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# -*- coding: utf-8 -*- 

 

u'''3-Point resection functions L{cassini}, L{collins}, L{pierlot} and L{tienstra}, 

survey functions L{snellius3} and L{wildberger3} and triangle functions L{triAngle}, 

L{triAngle4}, L{triSide}, L{triSide2} and L{triSide4}. 

 

@note: Function L{pierlot} transcoded with permission from U{triangulationPierlot 

<http://www.Telecom.ULg.ac.BE/triangulation/doc/total_8c.html>} and U{Pierlot 

<http://www.Telecom.ULg.ac.BE/publi/publications/pierlot/Pierlot2014ANewThree>}. 

''' 

 

from pygeodesy.basics import isnear0, map1 

from pygeodesy.errors import _and, _or, ResectionError, TriangleError, _xkwds 

from pygeodesy.fmath import favg, fdot, fidw, fmean, fsum, fsum_, fsum1, fsum1_, \ 

hypot, hypot2_ 

from pygeodesy.interns import EPS, EPS0, EPS02, PI, PI2, PI_2, PI_4, _a_, _A_, \ 

_b_, _B_, _c_, _C_, _coincident_, _colinear_, _d_, \ 

_invalid_, _negative_, _not_, _rIn_, _SPACE_, \ 

_0_0, _0_5, _1_0, _N_1_0, _2_0, _N_2_0, _4_0, _360_0 

from pygeodesy.lazily import _ALL_LAZY 

from pygeodesy.named import Fmt, _NamedTuple, _Pass 

# from pygeodesy.streprs import Fmt # from .named 

from pygeodesy.units import Degrees, Distance, Radians 

from pygeodesy.utily import acos1, asin1, sincos2, sincos2_, sincos2d, sincos2d_ 

from pygeodesy.vector3d import _otherV3d, Vector3d 

 

from math import cos, atan2, degrees, radians, sin, sqrt 

 

__all__ = _ALL_LAZY.resections 

__version__ = '21.11.01' 

 

_concyclic_ = 'concyclic' 

_PA_ = 'PA' 

_PB_ = 'PB' 

_PC_ = 'PC' 

_pointH_ = 'pointH' 

_pointP_ = 'pointP' 

_R3_ = 'R3' 

_radA_ = 'radA' 

_radB_ = 'radB' 

_radC_ = 'radC' 

 

 

class Collins5Tuple(_NamedTuple): 

'''5-Tuple C{(pointP, pointH, a, b, c)} with survey C{pointP}, auxiliary 

C{pointH}, each an instance of B{C{pointA}}'s (sub-)class and triangle 

sides C{a}, C{b} and C{c} in C{meter}, conventionally. 

''' 

_Names_ = (_pointP_, _pointH_, _a_, _b_, _c_) 

_Units_ = (_Pass, _Pass, Distance, Distance, Distance) 

 

 

class Survey3Tuple(_NamedTuple): 

'''3-Tuple C{(PA, PB, PC)} with distance from survey point C{P} to each 

of the triangle corners C{A}, C{B} and C{C}. 

''' 

_Names_ = (_PA_, _PB_, _PC_) 

_Units_ = ( Distance, Distance, Distance) 

 

 

class Tienstra7Tuple(_NamedTuple): 

'''7-Tuple C{(pointP, A, B, C, a, b, c)} with survey C{pointP}, interior 

triangle angles C{A}, C{B} and C{C} in C{degrees} and triangle sides 

C{a}, C{b} and C{c} in C{meter}, conventionally. 

''' 

_Names_ = (_pointP_, _A_, _B_, _C_, _a_, _b_, _c_) 

_Units_ = (_Pass, Degrees, Degrees, Degrees, Distance, Distance, Distance) 

 

 

class TriAngle4Tuple(_NamedTuple): 

'''4-Tuple C{(radA, radB, radC, rIn)} with the interior angles at triangle 

corner C{A}, C{B} and C{C} and the C{InCircle} radius C{rIn} aka C{inradius}. 

''' 

_Names_ = (_radA_, _radB_, _radC_, _rIn_) 

_Units_ = ( Radians, Radians, Radians, Distance) 

 

 

class TriSide2Tuple(_NamedTuple): 

'''2-Tuple C{(a, radA)} with triangle side C{a} (C{meter} conventionally) 

and the opposite triangle angle C{radA} (C{radians}). 

''' 

_Names_ = (_a_, _radA_) 

_Units_ = ( Distance, Radians) 

 

 

class TriSide4Tuple(_NamedTuple): 

'''4-Tuple C{(a, b, radC, d)} with the length of triangle sides C{a} and 

C{b}, the interior angle C{radC} at triangle corner C{radC} (C{radians}) 

and triangle height C{d}, perpendicular to triangle side C{c}. 

