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# -*- coding: utf-8 -*- 

 

u'''2- or 3-D vectorial functions L{circin6}, L{circum3}, L{circum4_}, 

L{iscolinearWith}, L{meeus2}, L{nearestOn}, L{radii11} and L{soddy4}. 

''' 

 

from pygeodesy.basics import len2, map1, map2, _xnumpy 

from pygeodesy.constants import EPS, EPS0, EPS02, EPS4, INF, INT0, \ 

_EPS4e8, isnear0, _0_0, _0_5, _1_0, _1_0_1T, \ 

_N_1_0, _2_0, _N_2_0, _4_0 

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

PointsError, TriangleError, _xError, _xkwds 

from pygeodesy.fmath import fdot, hypot, hypot2_ 

from pygeodesy.fsums import fsum_, fsum1_ 

from pygeodesy.interns import NN, _a_, _and_, _b_, _c_, _center_, _coincident_, \ 

_colinear_, _concentric_, _COMMASPACE_, _few_, \ 

_intersection_, _invalid_, _near_, _no_, _radius_, \ 

_rIn_, _s_, _SPACE_, _too_ 

# from pygeodesy.lazily import _ALL_LAZY # from .named 

from pygeodesy.named import _ALL_LAZY, Fmt, _NamedTuple, _Pass 

from pygeodesy.namedTuples import LatLon3Tuple, Vector2Tuple 

# from pygeodesy.streprs import Fmt # from .named 

from pygeodesy.units import Float, Int, Meter, Radius, Radius_ 

from pygeodesy.vector3d import iscolinearWith, nearestOn, _nearestOn2, _nVc, _otherV3d, \ 

trilaterate2d2, trilaterate3d2, Vector3d # PYCHOK unused 

 

from contextlib import contextmanager 

from math import sqrt 

 

__all__ = _ALL_LAZY.vector2d 

__version__ = '22.09.15' 

 

_cA_ = 'cA' 

_cB_ = 'cB' 

_cC_ = 'cC' 

_deltas_ = 'deltas' 

_numpy_1_10 = None # PYCHOK import numpy once 

_of_ = 'of' 

_outer_ = 'outer' 

_raise_ = 'raise' # PYCHOK used! 

_rank_ = 'rank' 

_residuals_ = 'residuals' 

_Type_ = 'Type' 

_with_ = 'with' 

 

 

class Circin6Tuple(_NamedTuple): 

'''6-Tuple C{(radius, center, deltas, cA, cB, cC)} with the C{radius}, the 

trilaterated C{center} and contact points of the I{inscribed} aka I{In- 

circle} of a triangle. The C{center} is I{un}ambiguous if C{deltas} is 

C{None}, otherwise C{center} is the mean and C{deltas} the differences of 

the L{pygeodesy.trilaterate3d2} results. Contact points C{cA}, C{cB} and 

C{cC} are the points of tangency, aka the corners of the U{Contact Triangle 

<https://MathWorld.Wolfram.com/ContactTriangle.html>}. 

''' 

_Names_ = (_radius_, _center_, _deltas_, _cA_, _cB_, _cC_) 

_Units_ = ( Radius, _Pass, _Pass, _Pass, _Pass, _Pass) 

 

 

class Circum3Tuple(_NamedTuple): # in .latlonBase 

'''3-Tuple C{(radius, center, deltas)} with the C{circumradius} and trilaterated 

C{circumcenter} of the C{circumcircle} through 3 points (aka {Meeus}' Type II 

circle) or the C{radius} and C{center} of the smallest I{Meeus}' Type I circle. 

The C{center} is I{un}ambiguous if C{deltas} is C{None}, otherwise C{center} 

is the mean and C{deltas} the differences of the L{pygeodesy.trilaterate3d2} 

results. 

''' 

_Names_ = (_radius_, _center_, _deltas_) 

_Units_ = ( Radius, _Pass, _Pass) 

 

 

class Circum4Tuple(_NamedTuple): 

'''4-Tuple C{(radius, center, rank, residuals)} with C{radius} and C{center} 

of a sphere I{least-squares} fitted through given points and the C{rank} 

and C{residuals} -if any- from U{numpy.linalg.lstsq 

<https://NumPy.org/doc/stable/reference/generated/numpy.linalg.lstsq.html>}. 

