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PyGeodesy
Package pygeodesy :: Module resections
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Module resections

3-Point resection functions cassini, collins5, pierlot, pierlotx and tienstra7, survey functions snellius3 and wildberger3 and triangle functions triAngle, triAngle5, triSide, triSide2 and triSide4.

Note: Functions pierlot and pierlotx are transcoded to Python with permission from triangulationPierlot and Pierlot.

Version: 24.09.23

Classes
  Collins5Tuple
5-Tuple (pointP, pointH, a, b, c) with survey pointP, auxiliary pointH, each an instance of pointA's (sub-)class and triangle sides a, b and c in meter, conventionally.
  ResectionError
Error raised for issues in pygeodesy.resections.
  Survey3Tuple
3-Tuple (PA, PB, PC) with distance from survey point P to each of the triangle corners A, B and C in meter, conventionally.
  Tienstra7Tuple
7-Tuple (pointP, A, B, C, a, b, c) with survey pointP, interior triangle angles A, B and C in degrees and triangle sides a, b and c in meter, conventionally.
  TriAngle5Tuple
5-Tuple (radA, radB, radC, rIn, area) with the interior angles at triangle corners A, B and C in radians, the InCircle radius rIn aka inradius in meter and the triangle area in meter squared, conventionally.
  TriSide2Tuple
2-Tuple (a, radA) with triangle side a in meter, conventionally and angle radA at the opposite triangle corner in radians.
  TriSide4Tuple
4-Tuple (a, b, radC, d) with interior angle radC at triangle corner C in radianswith the length of triangle sides a and b and with triangle height d perpendicular to triangle side c, in the same units as triangle sides a and b.
Functions
 
cassini(pointA, pointB, pointC, alpha, beta, useZ=False, **Clas_and_kwds)
3-Point resection using Cassini's method.
 
collins5(pointA, pointB, pointC, alpha, beta, useZ=False, **Clas_and_kwds)
3-Point resection using Collins' method.
 
pierlot(point1, point2, point3, alpha12, alpha23, useZ=False, eps=2.220446049250313e-16, **Clas_and_kwds)
3-Point resection using Pierlot's method ToTal with approximate limits for the (pseudo-)singularities.
 
pierlotx(point1, point2, point3, alpha1, alpha2, alpha3, useZ=False, **Clas_and_kwds)
3-Point resection using Pierlot's method ToTal with exact limits for the (pseudo-)singularities.
 
snellius3(a, b, degC, alpha, beta)
Snellius' surveying using Snellius Pothenot.
 
tienstra7(pointA, pointB, pointC, alpha, beta=None, gamma=None, useZ=False, **Clas_and_kwds)
3-Point resection using Tienstra's formula.
 
triAngle(a, b, c)
Compute one angle of a triangle.
 
triAngle5(a, b, c)
Compute the angles of a triangle.
 
triArea(a, b, c)
Compute the area of a triangle using Heron's stable formula.
 
triSide(a, b, radC)
Compute one side of a triangle.
 
triSide2(b, c, radB)
Compute a side and its opposite angle of a triangle.
 
triSide4(radA, radB, c)
Compute two sides and the height of a triangle.
 
wildberger3(a, b, c, alpha, beta, R3=<built-in function min>)
Snellius' surveying using Rational Trigonometry.
Variables
  __all__ = _ALL_LAZY.resections
Function Details

cassini (pointA, pointB, pointC, alpha, beta, useZ=False, **Clas_and_kwds)

 

3-Point resection using Cassini's method.

Arguments:
  • pointA - First point (Cartesian, Vector3d, Vector3Tuple, Vector4Tuple or Vector2Tuple if useZ=False).
  • pointB - Second point (Cartesian, Vector3d, Vector3Tuple, Vector4Tuple or Vector2Tuple if useZ=False).
  • pointC - Center point (Cartesian, Vector3d, Vector3Tuple, Vector4Tuple or Vector2Tuple if useZ=False).
  • alpha - Angle subtended by triangle side pointA to pointC (degrees, non-negative).
  • beta - Angle subtended by triangle side pointB to pointC (degrees, non-negative).
  • useZ - If True, use and interpolate the Z component, otherwise force z=INT0 (bool).
  • Clas_and_kwds - Optional class Clas=pointA.classof to return the survey point with optionally other Clas keyword arguments to instantiate the survey point.
Returns:
The survey point, an instance of Clas or pointA's (sub-)class.
Raises:
  • ResectionError - Near-coincident, -colinear or -concyclic points or negative or invalid alpha or beta.
  • TypeError - Invalid pointA, pointB or pointM.

Note: Typically, pointC is between pointA and pointB.

