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# -*- coding: utf-8 -*- 

 

u'''Generic 3-D vector base class L{Vector3d}, class L{VectorError} 

and functions L{intersections2}, L{iscolinearWith}, L{nearestOn}, 

L{parse3d}, L{sumOf} and L{trilaterate3d2}. 

 

Pure Python implementation of vector-based functions by I{(C) Chris 

Veness 2011-2015} published under the same MIT Licence**, see 

U{Vector-based geodesy 

<https://www.Movable-Type.co.UK/scripts/latlong-vectors.html>}. 

 

@newfield example: Example, Examples 

''' 

 

from pygeodesy.basics import isscalar, len2, map1, property_doc_, \ 

property_RO, _xnumpy, _xkwds 

from pygeodesy.errors import _AssertionError, CrossError, IntersectionError, \ 

_IsnotError, _TypeError, _ValueError 

from pygeodesy.fmath import euclid_, fdot, fsum, fsum_, hypot_, hypot2_ 

from pygeodesy.formy import n_xyz2latlon, n_xyz2philam, _radical2 

from pygeodesy.interns import EPS, EPS1, NN, PI, PI2, _coincident_, _colinear_, \ 

_COMMA_, _COMMA_SPACE_, _datum_, _h_, _height_, \ 

_invalid_, joined_, _Missing, _name_, \ 

_near_concentric_, _no_intersection_, _PARENTH_, \ 

_scalar_, _too_distant_fmt_, _y_, _z_, _0_0, _1_0 

from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY 

from pygeodesy.named import modulename, _NamedBase, _xnamed, _xother3, _xotherError 

from pygeodesy.namedTuples import Vector3Tuple # Vector4Tuple 

from pygeodesy.streprs import strs 

from pygeodesy.units import Radius, Radius_ 

 

from math import atan2, copysign, cos, sin, sqrt 

 

__all__ = _ALL_LAZY.vector3d 

__version__ = '20.10.08' 

 

 

def _xyzn4(xyz, y, z, Error=_TypeError): # imported by .ecef 

'''(INTERNAL) Get an C{(x, y, z, name)} 4-tuple. 

''' 

try: 

t = xyz.x, xyz.y, xyz.z 

except AttributeError: 

t = xyz, y, z 

try: 

x, y, z = map1(float, *t) 

except (TypeError, ValueError) as x: 

d = dict(zip(('xyz', _y_, _z_), t)) 

raise Error(txt=str(x), **d) 

 

return x, y, z, getattr(xyz, _name_, NN) 

 

 

def _xyzhdn6(xyz, y, z, height, datum, ll, Error=_TypeError): # by .cartesianBase, .nvectorBase 

'''(INTERNAL) Get an C{(x, y, z, h, d, name)} 6-tuple. 

''' 

x, y, z, n = _xyzn4(xyz, y, z, Error=Error) 

 

h = height or getattr(xyz, _height_, None) \ 

or getattr(xyz, _h_, None) \ 

or getattr(ll, _height_, None) 

 

d = datum or getattr(xyz, _datum_, None) \ 

or getattr(ll, _datum_, None) 

 

return x, y, z, h, d, n 

 

 

class VectorError(_ValueError): 

'''L{Vector3d} or C{*Nvector} issue. 

''' 

pass 

 

 

class Vector3d(_NamedBase): # XXX or _NamedTuple or Vector3Tuple? 

'''Generic 3-D vector manipulation. 

 

In a geodesy context, these may be used to represent: 

- n-vector representing a normal to point on earth's surface 

- earth-centered, earth-fixed vector (= n-vector for spherical model) 

- great circle normal to vector 

- motion vector on earth's surface 

- etc. 

''' 

_crosserrors = True # un/set by .errors.crosserrors 

 

_fromll = None # original latlon, '_fromll' 

_length = None # cached length 

_length2 = None # cached length2 

_numpy = None # module numpy iff imported by trilaterate3d2 below 

_united = None # cached norm, unit (L{Vector3d}) 

_xyz = None # cached xyz (L{Vector3Tuple}) 

 

_x = 0 # X component 

_y = 0 # Y component 

_z = 0 # Z component 

 

def __init__(self, x, y, z, ll=None, name=NN): 

'''New 3-D L{Vector3d}. 

 

The vector may be normalised or use x/y/z values 

for height relative to the surface of the sphere 

or ellipsoid, distance from earth centre, etc. 

 

@arg x: X component of vector (C{scalar}). 

@arg y: Y component of vector (C{scalar}). 

@arg z: Z component of vector (C{scalar}). 

@kwarg ll: Optional latlon reference (C{LatLon}). 

@kwarg name: Optional name (C{str}). 

''' 

self._x = x 

self._y = y 

self._z = z 

if ll: 

self._fromll = ll 

if name: 

self.name = name 

 

def __abs__(self): 

'''Return the norm of this vector. 

 

@return: Norm, unit length (C{float}); 

''' 

return self.length 

 

def __add__(self, other): 

'''Add this to an other vector (L{Vector3d}). 

 

@return: Vectorial sum (L{Vector3d}). 

 

@raise TypeError: Incompatible B{C{other}} C{type}. 

''' 

return self.plus(other) 

# __iadd__ = __add__ 

__radd__ = __add__ 

 

def __cmp__(self, other): # Python 2- 

'''Compare this and an other vector 

 

@arg other: The other vector (L{Vector3d}). 

 

@return: -1, 0 or +1 (C{int}). 

 

@raise TypeError: Incompatible B{C{other}} C{type}. 

''' 

self.others(other) 

return -1 if self.length < other.length else ( 

+1 if self.length > other.length else 0) 

 

cmp = __cmp__ 

 

def __div__(self, scalar): 

'''Divide this vector by a scalar. 