''' 

_Names_ = (_radius_, _center_, _rank_, _residuals_) 

_Units_ = ( Radius, _Pass, Int, _Pass) 

 

 

class Meeus2Tuple(_NamedTuple): 

'''2-Tuple C{(radius, Type)} with C{radius} and I{Meeus}' C{Type} of the smallest 

circle I{containing} 3 points. C{Type} is C{None} for a I{Meeus}' Type II 

C{circumcircle} passing through all 3 points. Otherwise C{Type} is the center 

of a I{Meeus}' Type I circle with 2 points on (a diameter of) and 1 point 

inside the circle. 

''' 

_Names_ = (_radius_, _Type_) 

_Units_ = ( Radius, _Pass) 

 

 

class Radii11Tuple(_NamedTuple): 

'''11-Tuple C{(rA, rB, rC, cR, rIn, riS, roS, a, b, c, s)} with the C{Tangent} 

circle radii C{rA}, C{rB} and C{rC}, the C{circumradius} C{cR}, the C{Incircle} 

radius C{rIn} aka C{inradius}, the inner and outer I{Soddy} circle radii C{riS} 

and C{roS} and the sides C{a}, C{b} and C{c} and semi-perimeter C{s} of a 

triangle, all in C{meter} conventionally. 

 

@note: C{Circumradius} C{cR} and outer I{Soddy} radius C{roS} may be C{INF}. 

''' 

_Names_ = ('rA', 'rB', 'rC', 'cR', _rIn_, 'riS', 'roS', _a_, _b_, _c_, _s_) 

_Units_ = ( Meter,) * len(_Names_) 

 

 

class Soddy4Tuple(_NamedTuple): 

'''4-Tuple C{(radius, center, deltas, outer)} with C{radius} and trilaterated 

C{center} of the I{inner} I{Soddy} circle and the radius of the C{outer} 

I{Soddy} circle. The C{center} is I{un}ambiguous if C{deltas} is C{None}, 

otherwise C{center} is the mean and C{deltas} the differences of the 

L{pygeodesy.trilaterate3d2} results. 

 

@note: The outer I{Soddy} radius C{outer} may be C{INF}. 

''' 

_Names_ = (_radius_, _center_, _deltas_, _outer_) 

_Units_ = ( Radius, _Pass, _Pass, Radius) 

 

 

def circin6(point1, point2, point3, eps=EPS4, useZ=True): 

'''Return the radius and center of the I{inscribed} aka I{In- circle} 

of a (2- or 3-D) triangle. 

 

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

C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}). 

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

C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}). 

@arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, 

C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}). 

@kwarg eps: Tolerance for function L{pygeodesy.trilaterate3d2} if 

C{B{useZ} is True} otherwise L{pygeodesy.trilaterate2d2}. 

@kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}). 

 

@return: L{Circin6Tuple}C{(radius, center, deltas, cA, cB, cC)}. The 

C{center} and contact points C{cA}, C{cB} and C{cC}, each an 

instance of B{C{point1}}'s (sub-)class, are co-planar with 

the three given points. 

 

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

than version 1.10 and C{B{useZ} is True}. 

 

@raise IntersectionError: Near-coincident or -colinear points or 

a trilateration or C{numpy} issue. 

 

@raise TypeError: Invalid B{C{point1}}, B{C{point2}} or B{C{point3}}. 

 

@see: Functions L{radii11} and L{circum3}, U{Incircle 

<https://MathWorld.Wolfram.com/Incircle.html>} and U{Contact Triangle 

<https://MathWorld.Wolfram.com/ContactTriangle.html>}. 

