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# -*- coding: utf-8 -*- 

 

u'''Extended 3-D vector class L{Vector3d} and functions. 

 

Function L{intersection3d3}, L{intersections2}, L{iscolinearWith}, 

L{nearestOn}, L{parse3d}, L{sumOf}, L{trilaterate2d2} and 

L{trilaterate3d2}. 

''' 

 

from pygeodesy.basics import isnear0, len2, map2, _xnumpy 

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

NumPyError, _TypeError, _ValueError, \ 

VectorError, _xError, _xkwds_popitem 

from pygeodesy.fmath import fdot, fsum, fsum_, hypot, hypot2_ 

from pygeodesy.formy import _radical2 

from pygeodesy.interns import EPS, EPS0, EPS1, MISSING, NN, _EPSqrt, \ 

_and_, _colinear_, _COMMA_, _COMMASPACE_, \ 

_datum_, _h_, _height_, _intersection_, \ 

_invalid_, _name_, _near_concentric_, \ 

_no_, _SPACE_, _too_, _xyz_, _y_, _z_, \ 

_0_0, _1_0, _2_0 

from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY 

from pygeodesy.named import modulename, _xnamed, _xotherError 

from pygeodesy.namedTuples import Intersection3Tuple, Vector2Tuple, \ 

Vector3Tuple # Vector4Tuple 

from pygeodesy.streprs import Fmt 

from pygeodesy.units import Float, Radius, Radius_ 

from pygeodesy.vector3dBase import Vector3dBase 

 

from math import sqrt 

 

__all__ = _ALL_LAZY.vector3d 

__version__ = '21.06.30' 

 

_raise_ = 'raise' 

 

 

class Vector3d(Vector3dBase): 

'''Extended 3-D vector. 

 

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 cartesian (= spherical n-vector) 

- great circle normal to vector 

- motion vector on earth's surface 

- etc. 

''' 

_numpy = None # module numpy iff imported by trilaterate3d2 below 

 

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 this point is colinear with B{C{point1}} 

and B{C{point2}}, 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 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, _otherV3d(other1=other1), 

_otherV3d(other2=other2), within=within) 

 

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

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

 

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

''' 

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

name=name or self.name) 

 

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

'''Trilaterate this and two other spheres, each given as a (3-D) center 

and a 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: Near-concentric, colinear, too distant or 

non-intersecting spheres or C{numpy} issue. 

 

@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, TypeError, ValueError) as x: 

raise _xError(x, center=self, radius=radius, 

center2=center2, radius2=radius2, 

center3=center3, radius3=radius3) 

 

 

def intersection3d3(start1, end1, start2, end2, eps=EPS, useZ=True, 

Vector=None, **Vector_kwds): 

'''Compute the intersection point of two lines, each defined by 

or through a start and end point. 

 

@arg start1: Start point of the first line (L{Vector3d}, 

C{Vector3Tuple} or C{Vector4Tuple}). 

@arg end1: End point of the first line (L{Vector3d}, 

C{Vector3Tuple} or C{Vector4Tuple}). 

@arg start2: Start point of the second line (L{Vector3d}, 

C{Vector3Tuple} or C{Vector4Tuple}). 

@arg end2: End point of the second line (L{Vector3d}, 

C{Vector3Tuple} or C{Vector4Tuple}). 

@kwarg eps: Tolerance for skew line distance and length (C{EPS}). 

@kwarg useZ: If C{True} use the Z component, if C{False} force 

C{z=0} (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: An L{Intersection3Tuple}C{(point, outside1, outside2)} with 

C{point} a C{Vector3d} or B{C{Vector}}. 

 

@raise IntersectionError: Invalid, skew, non-coplanar or otherwise 

non-intersecting lines. 

 

@see: U{Line-line intersection<https://MathWorld.Wolfram.com/Line-LineIntersection.html>} 

and U{line-line distance<https://MathWorld.Wolfram.com/Line-LineDistance.html>}, 

U{skew lines<https://MathWorld.Wolfram.com/SkewLines.html>} and U{point-line 

distance<https://MathWorld.Wolfram.com/Point-LineDistance3-Dimensional.html>}. 