 

@arg scalar: The divisor (C{scalar}). 

 

@return: Quotient (L{Vector3d}). 

 

@raise TypeError: Non-scalar B{C{scalar}}. 

''' 

return self.dividedBy(scalar) 

# __itruediv__ = __div__ 

__truediv__ = __div__ 

 

def __eq__(self, other): 

'''Is this vector equal to an other vector? 

 

@arg other: The other vector (L{Vector3d}). 

 

@return: C{True} if equal, C{False} otherwise. 

 

@raise TypeError: Incompatible B{C{other}} C{type}. 

''' 

self.others(other) 

return self.isequalTo(other) 

 

def __ge__(self, other): 

'''Is this vector longer than or equal to an other vector? 

 

@arg other: The other vector (L{Vector3d}). 

 

@return: C{True} if so, C{False} otherwise. 

 

@raise TypeError: Incompatible B{C{other}} C{type}. 

''' 

self.others(other) 

return self.length >= other.length 

 

def __gt__(self, other): 

'''Is this vector longer than an other vector? 

 

@arg other: The other vector (L{Vector3d}). 

 

@return: C{True} if so, C{False} otherwise. 

 

@raise TypeError: Incompatible B{C{other}} C{type}. 

''' 

self.others(other) 

return self.length > other.length 

 

def __le__(self, other): # Python 3+ 

'''Is this vector shorter than or equal to an other vector? 

 

@arg other: The other vector (L{Vector3d}). 

 

@return: C{True} if so, C{False} otherwise. 

 

@raise TypeError: Incompatible B{C{other}} C{type}. 

''' 

self.others(other) 

return self.length <= other.length 

 

def __lt__(self, other): # Python 3+ 

'''Is this vector shorter than an other vector? 

 

@arg other: The other vector (L{Vector3d}). 

 

@return: C{True} if so, C{False} otherwise. 

 

@raise TypeError: Incompatible B{C{other}} C{type}. 

''' 

self.others(other) 

return self.length < other.length 

 

# Luciano Ramalho, "Fluent Python", page 397, O'Reilly 2016 

def __matmul__(self, other): # PYCHOK Python 3.5+ ... c = a @ b 

'''Compute the cross product of this and an other vector. 

 

@arg other: The other vector (L{Vector3d}). 

 

@return: Cross product (L{Vector3d}). 

 

@raise TypeError: Incompatible B{C{other}} C{type}. 

''' 

return self.cross(other) 

# __imatmul__ = __matmul__ 

 

def __mul__(self, scalar): 

'''Multiply this vector by a scalar 

 

@arg scalar: Factor (C{scalar}). 

 

@return: Product (L{Vector3d}). 

''' 

return self.times(scalar) 

# __imul__ = __mul__ 

# __rmul__ = __mul__ 

 

def __ne__(self, other): 

'''Is this vector not equal to an other vector? 

 

@arg other: The other vector (L{Vector3d}). 

 

@return: C{True} if so, C{False} otherwise. 

 

@raise TypeError: Incompatible B{C{other}} C{type}. 

''' 

self.others(other) 

return not self.isequalTo(other) 

 

def __neg__(self): 

'''Negate this vector. 

 

@return: Negative (L{Vector3d}) 

''' 

return self.negate() 

 

def __pos__(self): 

'''Copy this vector. 

 

@return: Positive (L{Vector3d}) 

''' 

return self.copy() 

 

# Luciano Ramalho, "Fluent Python", page 397, O'Reilly 2016 

def __rmatmul__(self, other): # PYCHOK Python 3.5+ ... c = a @ b 

'''Compute the cross product of an other and this vector. 

 

@arg other: The other vector (L{Vector3d}). 

 

@return: Cross product (L{Vector3d}). 

 

@raise TypeError: Incompatible B{C{other}} C{type}. 

''' 

self.others(other) 

return other.cross(self) 

 

def __rsub__(self, other): 

'''Subtract this vector from an other vector. 

 

@arg other: The other vector (L{Vector3d}). 

 

@return: Difference (L{Vector3d}). 

 

@raise TypeError: Incompatible B{C{other}} C{type}. 

''' 

self.others(other) 

return other.minus(self) 

 

def __sub__(self, other): 

'''Subtract an other vector from this vector. 

 

@arg other: The other vector (L{Vector3d}). 

 

@return: Difference (L{Vector3d}). 

 

@raise TypeError: Incompatible B{C{other}} C{type}. 

''' 

return self.minus(other) 

# __isub__ = __sub__ 

 

def _update(self, updated, *attrs): 

'''(INTERNAL) Zap cached attributes if updated. 

''' 

if updated: 

_NamedBase._update(self, updated, # '_fromll' 

'_length', '_length2', 

'_united', '_xyz', *attrs) 

 

def angleTo(self, other, vSign=None, wrap=False): 

'''Compute the angle between this and an other vector. 

 

@arg other: The other vector (L{Vector3d}). 

@kwarg vSign: Optional vector, if supplied (and out of the 

plane of this and the other), angle is signed 

positive if this->other is clockwise looking 

along vSign or negative in opposite direction, 

otherwise angle is unsigned. 

@kwarg warp: Wrap/unroll the angle to +/-PI (c{bool}). 

 

@return: Angle (C{radians}). 

 

@raise TypeError: If B{C{other}} or B{C{vSign}} not a L{Vector3d}. 

''' 

x = self.cross(other) 

s = x.length 

if s < EPS: 

return _0_0 

# use vSign as reference to get sign of s 

if vSign and x.dot(vSign) < 0: 

s = -s 

 

a = atan2(s, self.dot(other)) 

if wrap and abs(a) > PI: 

a -= copysign(PI2, a) 

return a 

 

def cross(self, other, raiser=None): # raiser=NN 

'''Compute the cross product of this and an other vector. 