''' 

try: 

return _circin6(point1, point2, point3, eps=eps, useZ=useZ) 

except (AssertionError, TypeError, ValueError) as x: 

raise _xError(x, point1=point1, point2=point2, point3=point3) 

 

 

def _circin6(point1, point2, point3, eps=EPS4, useZ=True, dLL3=False, **Vector_kwds): 

# (INTERNAL) Radius, center, deltas, 3 contact points 

 

def _fraction(r, a): 

return (r / a) if a > EPS0 else _0_5 

 

def _contact2(a, p2, r2, p3, r3, V, V_kwds): 

c = p2.intermediateTo(p3, _fraction(r2, a)) if r2 > r3 else \ 

p3.intermediateTo(p2, _fraction(r3, a)) 

C = V(c.x, c.y, c.z, **V_kwds) 

return c, C 

 

t, p1, p2, p3 = _radii11ABC(point1, point2, point3, useZ=useZ) 

V, r1, r2, r3 = point1.classof, t.rA, t.rB, t.rC 

 

c1, cA = _contact2(t.a, p2, r2, p3, r3, V, _xkwds(Vector_kwds, name=_cA_)) 

c2, cB = _contact2(t.b, p3, r3, p1, r1, V, _xkwds(Vector_kwds, name=_cB_)) 

c3, cC = _contact2(t.c, p1, r1, p2, r2, V, _xkwds(Vector_kwds, name=_cC_)) 

 

r = t.rIn 

c, d = _tricenter3d2(c1, r, c2, r, c3, r, eps=eps, useZ=useZ, dLL3=dLL3, 

**_xkwds(Vector_kwds, Vector=V, 

name=circin6.__name__)) 

return Circin6Tuple(r, c, d, cA, cB, cC) 

 

 

def circum3(point1, point2, point3, circum=True, eps=EPS4, useZ=True): 

'''Return the radius and center of the smallest circle I{through} or I{containing} 

three (2- or 3-D) points. 

 

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

C{Vector4Tuple}). 

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

C{Vector4Tuple}). 

@arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or 

C{Vector4Tuple}). 

@kwarg circum: If C{True} return the C{circumradius} and C{circumcenter} 

always, ignoring the I{Meeus}' Type I case (C{bool}). 

@kwarg eps: Tolerance for function L{pygeodesy.trilaterate3d2} if C{B{useZ} 

is True} otherwise L{pygeodesy.trilaterate2d2}. 

@kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}). 

 

@return: A L{Circum3Tuple}C{(radius, center, deltas)}. The C{center}, an 

instance of B{C{point1}}'s (sub-)class, is co-planar with the three 

given points. 

 

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

than version 1.10 and C{B{useZ} is True}. 

 

@raise IntersectionError: Near-coincident or -colinear points or 

a trilateration or C{numpy} issue. 

 

@raise TypeError: Invalid B{C{point1}}, B{C{point2}} or B{C{point3}}. 

 

@see: U{Jean Meeus, "Astronomical Algorithms", 2nd Ed. 1998, page 127ff 

<http://www.Agopax.IT/Libri_astronomia/pdf/Astronomical%20Algorithms.pdf>}, 

U{circumradius<https://MathWorld.Wolfram.com/Circumradius.html>}, 

U{circumcircle<https://MathWorld.Wolfram.com/Circumcircle.html>} and 

functions L{pygeodesy.circum4_} and L{pygeodesy.meeus2}. 

''' 

try: 

p1 = _otherV3d(useZ=useZ, point1=point1) 

return _circum3(p1, point2, point3, circum=circum, eps=eps, useZ=useZ, 

clas=point1.classof) 

except (AssertionError, TypeError, ValueError) as x: 

raise _xError(x, point1=point1, point2=point2, point3=point3, circum=circum) 

 

 

def _circum3(p1, point2, point3, circum=True, eps=EPS4, useZ=True, dLL3=False, 

clas=Vector3d, **clas_kwds): # in .latlonBase 

# (INTERNAL) Radius, center, deltas 

r, d, p2, p3 = _meeus4(p1, point2, point3, circum=circum, useZ=useZ, 

clas=clas, **clas_kwds) 

if d is None: # Meeus' Type II or circum=True 

kwds = _xkwds(clas_kwds, eps=eps, Vector=clas, name=circum3.__name__) 

c, d = _tricenter3d2(p1, r, p2, r, p3, r, useZ=useZ, dLL3=dLL3, **kwds) 

else: # Meeus' Type I 

c, d = d, None 

return Circum3Tuple(r, c, d) 

 

 

def circum4_(*points, **Vector_and_kwds): 

'''Best-fit a sphere through three or more (3-D) points. 

 

@arg points: The points (each a C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, 

or C{Vector4Tuple}). 