''' 

_Names_ = (_a_, _b_, _radC_, _d_) 

_Units_ = ( Distance, Distance, Radians, Distance) 

 

 

def cassini(pointA, pointB, pointC, alpha, beta, useZ=False, Clas=None, **Clas_kwds): 

'''3-Point resection using U{Cassini<https://NL.WikiPedia.org/wiki/Achterwaartse_insnijding>}'s method. 

 

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

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

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

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

@arg pointC: Center point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, 

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

@arg alpha: Angle subtended by triangle side B{C{pointA}} to B{C{pointC}} 

(C{degrees}, non-negative). 

@arg beta: Angle subtended by triangle side B{C{pointB}} to B{C{pointC}} 

(C{degrees}, non-negative). 

@kwarg useZ: If C{True}, use and interpolate the Z component, otherwise 

force C{z=0} (C{bool}). 

@kwarg Clas: Optional class to return the survey and auxiliary point 

or C{None} for B{C{pointA}}'s (sub-)class. 

@kwarg Clas_kwds: Optional additional keyword argument for the survey 

and auxiliary point instance. 

 

@note: Typically, B{C{pointC}} is between B{C{pointA}} and B{C{pointB}}. 

 

@return: The survey point, an instance of B{C{Clas}} or if C{B{Clas} is 

None} of B{C{pointA}}'s (sub-)class. 

 

@raise ResectionError: Near-coincident, -colinear or -concyclic points 

or negative or invalid B{C{alpha}} or B{C{beta}}. 

 

@raise TypeError: Invalid B{C{pointA}}, B{C{pointB}} or B{C{pointM}}. 

 

@see: U{Three Point Resection Problem<https://Dokumen.tips/documents/ 

three-point-resection-problem-introduction-kaestner-burkhardt-method.html>} 

and functions L{pygeodesy.collins}, L{pygeodesy.pierlot} and 

L{pygeodesy.tienstra}. 

''' 

 

def _H(A, C, sa): 

s, c = sincos2d(sa) 

if isnear0(s): 

raise ValueError(_or(_coincident_, _colinear_)) 

t = s, c, c 

x = fdot(t, A.x, C.y, -A.y) / s 

y = fdot(t, A.y, -C.x, A.x) / s 

return Vector3d(x, y, 0) 

 

A = _otherV3d(useZ=useZ, pointA=pointA) 

B = _otherV3d(useZ=useZ, pointB=pointB) 

C = _otherV3d(useZ=useZ, pointC=pointC) 

 

try: 

sa, sb = map1(float, alpha, beta) 

if min(sa, sb) < 0: 

raise ValueError(_negative_) 

if fsum_(_360_0, -sa, -sb) < EPS0: 

raise ValueError(_invalid_) 

 

H1 = _H(A, C, sa) 

H2 = _H(B, C, -sb) 

 

x = H1.x - H2.x 

y = H1.y - H2.y 

if isnear0(x) or isnear0(y): 

raise ValueError(_SPACE_(_concyclic_, (x, y))) 

 

m = y / x 

n = x / y 

N = n + m 

if isnear0(N): 

raise ValueError(_SPACE_(_concyclic_, (m, n, N))) 

 

t = n, m, _1_0, _N_1_0 

x = fdot(t, C.x, H1.x, C.y, H1.y) / N 

y = fdot(t, H1.y, C.y, C.x, H1.x) / N 

z = _zidw(A, B, C, x, y) if useZ else 0 

 

clas = Clas or pointA.classof 

return clas(x, y, z, **_xkwds(Clas_kwds, name=cassini.__name__)) 

 

except (TypeError, ValueError) as x: 

raise ResectionError(pointA=pointA, pointB=pointB, pointC=pointC, 

alpha=alpha, beta=beta, txt=str(x)) 

 

 

def collins(pointA, pointB, pointC, alpha, beta, useZ=False, Clas=None, **Clas_kwds): 

'''3-Point resection using U{Collins<https://Dokumen.tips/documents/ 

three-point-resection-problem-introduction-kaestner-burkhardt-method.html>}' method. 

 

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

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

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

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

@arg pointC: Center point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, 

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

@arg alpha: Angle subtended by triangle side C{b} from B{C{pointA}} to 

B{C{pointC}} (C{degrees}, non-negative). 

@arg beta: Angle subtended by triangle side C{a} from B{C{pointB}} to 

B{C{pointC}} (C{degrees}, non-negative). 

@kwarg useZ: If C{True}, use and interpolate the Z component, otherwise 

force C{z=0} (C{bool}). 

@kwarg Clas: Optional class to return the survey point or C{None} for 

B{C{pointA}}'s (sub-)class. 

@kwarg Clas_kwds: Optional additional keyword argument for the survey 

point instance. 

 

@note: Typically, B{C{pointC}} is between B{C{pointA}} and B{C{pointB}}. 