 

@arg other: The other vector (L{Vector3d}). 

@kwarg raiser: Optional, L{CrossError} label if raised (C{str}). 

 

@return: Cross product (L{Vector3d}). 

 

@raise CrossError: Zero or near-zero cross product and both 

B{C{raiser}} and L{crosserrors} set. 

 

@raise TypeError: Incompatible B{C{other}} C{type}. 

''' 

self.others(other) 

 

x = self.y * other.z - self.z * other.y 

y = self.z * other.x - self.x * other.z 

z = self.x * other.y - self.y * other.x 

 

if raiser and self.crosserrors and max(map1(abs, x, y, z)) < EPS: 

t = _coincident_ if self.isequalTo(other) else _colinear_ 

r = getattr(other, '_fromll', None) or other 

raise CrossError(raiser, r, txt=t) 

 

return self.classof(x, y, z) 

 

@property_doc_('''raise or ignore L{CrossError} exceptions (C{bool}).''') 

def crosserrors(self): 

'''Get L{CrossError} exceptions (C{bool}). 

''' 

return self._crosserrors 

 

@crosserrors.setter # PYCHOK setter! 

def crosserrors(self, raiser): 

'''Raise L{CrossError} exceptions (C{bool}). 

''' 

self._crosserrors = bool(raiser) 

 

def dividedBy(self, factor): 

'''Divide this vector by a scalar. 

 

@arg factor: The divisor (C{scalar}). 

 

@return: New, scaled vector (L{Vector3d}). 

 

@raise TypeError: Non-scalar B{C{factor}}. 

 

@raise VectorError: Invalid or zero B{C{factor}}. 

''' 

if not isscalar(factor): 

raise _IsnotError(_scalar_, factor=factor) 

try: 

return self.times(_1_0 / factor) 

except (ValueError, ZeroDivisionError) as x: 

raise VectorError(factor=factor, txt=str(x)) 

 

def dot(self, other): 

'''Compute the dot (scalar) product of this and an other vector. 

 

@arg other: The other vector (L{Vector3d}). 

 

@return: Dot product (C{float}). 

 

@raise TypeError: Incompatible B{C{other}} C{type}. 

''' 

if other is self: 

d = self.length2 

else: 

self.others(other) 

d = fdot(self.xyz, *other.xyz) 

return d 

 

def equals(self, other, units=False): # PYCHOK no cover 

'''DEPRECATED, use method C{isequalTo}. 

''' 

return self.isequalTo(other, units=units) 

 

@property_RO 

def euclid(self): 

'''Approximate the length (norm, magnitude) of this vector. 

 

@see: Function L{euclid_} and properties C{length} and C{length2}. 

''' 

return euclid_(self.x, self.y, self.z) 

 

def iscolinearWith(self, point1, point2, eps=EPS): 

'''Check whether this and two other points are colinear. 

 

@arg point1: One point (L{Vector3d}, C{Vector3Tuple} 

or C{Vector4Tuple}). 

@arg point2: Another point (L{Vector3d}, C{Vector3Tuple} 

or C{Vector4Tuple}). 

@kwarg eps: Tolerance (C{scalar}), same units as C{x}, 

C{y}, and C{z}. 

 

@return: C{True} if colinear, C{False} otherwise. 

 

@raise TypeError: Invalid B{C{point1}} or B{C{point2}}. 

 

@see: Method L{nearestOn}. 

''' 

v = self if self.name else _otherV3d(this=self) 

return _iscolinearWith(v, point1, point2, eps=eps) 

 

def isequalTo(self, other, units=False, eps=EPS): 

'''Check if this and an other vector are equal or equivalent. 

 

@arg other: The other vector (L{Vector3d}). 

@kwarg units: Optionally, compare the normalized, unit 

version of both vectors. 

@kwarg eps: Tolerance (C{scalar}), same units as C{x}, 

C{y}, and C{z}. 

 

@return: C{True} if vectors are identical, C{False} otherwise. 

 

@raise TypeError: Incompatible B{C{other}} C{type}. 

 

@example: 

 

>>> v1 = Vector3d(52.205, 0.119) 

>>> v2 = Vector3d(52.205, 0.119) 

>>> e = v1.isequalTo(v2) # True 

''' 

self.others(other) 

 

if units: 

d = self.unit().minus(other.unit()) 

else: 

d = self.minus(other) 

return max(map(abs, d.xyz)) < eps 

 

@property_RO 

def length(self): 

'''Get the length (norm, magnitude) of this vector (C{float}). 

 

@see: Properties C{length2} and C{euclid}. 

''' 

if self._length is None: 

self._length = hypot_(self.x, self.y, self.z) 

return self._length 

 

@property_RO 

def length2(self): 

'''Get the length I{squared} of this vector (C{float}). 

 

@see: Property C{length}. 

''' 

if self._length2 is None: 

self._length2 = hypot2_(self.x, self.y, self.z) 

return self._length2 

 

def minus(self, other): 

'''Subtract an other vector from this vector. 

 

@arg other: The other vector (L{Vector3d}). 

 

@return: New vector difference (L{Vector3d}). 

 

@raise TypeError: Incompatible B{C{other}} C{type}. 

''' 

self.others(other) 

 

return self.classof(self.x - other.x, 

self.y - other.y, 

self.z - other.z) 

 

def nearestOn(self, other1, other2, within=True): 

'''Locate the point between two points closest to this point. 

 

@arg other1: Start point (L{Vector3d}). 

@arg other2: End point (L{Vector3d}). 

@kwarg within: If C{True} return the closest point between 

the given points, otherwise the closest 

point on the extended line through both 

points (C{bool}). 

 

@return: Closest point (L{Vector3d}). 