@kwarg Vector_and_kwds: Optional class C{B{Vector}=None} to return the center 

and optional, additional B{C{Vector}} keyword arguments, 

otherwise the first B{C{points}}' (sub-)class. 

 

@return: L{Circum4Tuple}C{(radius, center, rank, residuals)} with C{center} an 

instance of C{B{points}[0]}' (sub-)class or B{C{Vector}} if specified. 

 

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

version 1.10. 

 

@raise NumPyError: Some C{numpy} issue. 

 

@raise PointsError: Too few B{C{points}}. 

 

@raise TypeError: One of the B{C{points}} is invalid. 

 

@see: U{Charles F. Jekel, "Least Squares Sphere Fit", Sep 13, 2015 

<https://Jekel.me/2015/Least-Squares-Sphere-Fit/>} and U{Appendix A 

<https://hdl.handle.net/10019.1/98627>}, U{numpy.linalg.lstsq 

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

U{Eberly 6<https://www.sci.Utah.EDU/~balling/FEtools/doc_files/LeastSquaresFitting.pdf>} 

and functions L{pygeodesy.circum3} and L{pygeodesy.meeus2}. 

''' 

n, ps = len2(points) 

if n < 3: 

raise PointsError(points=n, txt=_too_(_few_)) 

 

A, b = [], [] 

for i, p in enumerate(ps): 

v = _otherV3d(useZ=True, i=i, points=p) 

A.append(v.times(_2_0).xyz + _1_0_1T) 

b.append(v.length2) 

 

with _numpy(n, circum4_) as np: 

A = np.array(A).reshape((n, 4)) 

b = np.array(b).reshape((n, 1)) 

C, R, rk, _ = np.linalg.lstsq(A, b, rcond=None) # to silence warning 

C = map1(float, *C) 

R = map1(float, *R) # empty if rk < 4 or n <= 4 

 

n = circum4_.__name__ 

c = Vector3d(*C[:3], name=n) 

r = Radius(sqrt(fsum_(C[3], *c.x2y2z2)), name=n) 

 

c = _nVc(c, **_xkwds(Vector_and_kwds, clas=ps[0].classof, name=n)) 

return Circum4Tuple(r, c, rk, R) 

 

 

def _iscolinearWith(p, point1, point2, eps=EPS, useZ=True): 

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

# separated to allow callers to embellish any exceptions 

p1 = _otherV3d(useZ=useZ, point1=point1) 

p2 = _otherV3d(useZ=useZ, point2=point2) 

n, _ = _nearestOn2(p, p1, p2, within=False, eps=eps) 

return n is p1 or n.minus(p).length2 < eps 

 

 

def meeus2(point1, point2, point3, circum=False, useZ=True): 

'''Return the radius and I{Meeus}' Type of the smallest circle I{through} 

or I{containing} three (2- or 3-D) points. 

 

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

C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}). 

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

C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}). 

@arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, 

C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}). 

@kwarg circum: If C{True} return the non-zero C{circumradius} always, 

ignoring the I{Meeus}' Type I case (C{bool}). 

@kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}). 

 

@return: L{Meeus2Tuple}C{(radius, Type)}. 

 

@raise IntersectionError: Near-coincident or -colinear points, iff C{B{circum}=True}. 

 

@raise TypeError: Invalid B{C{point1}}, B{C{point2}} or B{C{point3}}. 

 

@see: U{Jean Meeus, "Astronomical Algorithms", 2nd Ed. 1998, page 127ff 

<http://www.Agopax.IT/Libri_astronomia/pdf/Astronomical%20Algorithms.pdf>}, 

U{circumradius<https://MathWorld.Wolfram.com/Circumradius.html>}, 

U{circumcircle<https://MathWorld.Wolfram.com/Circumcircle.html>} and 

functions L{pygeodesy.circum3} and L{pygeodesy.circum4_}. 

