Module pymap3d.rsphere
compute radii of auxiliary spheres
Source code
""" compute radii of auxiliary spheres"""
try:
from numpy import radians, sin, cos, log, sqrt, degrees, asarray
except ImportError:
from math import radians, sin, cos, log, sqrt, degrees
asarray = None
from .ellipsoid import Ellipsoid
from .rcurve import rcurve_meridian, rcurve_transverse
from .vincenty import vdist
__all__ = [
"rsphere_eqavol",
"rsphere_authalic",
"rsphere_rectifying",
"rsphere_euler",
"rsphere_curve",
"rsphere_triaxial",
"rsphere_biaxial",
]
def rsphere_eqavol(ell: Ellipsoid = None) -> float:
"""computes the radius of the sphere with equal volume as the ellipsoid
Parameters
----------
ell : Ellipsoid, optional
reference ellipsoid
Returns
-------
radius: float
radius of sphere
"""
if ell is None:
ell = Ellipsoid()
f = ell.flattening
return ell.semimajor_axis * (1 - f / 3 - f ** 2 / 9)
def rsphere_authalic(ell: Ellipsoid = None) -> float:
"""computes the radius of the sphere with equal surface area as the ellipsoid
Parameters
----------
ell : Ellipsoid, optional
reference ellipsoid
Returns
-------
radius: float
radius of sphere
"""
if ell is None:
ell = Ellipsoid()
e = ell.eccentricity
if e > 0:
f1 = ell.semimajor_axis ** 2 / 2
f2 = (1 - e ** 2) / (2 * e)
f3 = log((1 + e) / (1 - e))
return sqrt(f1 * (1 + f2 * f3))
else:
return ell.semimajor_axis
def rsphere_rectifying(ell: Ellipsoid = None) -> float:
"""computes the radius of the sphere with equal meridional distances as the ellipsoid
Parameters
----------
ell : Ellipsoid, optional
reference ellipsoid
Returns
-------
radius: float
radius of sphere
"""
if ell is None:
ell = Ellipsoid()
return ((ell.semimajor_axis ** (3 / 2) + ell.semiminor_axis ** (3 / 2)) / 2) ** (2 / 3)
def rsphere_euler(lat1, lon1, lat2, lon2, ell: Ellipsoid = None, deg: bool = True) -> float:
"""computes the Euler radii of curvature at the midpoint of the
great circle arc defined by the endpoints (lat1,lon1) and (lat2,lon2)
Parameters
----------
lat1, lat2 : float
geodetic latitudes (degrees)
lon1, lon2 : float
geodetic longitudes (degrees)
ell : Ellipsoid, optional
reference ellipsoid
deg : bool, optional
degrees input/output (False: radians in/out)
Returns
-------
radius: float
radius of sphere
"""
if not deg:
lat1, lon1, lat2, lon2 = degrees(lat1), degrees(lon1), degrees(lat2), degrees(lon2)
if asarray is not None:
lat1, lat2 = asarray(lat1), asarray(lat2)
latmid = lat1 + (lat2 - lat1) / 2 # compute the midpoint
# compute azimuth
az = vdist(lat1, lon1, lat2, lon2, ell=ell)[1]
# compute meridional and transverse radii of curvature
rho = rcurve_meridian(latmid, ell, deg=True)
nu = rcurve_transverse(latmid, ell, deg=True)
az = radians(az)
den = rho * sin(az) ** 2 + nu * cos(az) ** 2
# compute radius of the arc from point 1 to point 2
return rho * nu / den
def rsphere_curve(lat, ell: Ellipsoid = None, deg: bool = True, method: str = "mean") -> float:
"""computes the arithmetic average of the transverse and meridional
radii of curvature at a specified latitude point
Parameters
----------
ell : Ellipsoid, optional
reference ellipsoid
method: str, optional
"mean" or "norm"
deg : bool, optional
degrees input/output (False: radians in/out)
Returns
-------
radius: float
radius of sphere
"""
if deg:
lat = radians(lat)
rho = rcurve_meridian(lat, ell, deg=False)
nu = rcurve_transverse(lat, ell, deg=False)
if method == "mean":
return (rho + nu) / 2
elif method == "norm":
return sqrt(rho * nu)
else:
raise Exception("pymap3d.rsphere.curve: method must be mean or norm")
def rsphere_triaxial(ell: Ellipsoid = None, method: str = "mean") -> float:
"""computes triaxial average of the semimajor and semiminor axes of the ellipsoid
Parameters
----------
ell : Ellipsoid, optional
reference ellipsoid
method: str, optional
"mean" or "norm"
Returns
-------
radius: float
radius of sphere
"""
if ell is None:
ell = Ellipsoid()
if method == "mean":
return (2 * ell.semimajor_axis + ell.semiminor_axis) / 3
elif method == "norm":
return (ell.semimajor_axis ** 2 * ell.semiminor_axis) ** (1 / 3)
else:
raise Exception("pymap3d.rsphere.rsphere_triaxial: method must be mean or norm")
def rsphere_biaxial(ell: Ellipsoid = None, method: str = "mean") -> float:
"""computes biaxial average of the semimajor and semiminor axes of the ellipsoid
Parameters
----------
ell : Ellipsoid, optional
reference ellipsoid
method: str, optional
"mean" or "norm"
Returns
-------
radius: float
radius of sphere
"""
if ell is None:
ell = Ellipsoid()
if method == "mean":
return (ell.semimajor_axis + ell.semiminor_axis) / 2
elif method == "norm":
return sqrt(ell.semimajor_axis * ell.semiminor_axis)
else:
raise Exception("pymap3d.rsphere.rsphere_biaxial: method must be mean or norm")Functions
def rsphere_authalic(ell=None)-
computes the radius of the sphere with equal surface area as the ellipsoid
Parameters
ell:Ellipsoid, optional- reference ellipsoid
Returns
radius:float- radius of sphere
Source code
def rsphere_authalic(ell: Ellipsoid = None) -> float: """computes the radius of the sphere with equal surface area as the ellipsoid Parameters ---------- ell : Ellipsoid, optional reference ellipsoid Returns ------- radius: float radius of sphere """ if ell is None: ell = Ellipsoid() e = ell.eccentricity if e > 0: f1 = ell.semimajor_axis ** 2 / 2 f2 = (1 - e ** 2) / (2 * e) f3 = log((1 + e) / (1 - e)) return sqrt(f1 * (1 + f2 * f3)) else: return ell.semimajor_axis def rsphere_biaxial(ell=None, method='mean')-
computes biaxial average of the semimajor and semiminor axes of the ellipsoid
Parameters
ell:Ellipsoid, optional- reference ellipsoid
method:str, optional- "mean" or "norm"
Returns
radius:float- radius of sphere
Source code
def rsphere_biaxial(ell: Ellipsoid = None, method: str = "mean") -> float: """computes biaxial average of the semimajor and semiminor axes of the ellipsoid Parameters ---------- ell : Ellipsoid, optional reference ellipsoid method: str, optional "mean" or "norm" Returns ------- radius: float radius of sphere """ if ell is None: ell = Ellipsoid() if method == "mean": return (ell.semimajor_axis + ell.semiminor_axis) / 2 elif method == "norm": return sqrt(ell.semimajor_axis * ell.semiminor_axis) else: raise Exception("pymap3d.rsphere.rsphere_biaxial: method must be mean or norm") def rsphere_curve(lat, ell=None, deg=True, method='mean')-
computes the arithmetic average of the transverse and meridional radii of curvature at a specified latitude point
Parameters
ell:Ellipsoid, optional- reference ellipsoid
method:str, optional- "mean" or "norm"
deg:bool, optional- degrees input/output (False: radians in/out)
Returns
radius:float- radius of sphere
Source code
def rsphere_curve(lat, ell: Ellipsoid = None, deg: bool = True, method: str = "mean") -> float: """computes the arithmetic average of the transverse and meridional radii of curvature at a specified latitude point Parameters ---------- ell : Ellipsoid, optional reference ellipsoid method: str, optional "mean" or "norm" deg : bool, optional degrees input/output (False: radians in/out) Returns ------- radius: float radius of sphere """ if deg: lat = radians(lat) rho = rcurve_meridian(lat, ell, deg=False) nu = rcurve_transverse(lat, ell, deg=False) if method == "mean": return (rho + nu) / 2 elif method == "norm": return sqrt(rho * nu) else: raise Exception("pymap3d.rsphere.curve: method must be mean or norm") def rsphere_eqavol(ell=None)-