 

@raise TypeError: If B{C{other1}} or B{C{other2}} is not L{Vector3d}. 

 

@see: Method L{sphericalTrigonometry.LatLon.nearestOn3} and 

U{3-D Point-Line distance<https://MathWorld.Wolfram.com/ 

Point-LineDistance3-Dimensional.html>}. 

''' 

return _nearestOn(self, _V3d(other1) or _otherV3d(other1=other1), 

_V3d(other2) or _otherV3d(other2=other2), 

within=within) 

 

def negate(self): 

'''Return this vector in opposite direction. 

 

@return: New, opposite vector (L{Vector3d}). 

''' 

return self.classof(-self.x, -self.y, -self.z) 

 

@property_RO 

def _N_vector(self): 

'''(INTERNAL) Get the (C{nvectorBase._N_vector_}) 

''' 

from pygeodesy.nvectorBase import _N_vector_ 

return _N_vector_(*(self._xyz or self.xyz)) 

 

def others(self, *other, **name_other_up): 

'''Refined class comparison. 

 

@arg other: The other vector (L{Vector3d}). 

@kwarg name_other_up: Overriding C{name=other} and C{up=1} 

keyword arguments. 

 

@return: The B{C{other}} if compatible. 

 

@raise TypeError: Incompatible B{C{other}} C{type}. 

''' 

other, name, up = _xother3(self, other, **name_other_up) 

if not isinstance(other, Vector3d): 

_NamedBase.others(self, other, name=name, up=up + 1) 

return other 

 

def parse(self, str3d, sep=_COMMA_, name=NN): 

'''Parse an C{"x, y, z"} string to a similar L{Vector3d} instance. 

 

@arg str3d: X, y and z string (C{str}), see function L{parse3d}. 

@kwarg sep: Optional separator (C{str}). 

@kwarg name: Optional instance name (C{str}), overriding this name. 

 

@return: The similar instance (L{Vector3d}). 

 

@raise VectorError: Invalid B{C{str3d}}. 

''' 

return parse3d(str3d, sep=sep, Vector=self.classof, 

name=name or self.name) 

 

def plus(self, other): 

'''Add this vector and an other vector. 

 

@arg other: The other vector (L{Vector3d}). 

 

@return: Vectorial sum (L{Vector3d}). 

 

@raise TypeError: Incompatible B{C{other}} C{type}. 

''' 

self.others(other) 

 

return self.classof(self.x + other.x, 

self.y + other.y, 

self.z + other.z) 

 

sum = plus # alternate name 

 

def rotate(self, axis, theta): 

'''Rotate this vector around an axis by a specified angle. 

 

See U{Rotation matrix from axis and angle 

<https://WikiPedia.org/wiki/Rotation_matrix#Rotation_matrix_from_axis_and_angle>} 

and U{Quaternion-derived rotation matrix 

<https://WikiPedia.org/wiki/Quaternions_and_spatial_rotation#Quaternion-derived_rotation_matrix>}. 

 

@arg axis: The axis being rotated around (L{Vector3d}). 

@arg theta: The angle of rotation (C{radians}). 

 

@return: New, rotated vector (L{Vector3d}). 

 

@JSname: I{rotateAround}. 

''' 

a = self.others(axis=axis).unit() # axis being rotated around 

 

c = cos(theta) 

b = a.times(1 - c) 

s = a.times(sin(theta)) 

 

p = self.unit().xyz # point being rotated 

 

# multiply p by a quaternion-derived rotation matrix 

return self.classof(fdot(p, a.x * b.x + c, a.x * b.y - s.z, a.x * b.z + s.y), 

fdot(p, a.y * b.x + s.z, a.y * b.y + c, a.y * b.z - s.x), 

fdot(p, a.z * b.x - s.y, a.z * b.y + s.x, a.z * b.z + c)) 

 

def rotateAround(self, axis, theta): # PYCHOK no cover 

'''DEPRECATED, use method C{rotate}. 

''' 

return self.rotate(axis, theta) 

 

def times(self, factor): 

'''Multiply this vector by a scalar. 

 

@arg factor: Scale factor (C{scalar}). 

 

@return: New, scaled vector (L{Vector3d}). 

 

@raise TypeError: Non-scalar B{C{factor}}. 

''' 

if not isscalar(factor): 

raise _IsnotError(_scalar_, factor=factor) 

return self.classof(self.x * factor, 

self.y * factor, 

self.z * factor) 

 

def to2ab(self): # PYCHOK no cover 

'''DEPRECATED, use property C{Nvector.philam}. 

 

@return: A L{PhiLam2Tuple}C{(phi, lam)}. 

''' 

return n_xyz2philam(self.x, self.y, self.z) 

 

def to2ll(self): # PYCHOK no cover 

'''DEPRECATED, use property C{Nvector.latlon}. 

 

@return: A L{LatLon2Tuple}C{(lat, lon)}. 

''' 

return n_xyz2latlon(self.x, self.y, self.z) 

 

def to3xyz(self): # PYCHOK no cover 

'''DEPRECATED, use property C{xyz}. 

 

@return: A L{Vector3Tuple}C{(x, y, z)}. 

''' 

return self.xyz 

 

def toStr(self, prec=5, fmt=_PARENTH_, sep=_COMMA_SPACE_): # PYCHOK expected 

'''Return a string representation of this vector. 

 

@kwarg prec: Optional number of decimal places (C{int}). 

@kwarg fmt: Optional, enclosing format to use (C{str}). 

@kwarg sep: Optional separator between components (C{str}). 

 

@return: Vector as "(x, y, z)" (C{str}). 