''' 

try: 

A = _otherV3d(useZ=useZ, point1=point1) 

r, t, _, _ = _meeus4(A, point2, point3, circum=circum, useZ=useZ, 

clas=point1.classof) 

except (TypeError, ValueError) as x: 

raise _xError(x, point1=point1, point2=point2, point3=point3, circum=circum) 

return Meeus2Tuple(r, t) 

 

 

def _meeus4(A, point2, point3, circum=False, useZ=True, clas=None, **clas_kwds): 

# (INTERNAL) Radius and Meeus' Type 

B = p2 = _otherV3d(useZ=useZ, point2=point2) 

C = p3 = _otherV3d(useZ=useZ, point3=point3) 

 

a = B.minus(C).length2 

b = C.minus(A).length2 

c = A.minus(B).length2 

if a < b: 

a, b, A, B = b, a, B, A 

if a < c: 

a, c, A, C = c, a, C, A 

 

if a > EPS02 and (circum or a < (b + c)): # circumradius 

b = sqrt(b / a) 

c = sqrt(c / a) 

r = fsum1_(_1_0, b, c) * fsum1_(_1_0, b, -c) * fsum1_(_1_0, -b, c) * fsum1_(_N_1_0, b, c) 

if r < EPS02: 

raise IntersectionError(_coincident_ if b < EPS0 or c < EPS0 else ( 

_colinear_ if _iscolinearWith(A, B, C) else _invalid_)) 

r = sqrt(a / r) * b * c 

t = None # Meeus' Type II 

else: # obtuse or right angle 

r = INT0 if a < EPS02 else (sqrt(a) * _0_5) 

t = B.plus(C).times(_0_5) # Meeus' Type I 

if clas is not None: 

t = clas(t.x, t.y, t.z, **_xkwds(clas_kwds, name=meeus2.__name__)) 

return r, t, p2, p3 

 

 

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=_null_space2.__name__) 

# 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=_null_space2.__name__) 

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

if e > t: # residual not near-zero 

raise _AssertionError(res=e, tol=t, txt=_null_space2.__name__) 

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

n = None 

# del A, s, vh # release numpy 

return n, r 

 

 

@contextmanager # <https://www.python.org/dev/peps/pep-0343/> Examples 

def _numpy(arg, where): 

# (INTERNAL) Yield numpy with any errors raised as NumPyError 

global _numpy_1_10 

np = _numpy_1_10 

if np is None: 

_numpy_1_10 = np = _xnumpy(where, 1, 10) 

 

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

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

yield np 

except Exception as x: # mostly FloatingPointError? 

raise NumPyError(x.__class__.__name__, arg, txt=str(x)) # _xError2? 

finally: # restore numpy error handling 

np.seterr(**e) 

 

 

def radii11(point1, point2, point3, useZ=True): 

'''Return the radii of the C{In-}, I{Soddy} and C{Tangent} circles of a 

(2- or 3-D) triangle. 

 

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

C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}). 

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

C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}). 

@arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, 

C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}). 

@kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}). 

 

@return: L{Radii11Tuple}C{(rA, rB, rC, cR, rIn, riS, roS, a, b, c, s)}. 

 

@raise TriangleError: Near-coincident or -colinear points. 

 

@raise TypeError: Invalid B{C{point1}}, B{C{point2}} or B{C{point3}}. 

 

@see: U{Circumradius<https://MathWorld.Wolfram.com/Circumradius.html>}, 

U{Incircle<https://MathWorld.Wolfram.com/Incircle.html>}, U{Soddy 

Circles<https://MathWorld.Wolfram.com/SoddyCircles.html>} and 

U{Tangent Circles<https://MathWorld.Wolfram.com/TangentCircles.html>}. 

''' 

try: 

return _radii11ABC(point1, point2, point3, useZ=useZ)[0] 

except (TypeError, ValueError) as x: 

raise _xError(x, point1=point1, point2=point2, point3=point3) 

 

 

def _radii11ABC(point1, point2, point3, useZ=True): 

# (INTERNAL) Tangent, Circum, Incircle, Soddy radii, sides and semi-perimeter 

A = _otherV3d(useZ=useZ, point1=point1, NN_OK=False) 

B = _otherV3d(useZ=useZ, point2=point2, NN_OK=False) 

C = _otherV3d(useZ=useZ, point3=point3, NN_OK=False) 