 

@return: L{Collins5Tuple}C{(pointP, pointH, a, b, c)} with survey C{pointP}, 

auxiliary C{pointH}, each an instance of B{C{Clas}} or if C{B{Clas} 

is None} of B{C{pointA}}'s (sub-)class and triangle sides C{a}, 

C{b} and C{c}. 

 

@raise ResectionError: Near-coincident, -colinear or -concyclic points 

or negative or invalid B{C{alpha}} or B{C{beta}}. 

 

@raise TypeError: Invalid B{C{pointA}}, B{C{pointB}} or B{C{pointM}}. 

 

@see: U{Collins' methode<https://NL.WikiPedia.org/wiki/Achterwaartse_insnijding>} 

and functions L{pygeodesy.cassini}, L{pygeodesy.pierlot} and 

L{pygeodesy.tienstra}. 

''' 

 

def _azi_len2(A, B, pi2): 

v = B.minus(A) 

r = atan2(v.x, v.y) 

if pi2 and r < 0: 

r += pi2 

return r, v.length 

 

def _cV3(d, r, A, B, C, useZ, V3, **kwds): 

s, c = sincos2(r) 

x = A.x + d * s 

y = A.y + d * c 

z = _zidw(A, B, C, x, y) if useZ else 0 

return V3(x, y, z, **kwds) 

 

A = _otherV3d(useZ=useZ, pointA=pointA) 

B = _otherV3d(useZ=useZ, pointB=pointB) 

C = _otherV3d(useZ=useZ, pointC=pointC) 

 

try: 

ra, rb = radians(alpha), radians(beta) 

if min(ra, rb) < 0: 

raise ValueError(_negative_) 

 

sra, srH = sin(ra), sin(ra + rb - PI) # rH = PI - ((PI - ra) + (PI - rb)) 

if isnear0(sra) or isnear0(srH): 

raise ValueError(_or(_coincident_, _colinear_, _concyclic_)) 

 

clas = Clas or pointA.classof 

kwds = _xkwds(Clas_kwds, name=collins.__name__) 

 

# za, a = _azi_len2(C, B, PI2) 

zb, b = _azi_len2(C, A, PI2) 

zc, c = _azi_len2(A, B, 0) 

 

# d = c * sin(PI - rb) / srH # B.minus(H).length 

d = c * sin(PI - ra) / srH # A.minus(H).length 

r = zc + PI - rb # zh = zc + (PI - rb) 

H = _cV3(d, r, A, B, C, useZ, Vector3d) 

 

zh, _ = _azi_len2(C, H, PI2) 

 

# d = a * sin(za - zh) / sin(rb) # B.minus(P).length 

d = b * sin(zb - zh) / sra # A.minus(P).length 

r = zh - ra # zb - PI + (PI - ra - (zb - zh)) 

P = _cV3(d, r, A, B, C, useZ, clas, **kwds) 

 

H = clas(H.x, H.y, H.z, **kwds) 

a = B.minus(C).length 

return Collins5Tuple(P, H, a, b, c) 

 

except (TypeError, ValueError) as x: 

raise ResectionError(pointA=pointA, pointB=pointB, pointC=pointC, 

alpha=alpha, beta=beta, txt=str(x)) 

 

 

def pierlot(point1, point2, point3, alpha12, alpha23, useZ=False, Clas=None, **Clas_kwds): 

'''3-Point resection using U{Pierlot<http://www.Telecom.ULg.ac.BE/publi/publications/ 

pierlot/Pierlot2014ANewThree>}'s method C{ToTal}. 

 

@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}). 

@arg alpha12: Angle subtended from B{C{point1}} to B{C{point2}} (C{degrees}). 

@arg alpha23: Angle subtended from B{C{point2}} to B{C{point3}} (C{degrees}). 

@kwarg useZ: If C{True}, interpolate the Z component, otherwise use C{z=0} 

(C{bool}). 

@kwarg Clas: Optional class to return the survey point or C{None} for 

B{C{point1}}'s (sub-)class. 

@kwarg Clas_kwds: Optional additional keyword arguments for the survey 

point instance. 

 

@note: Points B{C{point1}}, B{C{point2}} and B{C{point3}} are ordered 

counter-clockwise, typically. 

 

@return: The survey (or robot) point, an instance of B{C{Clas}} or if 

C{B{Clas} is None} of B{C{point1}}'s (sub-)class. 

 

@raise ResectionError: Near-coincident, -colinear or -concyclic points 

or invalid B{C{alpha12}} or B{C{alpha23}}. 