''' 

t = sep.join(strs(self.xyz, prec=prec)) 

if fmt: 

t = fmt % (t,) 

return t 

 

def trilaterate3d2(self, radius, center2, radius2, center3, radius3, eps=EPS): 

'''Trilaterate this and two other spheres, each given as a (3d) center and radius. 

 

@arg radius: Radius of this sphere (same C{units} as this C{x}, C{y} 

and C{z}). 

@arg center2: Center of the 2nd sphere (L{Vector3d}, C{Vector3Tuple} 

or C{Vector4Tuple}). 

@arg radius2: Radius of this sphere (same C{units} as this C{x}, C{y} 

and C{z}). 

@arg center3: Center of the 3rd sphere (L{Vector3d}, C{Vector3Tuple} 

or C{Vector4Tuple}). 

@arg radius3: Radius of the 3rd sphere (same C{units} as this C{x}, C{y} 

and C{z}). 

@kwarg eps: Tolerance (C{scalar}), same units as C{x}, C{y}, and C{z}. 

 

@return: 2-Tuple with two trilaterated points, each an instance of this 

L{Vector3d} (sub-)class. Both points are the same instance if 

all three spheres intersect or abut in a single point. 

 

@raise ImportError: Package C{numpy} not found, not installed or 

older than version 1.15. 

 

@raise IntersectionError: No intersection, colinear or near concentric 

centers or trilateration failed some other way. 

 

@raise TypeError: Invalid B{C{center2}} or B{C{center3}}. 

 

@raise UnitError: Invalid B{C{radius}}, B{C{radius2}} or B{C{radius3}}. 

 

@note: Package U{numpy<https://pypi.org/project/numpy>} is required, 

version 1.15 or later. 

 

@see: Norrdine, A. U{I{An Algebraic Solution to the Multilateration 

Problem}<https://www.ResearchGate.net/publication/ 

275027725_An_Algebraic_Solution_to_the_Multilateration_Problem>} 

and U{I{implementation}<https://www.ResearchGate.net/publication/ 

288825016_Trilateration_Matlab_Code>}. 

''' 

try: 

return _trilaterate3d2(self if self.name else _otherV3d(center=self), 

Radius_(radius, low=eps), 

center2, radius2, center3, radius3, 

eps=eps, Vector=self.classof) 

except (AssertionError, FloatingPointError) as x: 

raise IntersectionError(center=self, radius=radius, 

center2=center2, radius2=radius2, 

center3=center3, radius3=radius3, 

txt=str(x)) 

 

def unit(self, ll=None): 

'''Normalize this vector to unit length. 

 

@kwarg ll: Optional, original location (C{LatLon}). 

 

@return: Normalized vector (L{Vector3d}). 

''' 

if self._united is None: 

n = self.length 

if n > EPS and abs(n - 1) > EPS: 

u = self._xnamed(self.dividedBy(n)) 

u._length = 1 

else: 

u = self.copy() 

u._fromll = ll or self._fromll 

self._united = u._united = u 

return self._united 

 

@property_RO 

def x(self): 

'''Get the X component (C{float}). 

''' 

return self._x 

 

@property_RO 

def xyz(self): 

'''Get the X, Y and Z components (L{Vector3Tuple}C{(x, y, z)}). 

''' 

if self._xyz is None: 

self._xyz = Vector3Tuple(self.x, self.y, self.z) 

return self._xnamed(self._xyz) 

 

@property_RO 

def y(self): 

'''Get the Y component (C{float}). 

''' 

return self._y 

 

@property_RO 

def z(self): 

'''Get the Z component (C{float}). 

''' 

return self._z 

 

 

def intersections2(center1, radius1, center2, radius2, sphere=True, 

Vector=None, **Vector_kwds): 

'''Compute the intersection of two spheres or circles, each defined 

by a center point and a radius. 

 

@arg center1: Center of the first sphere or circle (L{Vector3d}, 

C{Vector3Tuple} or C{Vector4Tuple}). 

@arg radius1: Radius of the first sphere or circle (same units as 

the B{C{center1}} coordinates). 

@arg center2: Center of the second sphere or circle (L{Vector3d}, 

C{Vector3Tuple} or C{Vector4Tuple}). 

@arg radius2: Radius of the second sphere or circle (same units as 

the B{C{center1}} and B{C{center2}} coordinates). 

@kwarg sphere: If C{True} compute the center and radius of the 

intersection of two spheres. If C{False}, ignore the 

C{z}-component and compute the intersection of two 

circles (C{bool}). 

@kwarg Vector: Class to return intersections (L{Vector3d} or 

C{Vector3Tuple}) or C{None} for L{Vector3d}. 

@kwarg Vector_kwds: Optional, additional B{C{Vector}} keyword arguments, 

ignored if C{B{Vector}=None}. 

 

@return: If B{C{sphere}} is C{True}, a 2-Tuple of the C{center} and 

C{radius} of the intersection of the spheres. The C{radius} 

is C{0.0} for abutting spheres. 

 

If B{C{sphere}} is C{False}, a 2-tuple of the intersection 

points of two circles. For abutting circles, both points 

are the same B{C{Vector}} instance. 

 

@raise IntersectionError: Concentric, invalid or non-intersecting 

spheres or circles. 

 

@raise UnitError: Invalid B{C{radius1}} or B{C{radius2}}. 

 

@see: U{Sphere-Sphere<https://MathWorld.Wolfram.com/Sphere- 

SphereIntersection.html>} and U{circle-circle 

<https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>} 

intersections. 

