(lat0, lon0, lat, lon)[source]

Relative cartesian coordinates with respect to a reference location.

For two locations, a reference location (lat0, lon0) and another location B, given in geographical coordinates in degrees, the corresponding cartesian coordinates are calculated. Assisting functions are azimuth() and distance_accurate50m().

Parameters:

  • lat0 – Latitude of the reference location in [deg].

  • lon0 – Longitude of the reference location in [deg].

  • lat – Latitude of the absolute location in [deg].

  • lon – Longitude of the absolute location in [deg].

Returns:

(n, e): relative north and east positions in [m].

Return type:

numpy.ndarray, (2xN)

Implemented formulations:

D_{AB} &= \mathrm{distance\_accurate50m(}A, B \mathrm{)}, \quad \varphi_{\mathrm{azi},AB} = \mathrm{azimuth(}A,B \mathrm{)}\\[0.3cm] n &= D_{AB} \cdot \cos( \frac{\pi }{180} \varphi_{ \mathrm{azi},AB} )\\ e &= D_{AB} \cdot \sin( \frac{\pi }{180} \varphi_{ \mathrm{azi},AB} )

positive_region(region)[source]

Normalize parameterization of a rectangular geographical region.

Parameters:

region (tuple of float) – (west, east, south, north) in [deg].

Returns:

(west, east, south, north), where west <= east and where west and east are in the range [-180., 180.+360.].

Return type:

tuple of float

points_in_region(p, region)[source]

Check what points are contained in a rectangular geographical region.

Parameters:

  • p (numpy.ndarray (N, 2)) – (lat, lon) pairs in [deg].

  • region (tuple of float) – (west, east, south, north) region boundaries in [deg].

Returns:

Mask, returning True for each point within the region.

Return type:

numpy.ndarray of bool, shape (N)

point_in_region(p, region)[source]

Check if a point is contained in a rectangular geographical region.

Parameters:

  • p (tuple of float) – (lat, lon) in [deg].

  • region (tuple of float) – (west, east, south, north) region boundaries in [deg].

Returns:

True, if point is in region, else False.

Return type:

bool

radius_to_region(lat, lon, radius)[source]

Get a rectangular region which fully contains a given circular region.

Parameters:

  • lat (float) – Latitude of the center point of circular region in [deg].

  • lon (float) – Longitude of the center point of circular region in [deg].

  • radius (float) – Radius of circular region in [m].

Returns:

Rectangular region as (east, west, south, north) in [deg] or None.

Return type:

tuple[float, float, float, float]

geographic_midpoint(lats, lons, weights=None)[source]

Calculate geographic midpoints by finding the center of gravity.

This method suffers from instabilities if points are centered around the poles.

Parameters:
Returns:

Latitudes and longitudes of the midpoints in [deg].

Return type:

numpy.ndarray, (2xN)

geodetic_to_ecef(lat, lon, alt)[source]

Convert geodetic coordinates to Earth-Centered, Earth-Fixed (ECEF) Cartesian coordinates. [3] [4]

Parameters:

  • lat (float) – Geodetic latitude in [deg].

  • lon (float) – Geodetic longitude in [deg].

  • alt (float) – Geodetic altitude (height) in [m] (positive for points outside the geoid).

Returns:

ECEF Cartesian coordinates (X, Y, Z) in [m].

Return type:

tuple[float, float, float]

ecef_to_geodetic(X, Y, Z)[source]

Convert Earth-Centered, Earth-Fixed (ECEF) Cartesian coordinates to geodetic coordinates (Ferrari’s solution).

Parameters:

Z (X, Y,) – Cartesian coordinates in ECEF system in [m].

Returns:

Geodetic coordinates (lat, lon, alt). Latitude and longitude are in [deg] and altitude is in [m] (positive for points outside the geoid).

Return type:

tuple, float

See also

https://en.wikipedia.org/wiki/Geographic_coordinate_conversion #The_application_of_Ferrari.27s_solution

exception Farside[source]
contains_points(polygon, points)[source]

Test which points are inside polygon on a sphere.

The inside of the polygon is defined as the area which is to the left hand side of an observer walking the polygon line, points in order, on the sphere. Lines between the polygon points are treated as great circle paths. The polygon may be arbitrarily complex, as long as it does not have any crossings or thin parts with zero width. The polygon may contain the poles and is allowed to wrap around the sphere multiple times.

The algorithm works by consecutive cutting of the polygon into (almost) hemispheres and subsequent Gnomonic projections to perform the point-in-polygon tests on a 2D plane.

Parameters:

  • polygon (numpy.ndarray of shape (N, 2), second index 0=lat, 1=lon) – Point coordinates defining the polygon [deg].

  • points (numpy.ndarray of shape (N, 2), second index 0=lat, 1=lon) – Coordinates of points to test [deg].

Returns:

Boolean mask array.

Return type:

numpy.ndarray of shape (N,).

contains_point(polygon, point)[source]

Test if point is inside polygon on a sphere.

Convenience wrapper to contains_points() to test a single point.

Parameters:

  • polygon (numpy.ndarray of shape (N, 2), second index 0=lat, 1=lon) – Point coordinates defining the polygon [deg].

  • point (tuple of float) – Coordinates (lat, lon) of point to test [deg].

Returns:

True, if point is located within polygon, else False.

Return type:

bool