 

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

 

@see: U{V. Pierlot, M. Van Droogenbroeck, "A New Three Object Triangulation 

Algorithm for Mobile Robot Positioning"<https://ORBi.ULiege.BE/ 

bitstream/2268/157469/1/Pierlot2014ANewThree.pdf>}, U{Vincent Pierlot, 

Marc Van Droogenbroeck, "18 Triangulation Algorithms for 2D Positioning 

(also known as the Resection Problem)"<http://www.Telecom.ULg.ac.BE/ 

triangulation>} and functions L{pygeodesy.cassini}, L{pygeodesy.collins} 

and L{pygeodesy.tienstra}. 

''' 

B1 = _otherV3d(useZ=useZ, point1=point1) 

B2 = _otherV3d(useZ=useZ, point2=point2) 

B3 = _otherV3d(useZ=useZ, point3=point3) 

 

try: # (INTERNAL) Raises error for (pseudo-)singularities 

s12, c12, s23, c23 = sincos2d_(alpha12, alpha23) 

if isnear0(s12) or isnear0(s23): 

raise ValueError(_or(_coincident_, _colinear_)) 

cot12 = c12 / s12 

cot23 = c23 / s23 

# cot31 = (1 - cot12 * cot23) / (cot12 + cot32) 

d = fsum1_(c12 * s23, s12 * c23) 

if isnear0(d): 

raise ValueError(_or(_coincident_, _colinear_)) 

cot31 = fsum1_(s12 * s23, -c12 * c23) / d 

 

x1_, y1_, _ = B1.minus(B2).xyz 

x3_, y3_, _ = B3.minus(B2).xyz 

 

x23 = x3_ - cot23 * y3_ 

y23 = y3_ + cot23 * x3_ 

 

x12_23 = fsum_(x1_, cot12 * y1_, -x23) 

y12_23 = fsum_(y1_, -cot12 * x1_, -y23) 

 

x31_23 = fsum_(x1_, -cot31 * y1_, x3_, cot31 * y3_, -x23) 

y31_23 = fsum_(y1_, cot31 * x1_, y3_, -cot31 * x3_, -y23) 

 

d = x31_23 * y12_23 - x12_23 * y31_23 

if isnear0(d): 

raise ValueError(_or(_coincident_, _colinear_, _concyclic_)) 

k = fsum_(x3_ * x1_, cot31 * (y3_ * x1_), 

y3_ * y1_, -cot31 * (x3_ * y1_)) / d 

 

x = B2.x + k * y12_23 

y = B2.y - k * x12_23 

z = _zidw(B1, B2, B3, x, y) if useZ else 0 

 

clas = Clas or point1.classof 

return clas(x, y, z, **_xkwds(Clas_kwds, name=pierlot.__name__)) 

 

except (TypeError, ValueError) as x: 

raise ResectionError(point1=point1, point2=point2, point3=point3, 

alpha12=alpha12, alpha23=alpha23, txt=str(x)) 

 

 

def snellius3(a, b, degC, alpha, beta): 

'''Snellius' surveying using U{Snellius Pothenot<https://WikiPedia.org/wiki/Snellius–Pothenot_problem>}. 

 

@arg a: Length of triangle side C{BC}, opposite of triangle corner C{A} (C{scalar}, 

non-negative C{meter} conventionally). 

@arg b: Length of triangle side C{AC}, opposite of triangle corner C{B} (C{scalar}, 

non-negative C{meter} conventionally). 

@arg degC: Angle at triangle corner C{C}, opposite triangle side C{c} (C{degrees}, 

non-negative). 

@arg alpha: Angle subtended by triangle side B{C{b}} (C{degrees}, non-negative). 

@kwarg beta: Angle subtended by triangle side B{C{a}} (C{degrees}, non-negative). 

 

@return: L{Survey3Tuple}C{(PA, PB, PC)} with distance from survey point C{P} to 

each of the triangle corners C{A}, C{B} and C{C} (same units as B{C{a}}, 

B{C{b}} and B{C{c}}). 

 

@raise TriangleError: Invalid B{C{a}}, B{C{b}} or B{C{degC}} or negative B{C{alpha}} 

or B{C{beta}}. 

 

@see: Function L{pygeodesy.wildberger3}. 

''' 

try: 

a, b, degC, alpha, beta = map1(float, a, b, degC, alpha, beta) 

ra, rb, rC = map1(radians, alpha, beta, degC) 

if min(ra, rb, rC, a, b) < 0: 

raise ValueError(_negative_) 

 

r = fsum_(ra, rb, rC) * _0_5 

k = PI - r 

if min(k, r) < 0: 

raise ValueError(_or(_coincident_, _colinear_)) 

 

sa, _, sb, _ = sincos2_(ra, rb) 

p = atan2(a * sa, b * sb) 

sp, cp, sr, cr = sincos2_(PI_4 - p, r) 

w = atan2(sp * sr, cp * cr) 

x = k + w 

y = k - w 

 

pc = abs((a * sin(y) / sb) if abs(sb) > abs(sa) else 

(b * sin(x) / sa)) 

pa = _triSide(b, pc, fsum_(PI, -ra, -x)) 

pb = _triSide(a, pc, fsum_(PI, -rb, -y)) 

return Survey3Tuple(pa, pb, pc) 