''' 

try: 

return _intersects2(center1, Radius_(radius1=radius1), 

center2, Radius_(radius2=radius2), 

sphere=sphere, Vector=Vector, **Vector_kwds) 

except (TypeError, ValueError) as x: 

raise IntersectionError(center1=center1, radius1=radius1, 

center2=center2, radius2=radius2, txt=str(x)) 

 

 

def _intersects2(center1, r1, center2, r2, sphere=True, too_d=None, # in .ellipsoidalBase._intersections2 

Vector=None, **Vector_kwds): 

# (INTERNAL) Intersect two spheres or circles, see L{intersections2} 

# above, separated to allow callers to embellish any exceptions 

 

def _V3(x, y, z): 

v = Vector3d(x, y, z) 

n = intersections2.__name__ 

return _V_n(v, n, Vector, Vector_kwds) 

 

def _xV3(c1, u, x, y): 

xy1 = x, y, _1_0 # transform to original space 

return _V3(fdot(xy1, u.x, -u.y, c1.x), 

fdot(xy1, u.y, u.x, c1.y), _0_0) 

 

c1 = _otherV3d(sphere=sphere, center1=center1) 

c2 = _otherV3d(sphere=sphere, center2=center2) 

 

if r1 < r2: # r1, r2 == R, r 

c1, c2 = c2, c1 

r1, r2 = r2, r1 

 

m = c2.minus(c1) 

d = m.length 

if d < max(r2 - r1, EPS): 

raise ValueError(_near_concentric_) 

 

o = fsum_(-d, r1, r2) # overlap == -(d - (r1 + r2)) 

# compute intersections with c1 at (0, 0) and c2 at (d, 0), like 

# <https://MathWorld.Wolfram.com/Circle-CircleIntersection.html> 

if o > EPS: # overlapping, r1, r2 == R, r 

x = _radical2(d, r1, r2).xline 

y = _1_0 - (x / r1)**2 

if y > EPS: 

y = r1 * sqrt(y) # y == a / 2 

elif y < 0: 

raise ValueError(_invalid_) 

else: # abutting 

y = _0_0 

elif o < 0: 

t = d if too_d is None else too_d 

raise ValueError(_too_distant_fmt_ % (t,)) 

else: # abutting 

x, y = r1, _0_0 

 

u = m.unit() 

if sphere: # sphere center and radius 

c = c1 if x < EPS else ( 

c2 if x > EPS1 else c1.plus(u.times(x))) 

t = _V3(c.x, c.y, c.z), Radius(y) 

 

elif y > 0: # intersecting circles 

t = _xV3(c1, u, x, y), _xV3(c1, u, x, -y) 

else: # abutting circles 

t = _xV3(c1, u, x, 0) 

t = t, t 

return t 

 

 

def iscolinearWith(point, point1, point2, eps=EPS): 

'''Check whether a point is colinear with two other points. 

 

@arg point: The point (L{Vector3d}, C{Vector3Tuple} or 

C{Vector4Tuple}). 

@arg point1: Another point (L{Vector3d}, C{Vector3Tuple} 

or C{Vector4Tuple}). 

@arg point2: Another point (L{Vector3d}, C{Vector3Tuple} 

or C{Vector4Tuple}). 

@kwarg eps: Tolerance (C{scalar}), same units as C{x}, 

C{y} and C{z}. 

 

@return: C{True} if colinear, C{False} otherwise. 

 

@raise TypeError: Invalid B{C{point}}, B{C{point1}} or 

B{C{point2}}. 

 

@see: Function L{nearestOn}. 

''' 

return _iscolinearWith(_V3d(point) or _otherV3d(point=point), 

point1, point2, eps=eps) 

 

 

def _iscolinearWith(v, p1, p2, eps=EPS): 

# (INTERNAL) Check colinear, see L{isColinear} above, 

# separated to allow callers to embellish any exceptions 

 

v1 = _V3d(p1) or _otherV3d(point1=p1) 

v2 = _V3d(p2) or _otherV3d(point2=p2) 

n = _nearestOn(v, v1, v2, within=False, eps=eps) 

return n is v1 or n.minus(v).length2 < eps 

 

 

def nearestOn(point, point1, point2, within=True, 

Vector=None, **Vector_kwds): 

'''Locate the point between two points closest to a reference. 

 

@arg point: Reference point (L{Vector3d}, C{Vector3Tuple} 

or C{Vector4Tuple}). 

@arg point1: Start point (L{Vector3d}, C{Vector3Tuple} or 

C{Vector4Tuple}). 

@arg point2: End point (L{Vector3d}, C{Vector3Tuple} or 

C{Vector4Tuple}). 

@kwarg within: If C{True} return the closest point between 

both given points, otherwise the closest 

point on the extended line through both 

points (C{bool}). 

@kwarg Vector: Class to return closest point (L{Vector3d} or 

C{Vector3Tuple}) or C{None} for L{Vector3d}. 

@kwarg Vector_kwds: Optional, additional B{C{Vector}} keyword 

arguments, ignored if C{B{Vector}=None}. 

 

@return: Closest point (L{Vector3d} or C{Vector}). 

 

@raise TypeError: Invalid B{C{point}}, B{C{point1}} or B{C{point2}}. 

 

@see: Methods L{sphericalTrigonometry.LatLon.nearestOn3} and 

L{sphericalTrigonometry.LatLon.nearestOn3} 

U{3-D Point-Line distance<https://MathWorld.Wolfram.com/ 

Point-LineDistance3-Dimensional.html>}. 