 

a = B.minus(C).length 

b = C.minus(A).length 

c = A.minus(B).length 

 

s = fsum1_(a, b, c) * _0_5 # semi-perimeter 

if s > EPS0: 

rs = (s - a), (s - b), (s - c) 

r3, r2, r1 = sorted(rs) # r3 <= r2 <= r1 

if r3 > EPS0: # and r2 > EPS0 and r1 > EPS0 

r3_r1 = r3 / r1 

r3_r2 = r3 / r2 

# t = r1 * r2 * r3 * (r1 + r2 + r3) 

# = r1**2 * r2 * r3 * (1 + r2 / r1 + r3 / r1) 

# = (r1 * r2)**2 * (r3 / r2) * (1 + r2 / r1 + r3 / r1) 

t = r3_r2 * fsum1_(_1_0, r2 / r1, r3_r1) # * (r1 * r2)**2 

if t > EPS02: 

t = sqrt(t) * _2_0 # * r1 * r2 

# d = r1 * r2 + r2 * r3 + r3 * r1 

# = r1 * (r2 + r2 * r3 / r1 + r3) 

# = r1 * r2 * (1 + r3 / r1 + r3 / r2) 

d = fsum1_(_1_0, r3_r1, r3_r2) # * r1 * r2 

# si/o = r1 * r2 * r3 / (r1 * r2 * (d +/- t)) 

# = r3 / (d +/- t) 

si = r3 / (d + t) 

so = (r3 / (d - t)) if d > t else INF 

# ci = sqrt(r1 * r2 * r3 / s) 

# = r1 * sqrt(r2 * r3 / r1 / s) 

ci = r1 * sqrt(r2 * r3_r1 / s) 

# co = a * b * c / (4 * ci * s) 

t = ci * s * _4_0 

co = (a * b * c / t) if t > EPS0 else INF 

r1, r2, r3 = rs # original order 

t = Radii11Tuple(r1, r2, r3, co, ci, si, so, a, b, c, s) 

return t, A, B, C 

 

raise TriangleError(_near_(_coincident_) if min(a, b, c) < EPS0 else ( 

_colinear_ if _iscolinearWith(A, B, C) else _invalid_)) 

 

 

def soddy4(point1, point2, point3, eps=EPS4, useZ=True): 

'''Return the radius and center of the C{inner} I{Soddy} circle of a 

(2- or 3-D) triangle. 

 

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

C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}). 

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

C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}). 

@arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, 

C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}). 

@kwarg eps: Tolerance for function L{pygeodesy.trilaterate3d2} if 

C{B{useZ} is True} otherwise L{pygeodesy.trilaterate2d2}. 

@kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}). 

 

@return: L{Soddy4Tuple}C{(radius, center, deltas, outer)}. The C{center}, 

an instance of B{C{point1}}'s (sub-)class, is co-planar with the 

three given points. 

 

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

than version 1.10 and C{B{useZ} is True}. 

 

@raise IntersectionError: Near-coincident or -colinear points or 

a trilateration or C{numpy} issue. 

 

@raise TypeError: Invalid B{C{point1}}, B{C{point2}} or B{C{point3}}. 

 

@see: Functions L{radii11} and L{circum3}. 

''' 

t, p1, p2, p3 = _radii11ABC(point1, point2, point3, useZ=useZ) 

 

r = t.riS 

c, d = _tricenter3d2(p1, t.rA + r, 

p2, t.rB + r, 

p3, t.rC + r, eps=eps, useZ=useZ, 

Vector=point1.classof, name=soddy4.__name__) 

return Soddy4Tuple(r, c, d, t.roS) 

 

 

def _tricenter3d2(p1, r1, p2, r2, p3, r3, eps=EPS4, useZ=True, dLL3=False, **kwds): 

# (INTERNAL) Trilaterate and disambiguate the 3-D center 

d, kwds = None, _xkwds(kwds, eps=eps, coin=True) 

if useZ and p1.z != p2.z != p3.z: # ignore z if all match 

a, b = _trilaterate3d2(p1, r1, p2, r2, p3, r3, **kwds) 

if a is b: # no unambiguity 

c = a # == b 

else: 

c = a.plus(b).times(_0_5) # mean 

if not a.isconjugateTo(b, minum=0, eps=eps): 

if dLL3: # deltas as (lat, lon, height) 

a = a.toLatLon() 

b = b.toLatLon() 

d = LatLon3Tuple(b.lat - a.lat, 

b.lon - a.lon, 

b.height - a.height, name=_deltas_) 

else: 

d = b.minus(a) # vectorial deltas 

else: 

if useZ: # pass z to Vector if given 

kwds = _xkwds(kwds, z=p1.z) 

c = _trilaterate2d2(p1.x, p1.y, r1, 

p2.x, p2.y, r2, 

p3.x, p3.y, r3, **kwds) 

return c, d 

 