 

except (TypeError, ValueError) as x: 

raise TriangleError(a=a, b=b, degC=degC, alpha=alpha, beta=beta, txt=str(x)) 

 

 

def tienstra(pointA, pointB, pointC, alpha, beta=None, gamma=None, 

useZ=False, Clas=None, **Clas_kwds): 

'''3-Point resection using U{Tienstra<https://WikiPedia.org/wiki/Tienstra_formula>}'s formula. 

 

@arg pointA: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, C{Vector4Tuple} or 

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

@arg pointB: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, C{Vector4Tuple} or 

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

@arg pointC: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, C{Vector4Tuple} or 

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

@arg alpha: Angle subtended by triangle side C{a} from B{C{pointB}} to B{C{pointC}} 

(C{degrees}, non-negative). 

@kwarg beta: Angle subtended by triangle side C{b} from B{C{pointA}} to B{C{pointC}} 

(C{degrees}, non-negative) or C{None} if C{B{gamma} is not None}. 

@kwarg gamma: Angle subtended by triangle side C{c} from B{C{pointA}} to B{C{pointB}} 

(C{degrees}, non-negative) or C{None} if C{B{beta} is not None}. 

@kwarg useZ: If C{True}, use and interpolate the Z component, otherwise force C{z=0} 

(C{bool}). 

@kwarg Clas: Optional class to return the survey point or C{None} for B{C{pointA}}'s 

(sub-)class. 

@kwarg Clas_kwds: Optional additional keyword arguments for the survey point instance. 

 

@note: Points B{C{pointA}}, B{C{pointB}} and B{C{pointC}} are ordered clockwise. 

 

@return: L{Tienstra7Tuple}C{(pointP, A, B, C, a, b, c)} with survey C{pointP}, 

an instance of B{C{Clas}} or if C{B{Clas} is None} of B{C{pointA}}'s 

(sub-)class and triangle angle C{A} at B{C{pointA}}, C{B} at B{C{pointB}} 

and C{C} at B{C{pointC}} in C{degrees} and triangle sides C{a}, C{b} and C{c}. 

 

@raise ResectionError: Near-coincident, -colinear or -concyclic points or sum of 

B{C{alpha}}, B{C{beta}} and B{C{gamma}} not C{360} or 

negative B{C{alpha}}, B{C{beta}} or B{C{gamma}}. 

 

@raise TypeError: Invalid B{C{pointA}}, B{C{pointB}} or B{C{pointC}}. 

 

@see: U{3-Point Resection Solver<http://MesaMike.org/geocache/GC1B0Q9/tienstra/>}, 

U{V. Pierlot, M. Van Droogenbroeck, "A New Three Object Triangulation..." 

<http://www.Telecom.ULg.ac.BE/publi/publications/pierlot/Pierlot2014ANewThree/>}, 

U{18 Triangulation Algorithms...<http://www.Telecom.ULg.ac.BE/triangulation/>} and 

functions L{pygeodesy.cassini}, L{pygeodesy.collins} and L{pygeodesy.pierlot}. 

''' 

 

def _deg_ks(r, s, ks, N): 

if isnear0(fsum1_(PI, r, -s)): # r + (PI2 - s) == PI 

raise ValueError(Fmt.PARENSPACED(concyclic=N)) 

# k = 1 / (cot(r) - cot(s)) 

# = 1 / (cos(r) / sin(r) - cos(s) / sin(s)) 

# = 1 / (cos(r) * sin(s) - cos(s) * sin(r)) / (sin(r) * sin(s)) 

# = sin(r) * sin(s) / (cos(r) * sin(s) - cos(s) * sin(r)) 

sr, cr, ss, cs = sincos2_(r, s) 

c = cr * ss - cs * sr 

if isnear0(c): 

raise ValueError(Fmt.PARENSPACED(cotan=N)) 

ks.append(sr * ss / c) 

return Degrees(degrees(r), name=N) # C degrees 

 

A = _otherV3d(useZ=useZ, pointA=pointA) 

B = _otherV3d(useZ=useZ, pointB=pointB) 

C = _otherV3d(useZ=useZ, pointC=pointC) 

 

try: 

sa, sb, sc = map1(radians, alpha, (beta or 0), (gamma or 0)) 

if beta is None: 

if gamma is None: 

raise ValueError(_and(Fmt.EQUAL(beta=beta), Fmt.EQUAL(gamma=gamma))) 

sb = fsum1_(PI2, -sa, -sc) 

elif gamma is None: 

sc = fsum1_(PI2, -sa, -sb) 

else: # subtended angles must add to 360 degrees 

r = fsum1_(sa, sb, sc) 

if abs(r - PI2) > EPS: 

raise ValueError(Fmt.EQUAL(sum=degrees(r))) 

if min(sa, sb, sc) < 0: 

raise ValueError(_negative_) 