''' 

try: 

v, o1, o2 = _intersect3d3(start1, end1, start2, end2, eps=eps, useZ=useZ) 

except (TypeError, ValueError) as x: 

raise _xError(x, start1=start1, end1=end1, start2=start2, end2=end2) 

v = _V_n(v, intersection3d3.__name__, Vector, Vector_kwds) 

return Intersection3Tuple(v, o1, o2) 

 

 

def _intersect3d3(start1, end1, start2, end2, eps=EPS, useZ=False): 

# (INTERNAL) Intersect two lines, see L{intersection} above, 

# separated to allow callers to embellish any exceptions 

 

def _outside(t, d2): # -1 before start#, +1 after end# 

return -1 if t < 0 else (+1 if t > d2 else 0) # XXX d2 + eps? 

 

s1 = _otherV3d(useZ=useZ, start1=start1) 

e1 = _otherV3d(useZ=useZ, end1=end1) 

s2 = _otherV3d(useZ=useZ, start2=start2) 

e2 = _otherV3d(useZ=useZ, end2=end2) 

 

a = e1.minus(s1) 

b = e2.minus(s2) 

c = s2.minus(s1) 

 

ab = a.cross(b) 

d = abs(c.dot(ab)) 

e = max(EPS0, eps or _0_0) 

if d > EPS0 and ab.length > e: 

d = d / ab.length 

if d > e: # argonic, skew lines distance 

raise IntersectionError(skew_d=d, txt=_no_(_intersection_)) 

 

# co-planar, non-skew lines 

ab2 = ab.length2 

if ab2 < e: # colinear, parallel or null line(s) 

x = a.length2 > b.length2 

if x: # make a shortest 

a, b = b, a 

s1, s2 = s2, s1 

e1, e2 = e2, e1 

if b.length2 < e: # both null 

if c.length < e: 

return s1, 0, 0 

elif e2.minus(e1).length < e: 

return e1, 0, 0 

elif a.length2 < e: # null (s1, e1), non-null (s2, e2) 

# like _nearestOn(s1, s2, e2, within=False, eps=e) 

t = s1.minus(s2).dot(b) 

v = s2.plus(b.times(t / b.length2)) 

if s1.minus(v).length < e: 

o = _outside(t, b.length2) 

return (v, o, 0) if x else (v, 0, (o * 2)) 

raise IntersectionError(length2=ab2, txt=_no_(_intersection_)) 

 

cb = c.cross(b) 

t = cb.dot(ab) 

o1 = _outside(t, ab2) 

v = s1.plus(a.times(t / ab2)) 

o2 = _outside(v.minus(s2).dot(b), b.length2) * 2 

return v, o1, o2 

 

 

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 _xError(x, center1=center1, radius1=radius1, 

center2=center2, radius2=radius2) 

 

 

def _intersects2(center1, r1, center2, r2, sphere=True, too_d=None, # in .ellipsoidalBaseDI._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(useZ=sphere, center1=center1) 

c2 = _otherV3d(useZ=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 IntersectionError(_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 IntersectionError(_invalid_) 

else: # abutting 

y = _0_0 

elif o < 0: 

t = d if too_d is None else too_d 

raise IntersectionError(_too_(Fmt.distant(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: First point (L{Vector3d}, C{Vector3Tuple} 

or C{Vector4Tuple}). 

@arg point2: Second 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 B{C{point}} is colinear B{C{point1}} 

and B{C{point2}}, C{False} otherwise. 

 

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

B{C{point2}}. 

 

@see: Function L{nearestOn}. 

''' 

return _iscolinearWith(_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 = _otherV3d(point1=p1) 

v2 = _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(_otherV3d(point =point), 

_otherV3d(point1=point1), 

_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(useZ=True, **name_v): 

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

if not name_v: 

raise _AssertionError(name_v=MISSING) 

 

name, v = _xkwds_popitem(name_v) 

if useZ and isinstance(v, Vector3dBase): 

return v 

 

try: 

return Vector3d(v.x, v.y, v.z if useZ 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 trilaterate2d2(x1, y1, radius1, x2, y2, radius2, x3, y3, radius3, eps=None): 

'''Trilaterate three circles, each given as a (2d) center and a radius. 