computes the radius of the sphere with equal volume as the ellipsoid
Parameters
ell:Ellipsoid, optional- reference ellipsoid
Returns
radius:float- radius of sphere
Source code
def rsphere_eqavol(ell: Ellipsoid = None) -> float: """computes the radius of the sphere with equal volume as the ellipsoid Parameters ---------- ell : Ellipsoid, optional reference ellipsoid Returns ------- radius: float radius of sphere """ if ell is None: ell = Ellipsoid() f = ell.flattening return ell.semimajor_axis * (1 - f / 3 - f ** 2 / 9) def rsphere_euler(lat1, lon1, lat2, lon2, ell=None, deg=True)-
computes the Euler radii of curvature at the midpoint of the great circle arc defined by the endpoints (lat1,lon1) and (lat2,lon2)
Parameters
lat1,lat2:float- geodetic latitudes (degrees)
lon1,lon2:float- geodetic longitudes (degrees)
ell:Ellipsoid, optional- reference ellipsoid
deg:bool, optional- degrees input/output (False: radians in/out)
Returns
radius:float- radius of sphere
Source code
def rsphere_euler(lat1, lon1, lat2, lon2, ell: Ellipsoid = None, deg: bool = True) -> float: """computes the Euler radii of curvature at the midpoint of the great circle arc defined by the endpoints (lat1,lon1) and (lat2,lon2) Parameters ---------- lat1, lat2 : float geodetic latitudes (degrees) lon1, lon2 : float geodetic longitudes (degrees) ell : Ellipsoid, optional reference ellipsoid deg : bool, optional degrees input/output (False: radians in/out) Returns ------- radius: float radius of sphere """ if not deg: lat1, lon1, lat2, lon2 = degrees(lat1), degrees(lon1), degrees(lat2), degrees(lon2) if asarray is not None: lat1, lat2 = asarray(lat1), asarray(lat2) latmid = lat1 + (lat2 - lat1) / 2 # compute the midpoint # compute azimuth az = vdist(lat1, lon1, lat2, lon2, ell=ell)[1] # compute meridional and transverse radii of curvature rho = rcurve_meridian(latmid, ell, deg=True) nu = rcurve_transverse(latmid, ell, deg=True) az = radians(az) den = rho * sin(az) ** 2 + nu * cos(az) ** 2 # compute radius of the arc from point 1 to point 2 return rho * nu / den def rsphere_rectifying(ell=None)-
computes the radius of the sphere with equal meridional distances as the ellipsoid
Parameters
ell:Ellipsoid, optional- reference ellipsoid
Returns
radius:float- radius of sphere
Source code
def rsphere_rectifying(ell: Ellipsoid = None) -> float: """computes the radius of the sphere with equal meridional distances as the ellipsoid Parameters ---------- ell : Ellipsoid, optional reference ellipsoid Returns ------- radius: float radius of sphere """ if ell is None: ell = Ellipsoid() return ((ell.semimajor_axis ** (3 / 2) + ell.semiminor_axis ** (3 / 2)) / 2) ** (2 / 3) def rsphere_triaxial(ell=None, method='mean')-
computes triaxial average of the semimajor and semiminor axes of the ellipsoid
Parameters
ell:Ellipsoid, optional- reference ellipsoid
method:str, optional- "mean" or "norm"
Returns
radius:float- radius of sphere
Source code
def rsphere_triaxial(ell: Ellipsoid = None, method: str = "mean") -> float: """computes triaxial average of the semimajor and semiminor axes of the ellipsoid Parameters ---------- ell : Ellipsoid, optional reference ellipsoid method: str, optional "mean" or "norm" Returns ------- radius: float radius of sphere """ if ell is None: ell = Ellipsoid() if method == "mean": return (2 * ell.semimajor_axis + ell.semiminor_axis) / 3 elif method == "norm": return (ell.semimajor_axis ** 2 * ell.semiminor_axis) ** (1 / 3) else: raise Exception("pymap3d.rsphere.rsphere_triaxial: method must be mean or norm")