''' 

v = _nearestOn(_V3d(point) or _otherV3d(point =point), 

_V3d(point1) or _otherV3d(point1=point1), 

_V3d(point2) or _otherV3d(point2=point2), 

within=within) 

return _V_n(v, nearestOn.__name__, Vector, Vector_kwds) 

 

 

def _nearestOn(p0, p1, p2, within=True, eps=EPS): 

# (INTERNAL) Get closest point, see L{nearestOn} above, 

# separated to allow callers to embellish any exceptions 

 

p21 = p2.minus(p1) 

d2 = p21.length2 

if d2 < eps: # coincident 

p = p1 # ... or p2 

else: 

t = p0.minus(p1).dot(p21) / d2 

p = p1 if (within and t < eps) else ( 

p2 if (within and t > (_1_0 - eps)) else 

p1.plus(p21.times(t))) 

return p 

 

 

def _null_space2(numpy, A, eps): 

# (INTERNAL) Return the nullspace and rank of matrix A 

# @see: <https://SciPy-Cookbook.ReadTheDocs.io/items/RankNullspace.html>, 

# <https://NumPy.org/doc/stable/reference/generated/numpy.linalg.svd.html>, 

# <https://StackOverflow.com/questions/19820921>, 

# <https://StackOverflow.com/questions/2992947> and 

# <https://StackOverflow.com/questions/5889142> 

A = numpy.array(A) 

m = max(numpy.shape(A)) 

if m != 4: # for this usage 

raise _AssertionError(shape=m, txt=modulename(_null_space2, True)) 

# if needed, square A, pad with zeros 

A = numpy.resize(A, m * m).reshape(m, m) 

# try: # no numpy.linalg.null_space <https://docs.SciPy.org/doc/> 

# return scipy.linalg.null_space(A) # XXX no scipy.linalg? 

# except AttributeError: 

# pass 

_, s, v = numpy.linalg.svd(A) 

t = max(eps, eps * s[0]) # tol, s[0] is largest singular 

r = numpy.sum(s > t) # rank 

if r == 3: # get null_space 

n = numpy.transpose(v[r:]) 

s = numpy.shape(n) 

if s != (m, 1): # bad null_space shape 

raise _AssertionError(shape=s, txt=modulename(_null_space2, True)) 

e = float(numpy.max(numpy.abs(numpy.dot(A, n)))) 

if e > t: # residual not near-zero 

raise _AssertionError(eps=e, txt=modulename(_null_space2, True)) 

else: # coincident, colinear, concentric centers, ambiguous, etc. 

n = None 

# del A, s, vh # release numpy 

return n, r 

 

 

def _otherV3d(sphere=True, **name_v): 

# check B{C{name#}} vector instance, return Vector3d 

name, v = name_v.popitem() 

try: 

return Vector3d(v.x, v.y, v.z if sphere else _0_0, name=name) 

except AttributeError: # no _x_ or _y_ attr 

pass 

raise _xotherError(Vector3d(0, 0, 0), v, name=name, up=2) 

 

 

def parse3d(str3d, sep=_COMMA_, name=NN, Vector=Vector3d, **Vector_kwds): 

'''Parse an C{"x, y, z"} string. 

 

@arg str3d: X, y and z values (C{str}). 

@kwarg sep: Optional separator (C{str}). 

@kwarg name: Optional instance name (C{str}). 

@kwarg Vector: Optional class (L{Vector3d}). 

@kwarg Vector_kwds: Optional B{C{Vector}} keyword arguments, 

ignored if C{B{Vector}=None}. 

 

@return: New B{C{Vector}} or if B{C{Vector}} is C{None}, 

a L{Vector3Tuple}C{(x, y, z)}. 

 

@raise VectorError: Invalid B{C{str3d}}. 

''' 

try: 

v = [float(v.strip()) for v in str3d.split(sep)] 

n = len(v) 

if n != 3: 

raise _ValueError(len=n) 

except (TypeError, ValueError) as x: 

raise VectorError(str3d=str3d, txt=str(x)) 

return _V_n(Vector3Tuple(*v), name, Vector, Vector_kwds) 

 

 

def sumOf(vectors, Vector=Vector3d, **Vector_kwds): 

'''Compute the vectorial sum of several vectors. 

 

@arg vectors: Vectors to be added (L{Vector3d}[]). 

@kwarg Vector: Optional class for the vectorial sum (L{Vector3d}). 

@kwarg Vector_kwds: Optional B{C{Vector}} keyword arguments, 

ignored if C{B{Vector}=None}. 

 

@return: Vectorial sum as B{C{Vector}} or if B{C{Vector}} is 

C{None}, a L{Vector3Tuple}C{(x, y, z)}. 

 

@raise VectorError: No B{C{vectors}}. 

''' 

n, vectors = len2(vectors) 

if n < 1: 

raise VectorError(vectors=n, txt=_Missing) 

 

v = Vector3Tuple(fsum(v.x for v in vectors), 

fsum(v.y for v in vectors), 

fsum(v.z for v in vectors)) 

return _V_n(v, sumOf.__name__, Vector, Vector_kwds) 

 

 

def trilaterate3d2(center1, radius1, center2, radius2, center3, radius3, 

eps=EPS, Vector=None, **Vector_kwds): 

'''Trilaterate three spheres, each given as a (3d) center point and radius. 

 

@arg center1: Center of the 1st sphere (L{Vector3d}, C{Vector3Tuple} 

or C{Vector4Tuple}). 

@arg radius1: Radius of the 1st sphere (same C{units} as C{x}, C{y} 

and C{z}). 

@arg center2: Center of the 2nd sphere (L{Vector3d}, C{Vector3Tuple} 

or C{Vector4Tuple}). 

@arg radius2: Radius of this sphere (same C{units} as C{x}, C{y} 

and C{z}). 

@arg center3: Center of the 3rd sphere (L{Vector3d}, C{Vector3Tuple} 

or C{Vector4Tuple}). 

@arg radius3: Radius of the 3rd sphere (same C{units} as C{x}, C{y} 

and C{z}). 

@kwarg eps: Tolerance (C{scalar}), same units as C{x}, C{y}, and C{z}. 

@kwarg Vector: Class to return intersections (L{Vector3d} or 

C{Vector3Tuple}) or C{None} for L{Vector3d}. 