 

@arg x1: Center C{x} coordinate of the 1st circle (C{scalar}). 

@arg y1: Center C{y} coordinate of the 1st circle (C{scalar}). 

@arg radius1: Radius of the 1st circle (C{scalar}). 

@arg x2: Center C{x} coordinate of the 2nd circle (C{scalar}). 

@arg y2: Center C{y} coordinate of the 2nd circle (C{scalar}). 

@arg radius2: Radius of the 2nd circle (C{scalar}). 

@arg x3: Center C{x} coordinate of the 3rd circle (C{scalar}). 

@arg y3: Center C{y} coordinate of the 3rd circle (C{scalar}). 

@arg radius3: Radius of the 3rd circle (C{scalar}). 

@kwarg eps: Check the trilaterated point I{delta} on all 3 

circles (C{scalar}) or C{None}. 

 

@return: Trilaterated point as L{Vector2Tuple}C{(x, y)}. 

 

@raise IntersectionError: No intersection, colinear or near-concentric 

centers, trilateration failed some other way 

or the trilaterated point is off one circle 

by more than B{C{eps}}. 

 

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

 

@see: U{Issue #49<https://GitHub.com/mrJean1/PyGeodesy/issues/49>}, 

U{Find X location using 3 known (X,Y) location using trilateration 

<https://math.StackExchange.com/questions/884807>} and 

L{trilaterate3d2} 

''' 

def _abct4(x1, y1, r1, x2, y2, r2): 

a = x2 - x1 

b = y2 - y1 

t = _tri_r2h(r1, r2, hypot(a, b)) 

c = _0_0 if t else (hypot2_(r1, x2, y2) - hypot2_(r2, x1, y1)) 

return a, b, c, t 

 

def _astr(**kwds): # kwds as (name=value, ...) strings 

return Fmt.PAREN(_COMMASPACE_(*(Fmt.EQUAL(*t) for t in kwds.items()))) 

 

r1 = Radius_(radius1=radius1) 

r2 = Radius_(radius2=radius2) 

r3 = Radius_(radius3=radius3) 

 

a, b, c, t = _abct4(x1, y1, r1, x2, y2, r2) 

if t: 

raise IntersectionError(_and(_astr(x1=x1, y1=y1, radius1=r1), 

_astr(x2=x2, y2=y2, radius2=r2)), txt=t) 

 

d, e, f, t = _abct4(x2, y2, r2, x3, y3, r3) 

if t: 

raise IntersectionError(_and(_astr(x2=x2, y2=y2, radius2=r2), 

_astr(x3=x3, y3=y3, radius3=r3)), txt=t) 

 

_, _, _, t = _abct4(x3, y3, r3, x1, y1, r1) 

if t: 

raise IntersectionError(_and(_astr(x3=x3, y3=y3, radius3=r3), 

_astr(x1=x1, y1=y1, radius1=r1)), txt=t) 

 

q = _2_0 * (e * a - b * d) 

if isnear0(q): 

t = _no_(_intersection_) 

raise IntersectionError(_and(_astr(x1=x1, y1=y1, radius1=r1), 

_astr(x2=x2, y2=y2, radius2=r2), 

_astr(x3=x3, y3=y3, radius3=r3)), txt=t) 

t = Vector2Tuple((c * e - b * f) / q, 

(a * f - c * d) / q, name=trilaterate2d2.__name__) 

 

if eps and eps > _0_0: 

for x, y, r in ((x1, y1, r1), (x2, y2, r2), (x3, y3, r3)): 

d = hypot(x - t.x, y - t.y) 

e = abs(d - r) 

if e > eps: 

t = _and(Float(delta=e).toRepr(), r.toRepr(), 

Float(distance=d).toRepr()) 

raise IntersectionError(t, txt=Fmt.exceeds_eps(eps)) 

 

return t 

 

 

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

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

'''Trilaterate three spheres, each given as a (3-D) center and a 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: Near-concentric, colinear, too distant or 

non-intersecting spheres. 

 

@raise NumPyError: Some C{numpy} issue. 