 

def _trilaterate2d2(x1, y1, radius1, x2, y2, radius2, x3, y3, radius3, 

coin=False, eps=None, 

Vector=None, **Vector_kwds): 

# (INTERNAL) Trilaterate three circles, see L{pygeodesy.trilaterate2d2} 

 

def _abct4(x1, y1, r1, x2, y2, r2): 

a = x2 - x1 

b = y2 - y1 

t = _trinear2far(r1, r2, hypot(a, b), coin) 

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 = (a * e - b * d) * _2_0 

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: 

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(), t.toRepr()) 

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

 

if Vector is not None: 

t = Vector(t.x, t.y, **_xkwds(Vector_kwds, name=t.name)) 

return t 

 

 

def _trilaterate3d2(c1, r1, c2, r2, c3, r3, eps=EPS, coin=False, # MCCABE 14 

**clas_Vector_and_kwds): 

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

# separated to allow callers to embellish any exceptions, like 

# C{FloatingPointError}s from C{numpy} 

 

def _F3d2(F): 

# map numpy 4-vector to floats tuple and Vector3d 

T = map2(float, F) 

return T, Vector3d(*T[1:]) 

 

def _N3(t01, x, z): 

# compute x, y and z and return as B{C{clas}} or B{C{Vector}} 

v = x.plus(z.times(t01)) 

n = trilaterate3d2.__name__ 

return _nVc(v, **_xkwds(clas_Vector_and_kwds, name=n)) 

 

def _perturbe5(eps, r): 

# perturbe radii to handle corner cases like this 

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

yield _0_0 

if eps and eps > 0: 

p = max(eps, EPS) 

yield p 

yield -min(p, r) 

q = max(eps, _EPS4e8) 

if q > p: 

yield q 

q = min(q, r) 

if q != min(p, r): 

yield -q 

 

def _roots(numpy, *coeffs): 

# only real, non-complex roots of a polynomial, if any 

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

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

 

c2 = _otherV3d(center2=c2, NN_OK=False) 

c3 = _otherV3d(center3=c3, NN_OK=False) 

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

Radius_(radius3=r3, low=eps)] 

 

# get null_space Z, pseudo-inverse A and vector B, once 

A = [(_1_0_1T + c.times(_N_2_0).xyz) for c in (c1, c2, c3)] # 3 x 4 

with _numpy(None, trilaterate3d2) as np: 

Z, _ = _null_space2(np, A, eps) 

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

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

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

Z, z = _F3d2(Z) 

z2 = z.length2 

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

 

for p in _perturbe5(eps, min(rs)): 

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

with _numpy(p, trilaterate3d2) as np: 

X, x = _F3d2(np.dot(A, b)) 

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

t = _roots(np, fdot(X, _N_1_0, *x.xyz), 

(fdot(Z, -_0_5, *x.xyz) * _2_0), z2) 

if t: 

break 

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

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

 

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.x < v.x else (v, u) 

return t 

 

 

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

# return IntersectionError with the cause of the error 

 

def _no_intersection(): 

t = _no_(_intersection_) 

if coin: 

def _reprs(*crs): 

return _and(*map(repr, crs)) 

 

r = repr(r1) if r1 == r2 == r3 else _reprs(r1, r2, r3) 

t = _SPACE_(t, _of_, _reprs(c1, c2, c3), _with_, _radius_, r) 

return t 

 

def _txt(c1, r1, c2, r2): 

t = _trinear2far(r1, r2, c1.minus(c2).length, coin) 

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, txt=None) 

 

 

def _trinear2far(r1, r2, h, coin): 

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

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

_coincident_ if coin else _concentric_) if h < abs(r1 - r2) else NN) 

 

# **) MIT License 

# 

# Copyright (C) 2016-2022 -- 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.