 

# triangle sides 

a = B.minus(C).length 

b = A.minus(C).length 

c = A.minus(B).length 

 

ks = [] # 3 Ks and triangle angles 

dA = _deg_ks(_triAngle(b, c, a), sa, ks, _A_) 

dB = _deg_ks(_triAngle(a, c, b), sb, ks, _B_) 

dC = _deg_ks(_triAngle(a, b, c), sc, ks, _C_) 

 

k = fsum1(ks) 

if isnear0(k): 

raise ValueError(Fmt.EQUAL(K=k)) 

x = fdot(ks, A.x, B.x, C.x) / k 

y = fdot(ks, A.y, B.y, C.y) / k 

z = _zidw(A, B, C, x, y) if useZ else 0 

 

clas = Clas or pointA.classof 

P = clas(x, y, z, **_xkwds(Clas_kwds, name=tienstra.__name__)) 

return Tienstra7Tuple(P, dA, dB, dC, a, b, c) 

 

except (TypeError, ValueError) as x: 

raise ResectionError(pointA=pointA, pointB=pointB, pointC=pointC, 

alpha=alpha, beta=beta, gamma=gamma, txt=str(x)) 

 

 

def triAngle(a, b, c): 

'''Compute an interior angle of a triangle. 

 

@arg a: Adjacent triangle side length (C{scalar}, non-negative 

C{meter} conventionally). 

@arg b: Adjacent triangle side length (C{scalar}, non-negative 

C{meter} conventionally). 

@arg c: Opposite triangle side length (C{scalar}, non-negative 

C{meter} conventionally). 

 

@return: Angle at triangle corner C{C}, opposite triangle side 

B{C{c}} (C{radians}). 

 

@raise TriangleError: Invalid B{C{a}}, B{C{b}} or B{C{c}}. 

 

@see: Function L{pygeodesy.triSide}. 

''' 

try: 

return _triAngle(a, b, c) 

except (TypeError, ValueError) as x: 

raise TriangleError(a=a, b=b, c=c, tx=str(x)) 

 

 

def _triAngle(a, b, c): 

# (INTERNAL) To allow callers to embellish errors 

a, b, c = map1(float, a, b, c) 

if a < b: 

a, b = b, a 

if b < 0 or c < 0: 

raise ValueError(_negative_) 

if a < EPS0: 

raise ValueError(_coincident_) 

b_a = b / a 

if b_a < EPS0: 

raise ValueError(_coincident_) 

return acos1(fsum_(_1_0, b_a**2, -(c / a)**2) / (b_a * _2_0)) 

 

 

def triAngle4(a, b, c): 

'''Compute the angles of a triangle side. 

 

@arg a: Length of triangle side C{BC}, opposite of triangle corner C{A} (C{scalar}, 

non-negative C{meter} conventionally). 

@arg b: Length of triangle side C{AC}, opposite of triangle corner C{B} (C{scalar}, 

non-negative C{meter} conventionally). 

@arg c: Length of triangle side C{AB}, opposite of triangle corner C{C} (C{scalar}, 

non-negative C{meter} conventionally). 

 

@return: L{TriAngle4Tuple}C{(radA, radB, radC, rIn)} with the triangle angles at 

corner C{A}, C{B} and C{C} (each in radians) and the C{InCircle} radius 

C{rIn} aka C{inradius} (same units and triangle sides B{C{a}}, B{C{b}} 

and B{C{c}}). 

 

@raise TriangleError: Invalid B{C{a}}, B{C{b}} or B{C{b}}. 

''' 

try: 

a, b, c = map1(float, a, b, c) 

ab = a < b 

if ab: 

a, b = b, a 

bc = b < c 

if bc: 

b, c = c, b 

 

if c > EPS0: # c = min(a, b, c) 

s = fsum_(a, b, c) * _0_5 

if s < EPS0: 

raise ValueError(_negative_) 

sa, sb, sc = (s - a), (s - b), (s - c) 

r = sa * sb * sc / s 

if r < EPS02: 

raise ValueError(_coincident_) 

r = sqrt(r) 

rA = atan2(r, sa) * _2_0 

rB = atan2(r, sb) * _2_0 

rC = fsum_(PI, -rA, -rB) 

if min(rA, rB, rC) < 0: 

raise ValueError(_colinear_) 

elif c < 0: 

raise ValueError(_negative_) 

else: # 0 <= c <= EPS0 

rA = rB = PI_2 

rC = r = _0_0 

 

if bc: 

rB, rC = rC, rB 

if ab: 

rA, rB = rB, rA 

return TriAngle4Tuple(rA, rB, rC, r) 

 

except (TypeError, ValueError) as x: 

raise TriangleError(a=a, b=b, c=c, txt=str(x)) 

 

 

def triSide(a, b, radC): 

'''Compute the length of a triangle side. 