 

@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>} and L{trilaterate2d2}. 

''' 

try: 

return _trilaterate3d2(_otherV3d(center1=center1), 

Radius_(radius1=radius1, low=eps), 

center2, radius2, center3, radius3, 

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

except (AssertionError, NumPyError, TypeError, ValueError) as x: 

raise _xError(x, center1=center1, radius1=radius1, 

center2=center2, radius2=radius2, 

center3=center3, radius3=radius3) 

 

 

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

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

# above, separated to allow callers to embellish any exceptions, like 

# C{FloatingPointError}s from C{numpy} 

 

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)) 

 

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) 

R = [r1, Radius_(radius2=r2, low=eps), 

Radius_(radius3=r3, low=eps)] 

 

# get null_space Z and pseudo-inverse A, once 

t = -_2_0 

A = [(_1_0, t * c.x, t * c.y, t * c.z) for c in (c1, c2, c3)] # 3 x 4 

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

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

Z, _ = _null_space2(np, A, eps) 

A = np.linalg.pinv(A) # Moore-Penrose pseudo-inverse 

except Exception as X: # mostly FloatingPointError? 

raise NumPyError(X.__class__.__name__, txt=str(X)) 

finally: # restore numpy error handling 

np.seterr(**e) 

if Z is None: # coincident, colinear, concentric, etc. 

raise _trilaterror(c1, r1, c2, r2, c3, r3, eps) 

 

B = [c.length2 for c in (c1, c2, c3)] 

# perturbe radii to handle corner cases like this 

# <https://GitHub.com/mrJean1/PyGeodesy/issues/49> 

if eps > _0_0: 

p1 = max(eps, EPS) 

p2 = max(eps, _EPSqrt) 

if p2 > p1: 

ps = _0_0, p1, -p1, p2, -p2 

else: 

ps = _0_0, p1, -p1 

else: 

ps = _0_0, 

for p in ps: 

b = [((r + p)**2 - b) for r, b in zip(R, B)] # 1 x 3 or 3 x 1 

# t = () 

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

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

X = np.dot(A, b) 

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_0 - Z0, # fdot(Z, -_0_5, *x.xyz) * 2 

z.length2) # fdot(Z, _0_0, *z.xyz) 

except Exception as X: # mostly FloatingPointError? 

raise NumPyError(X.__class__.__name__, txt=str(X)) 

finally: # restore numpy error handling 

np.seterr(**e) 

if t: 

break 

else: # coincident, concentric, colinear, too distant, no intersection, etc. 

raise _trilaterror(c1, r1, c2, r2, c3, r3, eps) 

 

v = _N3(t[0], x, z) 

if len(t) < 2: # one intersection 

t = v, v 

elif 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 _trilaterror(c1, r1, c2, r2, c3, r3, eps): 

# return FloatingPointError with the cause of the error 

 

def _txt(c1, r1, c2, r2): 

t = _tri_r2h(r1, r2, c1.minus(c2).length) 

return _SPACE_(c1.name, _and_, c2.name, t) if t else t 

 

t = _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_)) 

return IntersectionError(t) 

 

 

def _tri_r2h(r1, r2, h): 

# check for near-concentric or too distant spheres/circles 

return _too_(Fmt.distant(h)) if h > (r1 + r2) else ( 

_near_concentric_ if h < abs(r1 - r2) else NN) 

 

 

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

# return a named Vector instance 

if Vector is not None: 

v = Vector(v.x, v.y, v.z, **Vector_kwds) 

return _xnamed(v, name) 

 

 

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

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

''' 

try: 

t = xyz.x, xyz.y, xyz.z 

n = getattr(xyz, _name_, NN) 

except AttributeError: 

t = xyz, y, z 

n = NN 

try: 

x, y, z = map2(float, t) 

except (TypeError, ValueError) as x: 

d = dict(zip((_xyz_, _y_, _z_), t)) 

raise Error(txt=str(x), **d) 

 

return x, y, z, n 

 

 

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 

 

 

__all__ += _ALL_DOCS(intersections2, sumOf, Vector3dBase) 

 

# **) MIT License 

# 

# Copyright (C) 2016-2021 -- 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.