@kwarg Vector_kwds: Optional, additional B{C{Vector}} keyword arguments, 

ignored if C{B{Vector}=None}. 

 

@return: 2-Tuple with two trilaterated points, each a B{C{Vector}} 

instance. Both points are the same instance if all three 

spheres abut/intersect in a single point. 

 

@raise ImportError: Package C{numpy} not found, not installed or 

older than version 1.15. 

 

@raise IntersectionError: No intersection, colinear or concentric 

centers or trilateration failed some other way. 

 

@raise TypeError: Invalid B{C{center1}}, B{C{center2}} or B{C{center3}}. 

 

@raise UnitError: Invalid B{C{radius1}}, B{C{radius2}} or B{C{radius3}}. 

 

@note: Package U{numpy<https://pypi.org/project/numpy>} is required, 

version 1.15 or later. 

 

@see: Norrdine, A. U{I{An Algebraic Solution to the Multilateration 

Problem}<https://www.ResearchGate.net/publication/ 

275027725_An_Algebraic_Solution_to_the_Multilateration_Problem>} 

and U{I{implementation}<https://www.ResearchGate.net/publication/ 

288825016_Trilateration_Matlab_Code>}. 

''' 

try: 

return _trilaterate3d2(_otherV3d(center1=center1), 

Radius_(radius1=radius1, low=eps), 

center2, radius2, center3, radius3, 

eps=eps, Vector=Vector, **Vector_kwds) 

except (AssertionError, FloatingPointError) as x: 

raise IntersectionError(center1=center1, radius1=radius1, 

center2=center2, radius2=radius2, 

center3=center3, radius3=radius3, 

txt=str(x)) 

 

 

def _trilaterate3d2(c1, r1, c2, r2, c3, r3, eps=EPS, Vector=None, **Vector_kwds): # MCCABE 13 

# (INTERNAL) Intersect three spheres or circles, see L{trilaterate3d2} 

# above, separated to allow callers to embellish any exceptions 

 

def _0f3d(F): 

# map numpy 4-vector to floats and split 

F0, x, y, z = map(float, F) 

return F0, Vector3d(x, y, z) 

 

def _N3(t01, x, z): 

# compute x, y and z and return as Vector 

v = x.plus(z.times(t01)) 

n = trilaterate3d2.__name__ 

return _V_n(v, n, Vector, Vector_kwds) 

 

def _real_roots(numpy, *coeffs): 

# non-complex roots of a polynomial 

rs = numpy.polynomial.polynomial.polyroots(coeffs) 

return tuple(float(r) for r in rs if not numpy.iscomplex(r)) 

 

def _txt(c1, r1, c2, r2): 

# check for concentric or too distant spheres 

d = c1.minus(c2).length 

if d < abs(r1 - r2): 

t = _near_concentric_ 

elif d > (r1 + r2): 

t = _too_distant_fmt_ % (d,) 

else: 

return NN 

return joined_(c1.name, 'and', c2.name, t) 

 

np = Vector3d._numpy 

if np is None: # get numpy, once or ImportError 

Vector3d._numpy = np = _xnumpy(trilaterate3d2, 1, 10) # macOS' Python 2.7 numpy 1.8 OK 

 

c2 = _otherV3d(center2=c2) 

c3 = _otherV3d(center3=c3) 

 

A = [] # 3 x 4 

b = [] # 1 x 3 or 3 x 1 

for c, d in ((c1, r1), 

(c2, Radius_(radius2=r2, low=eps)), 

(c3, Radius_(radius3=r3, low=eps))): 

A.append((_1_0, -2 * c.x, -2 * c.y, -2 * c.z)) 

b.append(d**2 - c.length2) 

 

try: # <https://NumPy.org/doc/stable/reference/generated/numpy.seterr.html> 

e = np.seterr(all='raise') # throw FloatingPointError for numpy errors 

 

X = np.dot(np.linalg.pinv(A), b) # Moore-Penrose pseudo-inverse 

Z, _ = _null_space2(np, A, eps) 

if Z is None: 

t = () # coincident, colinear, concentric, etc. 

else: 

X0, x = _0f3d(X) 

Z0, z = _0f3d(Z) 

# quadratic polynomial coefficients, ordered (^0, ^1, ^2) 

t = _real_roots(np, x.length2 - X0, # fdot(X, -_1_0, *x.xyz) 

z.dot(x) * 2 - Z0, # fdot(Z, -_0_5, *x.xyz) * 2 

z.length2) # fdot(Z, _0_0, *z.xyz) 

 

finally: # restore numpy error handling 

np.seterr(**e) 

 

if not t: # coincident, colinear, too distant, no intersection, etc. 

raise FloatingPointError(_txt(c1, r1, c2, r2) or 

_txt(c1, r1, c3, r3) or 

_txt(c2, r2, c3, r3) or (_colinear_ if 

_iscolinearWith(c1, c2, c3, eps=eps) else 

_no_intersection_)) 

elif len(t) < 2: # one intersection 

t *= 2 

 

v = _N3(t[0], x, z) 

if abs(t[0] - t[1]) < eps: # abutting 

t = v, v 

else: # "lowest" intersection first (to avoid test failures) 

u = _N3(t[1], x, z) 

t = (u, v) if u < v else (v, u) 

return t 

 

 

def _V3d(v3d): 

# return v3d if it's a L{Vector3d} instance 

return v3d if isinstance(v3d, Vector3d) else None 

 

 

def _V_n(v, name, Vector, Vector_kwds): 

# return a named Vector instance 

if Vector is None: 

v = _xnamed(v, name) 

else: 

kwds = _xkwds(Vector_kwds, name=name) 

v = Vector(v.x, v.y, v.z, **kwds) 

return v 

 

 

__all__ += _ALL_DOCS(intersections2, 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.