 

@arg a: Adjacent triangle side length (C{scalar}, 

non-negative C{meter} conventionally). 

@arg b: Adjacent triangle side length (C{scalar}, 

non-negative C{meter} conventionally). 

@arg radC: Angle included by sides B{C{a}} and B{C{b}}, 

opposite triangle side C{c} (C{radians}). 

 

@return: Length of triangle side C{c}, opposite angle B{C{rC}} 

(same units as B{C{a}} and B{C{b}}). 

 

@raise TriangleError: Invalid B{C{a}}, B{C{b}} or B{C{radC}}. 

''' 

try: 

return _triSide(a, b, radC) 

except (TypeError, ValueError) as x: 

raise TriangleError(a=a, b=b, radC=radC, txt=str(x)) 

 

 

def _triSide(a, b, radC): 

# (INTERNAL) To allow callers to embellish errors 

a, b, r = map1(float, a, b, radC) 

if min(a, b, r) < 0: 

raise ValueError(_negative_) 

 

if isnear0(r): 

return abs(a - b) 

ab = _N_2_0 * a * b * cos(r) 

if abs(ab) < EPS02: 

return hypot(a, b) 

 

c2 = fsum_(a**2, b**2, ab) 

if abs(c2) < EPS02: 

c = _0_0 

elif c2 < 0: 

raise ValueError(_invalid_) 

else: 

c = sqrt(c2) 

return c 

 

 

def triSide2(b, c, radB): 

'''Compute the length of a triangle side and the angle. 

 

@arg b: Adjacent triangle side length (C{scalar}, 

non-negative C{meter} conventionally). 

@arg c: Adjacent triangle side length (C{scalar}, 

non-negative C{meter} conventionally). 

@arg radB: Angle included by sides B{C{a}} and B{C{c}}, 

opposite triangle side C{b} (C{radians}). 

 

@return: L{TriSide2Tuple}C{(a, radA)} with triangle angle 

C{radA} (radians) and the length of the opposite 

triangle side C{a} (same units as B{C{b}} and B{C{c}}). 

 

@raise TriangleError: Invalid B{C{b}} or B{C{c}} or either 

B{C{b}} or B{C{radB}} near zero and 

not both. 

''' 

try: 

return _triSide2(b, c, radB) 

except (TypeError, ValueError) as x: 

raise TriangleError(b=b, c=c, radB=radB, txt=str(x)) 

 

 

def _triSide2(b, c, radB): 

# (INTERNAL) To allow callers to embellish errors 

b, c, rB = map1(float, b, c, radB) 

if min(b, c, rB) < 0: 

raise ValueError(_negative_) 

sB, cB = sincos2(rB) 

if isnear0(sB): 

if not isnear0(b): 

raise ValueError(_invalid_) 

if cB < 0: 

a, rA = (b + c), PI 

else: 

a, rA = abs(b - c), _0_0 

elif isnear0(b): 

raise ValueError(_invalid_) 

else: 

rA = fsum1_(PI, -rB, -asin1(c * sB / b)) 

a = sin(rA) * b / sB 

return TriSide2Tuple(a, rA) 

 

 

def triSide4(radA, radB, c): 

'''Compute the length of two triangle sides and the triangle height. 

 

@arg radA: Angle at triangle corner C{A}, opposite triangle side C{a} 

(C{scalar}, non-negative). 

@arg radB: Angle at triangle corner C{B}, opposite triangle side C{b} 

(C{scalar}, non-negative). 

@arg c: Length of triangle side between corners C{A} and C{B}, (C{scalar}, 

non-negative C{meter} conventionally). 

 

@return: L{TriSide4Tuple}C{(a, b, radC, d)} with triangle sides C{a} and 

C{b} and triangle height C{d} perpendicular to triangle side 

B{C{c}} (all in same units as B{C{c}}) and the interior angle 

at triangle corner C{C} (C{radians}), opposite of triangle 

side B{C{c}}. 

 

@raise TriangleError: Invalid B{C{a}} or B{C{b}}. 

 

@see: Function L{pygeodesy.triSide} and U{Triangulation, Surveying 

<https://WikiPedia.org/wiki/Triangulation_(surveying)>}. 

''' 

try: 

rA, rB, c = map1(float, radA, radB, c) 

rC = fsum_(PI, -rA, -rB) 

if min(rC, rA, rB, c) < 0: 

raise ValueError(_negative_) 

sa, ca, sb, cb = sincos2_(rA, rB) 

sc = fsum1_(sa * cb, sb * ca) 

if sc < EPS0 or min(sa, sb) < 0: 

raise ValueError(_invalid_) 

sc = c / sc 

return TriSide4Tuple((sa * sc), (sb * sc), rC, (sa * sb * sc)) 

 

except (TypeError, ValueError) as x: 

raise TriangleError(radA=radA, radB=radB, c=c, txt=str(x)) 

 

 

def wildberger3(a, b, c, alpha, beta, R3=min): 

'''Snellius' surveying using U{Rational Trigonometry<https://WikiPedia.org/wiki/Snellius–Pothenot_problem>}. 

 

@arg a: Length of triangle side C{BC}, opposite of triangle corner C{A} (C{scalar}, 

non-negative C{meter} conventionally). 

@arg b: Length of triangle side C{AC}, opposite of triangle corner C{B} (C{scalar}, 

non-negative C{meter} conventionally). 

@arg c: Length of triangle side C{AB}, opposite of triangle corner C{C} (C{scalar}, 

non-negative C{meter} conventionally). 

@arg alpha: Angle subtended by triangle side B{C{b}} (C{degrees}, non-negative). 

@arg beta: Angle subtended by triangle side B{C{a}} (C{degrees}, non-negative). 

@kwarg R3: Callable to determine C{R3} from C{(R3 - C)**2 = D}, typically standard 

function C{min} or C{max}, invoked with 2 arguments. 

 

@return: L{Survey3Tuple}C{(PA, PB, PC)} with distance from survey point C{P} to 

each of the triangle corners C{A}, C{B} and C{C} (same units as B{C{a}}, 

B{C{b}} and B{C{c}}). 

 

@raise TriangleError: Invalid B{C{a}}, B{C{b}} or B{C{c}} or negative B{C{alpha}} or 

B{C{beta}} or B{C{R3}} not C{callable}. 

 

@see: U{Wildberger, Norman J.<https://math.sc.chula.ac.th/cjm/content/ 

survey-article-greek-geometry-rational-trigonometry-and-snellius-–-pothenot-surveying>}, 

U{Devine Proportions, page 252<http://www.ms.LT/derlius/WildbergerDivineProportions.pdf>} 

and function L{pygeodesy.snellius3}. 

''' 

def _s(x): 

return sin(x)**2 

 

def _vpa(r1, r3, q2, q3_r3): 

r = r1 * r3 * _4_0 

n = r - fsum_(r1, r3, -q2)**2 

if n < 0 or isnear0(r): 

raise ValueError(_coincident_) 

return sqrt((n / r) * q3_r3) 

 

try: 

a, b, c, da, db = map1(float, a, b, c, alpha, beta) 

if min(a, b, c, da, db) < 0: 

raise ValueError(_negative_) 

 

ra, rb = radians(da), radians(db) 

s1, s2, s3 = s = map1(_s, rb, ra, ra + rb) # rb, ra! 

if min(s) < EPS02: 

raise ValueError(_or(_coincident_, _colinear_)) 

 

q1, q2, q3 = q = a**2, b**2, c**2 

if min(q) < EPS02: 

raise ValueError(_coincident_) 

q3_s3 = q3 / s3 

 

r1 = s2 * q3_s3 # s2! 

r2 = s1 * q3_s3 # s1! 

qs = fsum(q) # == hypot2_(a, b, c) 

ss = fsum1(s) 

s += (qs * _0_5), 

c0 = -fdot(s, q1, q2, q3, -ss) / s3 

d0 = (qs**2 - hypot2_(*q) * _2_0) * s1 * s2 / s3 

if d0 > EPS02: # > c0 

d0 = sqrt(d0) 

if not callable(R3): 

raise ValueError(_SPACE_(_R3_, _not_(callable.__name__))) 

r3 = R3(c0 + d0, c0 - d0) # XXX min or max 

elif d0 < 0: 

raise ValueError(_negative_) 

else: # isnear0(sqrt(d0)) 

r3 = c0 

 

pa = _vpa(r1, r3, q2, q3_s3) 

pb = _vpa(r2, r3, q1, q3_s3) 

pc = favg(_triSide2(b, pa, ra).a, 

_triSide2(a, pb, rb).a) 

return Survey3Tuple(pa, pb, pc) 

 

except (TypeError, ValueError) as x: 

raise TriangleError(a=a, b=b, c=c, alpha=alpha, beta=beta, R3=R3, txt=str(x)) 

 

 

def _zidw(A, B, C, x, y): 

# interpolate z or coplanar with A, B and C? 

t = A.z, B.z, C.z 

v = Vector3d(x, y, fmean(t)) 

return fidw(t, (v.minus(A).length, v.minus(B).length, v.minus(C).length)) 

 

# **) 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.