See Also: Three Point Resection Problem and functions collins5, pierlot, pierlotx and tienstra7.

collins5 (pointA, pointB, pointC, alpha, beta, useZ=False, **Clas_and_kwds)

 

3-Point resection using Collins' method.

Arguments:
  • pointA - First point (Cartesian, Vector3d, Vector3Tuple, Vector4Tuple or Vector2Tuple if useZ=False).
  • pointB - Second point (Cartesian, Vector3d, Vector3Tuple, Vector4Tuple or Vector2Tuple if useZ=False).
  • pointC - Center point (Cartesian, Vector3d, Vector3Tuple, Vector4Tuple or Vector2Tuple if useZ=False).
  • alpha - Angle subtended by triangle side b from pointA to pointC (degrees, non-negative).
  • beta - Angle subtended by triangle side a from pointB to pointC (degrees, non-negative).
  • useZ - If True, use and interpolate the Z component, otherwise force z=INT0 (bool).
  • Clas_and_kwds - Optional class Clas=pointA.classof to return the survey point with optionally other Clas keyword arguments to instantiate the survey point.
Returns:
Collins5Tuple(pointP, pointH, a, b, c) with survey pointP, auxiliary pointH, each an instance of Clas or pointA's (sub-)class and triangle sides a, b and c in meter, conventionally.
Raises:
  • ResectionError - Near-coincident, -colinear or -concyclic points or negative or invalid alpha or beta.
  • TypeError - Invalid pointA, pointB or pointM.

Note: Typically, pointC is between pointA and pointB.

See Also: Collins' methode and functions cassini, pierlot, pierlotx and tienstra7.

pierlot (point1, point2, point3, alpha12, alpha23, useZ=False, eps=2.220446049250313e-16, **Clas_and_kwds)

 

3-Point resection using Pierlot's method ToTal with approximate limits for the (pseudo-)singularities.

Arguments:
  • point1 - First point (Cartesian, Vector3d, Vector3Tuple, Vector4Tuple or Vector2Tuple if useZ=False).
  • point2 - Second point (Cartesian, Vector3d, Vector3Tuple, Vector4Tuple or Vector2Tuple if useZ=False).
  • point3 - Third point (Cartesian, Vector3d, Vector3Tuple, Vector4Tuple or Vector2Tuple if useZ=False).
  • alpha12 - Angle subtended from point1 to point2 or alpha2 - alpha1 (degrees).
  • alpha23 - Angle subtended from point2 to point3 or alpha3 - alpha2(degrees).
  • useZ - If True, interpolate the survey point's Z component, otherwise use z=INT0 (bool).
  • eps - Tolerance for cot (pseudo-)singularities (float).
  • Clas_and_kwds - Optional class Clas=point1.classof to return the survey point with optionally other Clas keyword arguments to instantiate the survey point.
Returns:
The survey (or robot) point, an instance of Clas or point1's (sub-)class.
Raises:
  • ResectionError - Near-coincident, -colinear or -concyclic points or invalid alpha12 or alpha23 or non-positive eps.
  • TypeError - Invalid point1, point2 or point3.

Note: Typically, point1, point2 and point3 are ordered by angle, modulo 360, counter-clockwise.

See Also: Pierlot's C function triangulationPierlot, V. Pierlot, M. Van Droogenbroeck, "A New Three Object Triangulation Algorithm for Mobile Robot Positioning", Vincent Pierlot, Marc Van Droogenbroeck, "18 Triangulation Algorithms for 2D Positioning (also known as the Resection Problem)" and functions pierlotx, cassini, collins5 and tienstra7.

pierlotx (point1, point2, point3, alpha1, alpha2, alpha3, useZ=False, **Clas_and_kwds)

 

3-Point resection using Pierlot's method ToTal with exact limits for the (pseudo-)singularities.

Arguments:
  • point1 - First point (Cartesian, Vector3d, Vector3Tuple, Vector4Tuple or Vector2Tuple if useZ=False).
  • point2 - Second point (Cartesian, Vector3d, Vector3Tuple, Vector4Tuple or Vector2Tuple if useZ=False).
  • point3 - Third point (Cartesian, Vector3d, Vector3Tuple, Vector4Tuple or Vector2Tuple if useZ=False).
  • alpha1 - Angle at point1 (degrees, counter-clockwise).
  • alpha2 - Angle at point2 (degrees, counter-clockwise).
  • alpha3 - Angle at point3 (degrees, counter-clockwise).
  • useZ - If True, interpolate the survey point's Z component, otherwise use z=INT0 (bool).
  • Clas_and_kwds - Optional class Clas=point1.classof to return the survey point with optionally other Clas keyword arguments to instantiate the survey point.
Returns:
The survey (or robot) point, an instance of Clas or point1's (sub-)class.
Raises:
  • ResectionError - Near-coincident, -colinear or -concyclic points or invalid alpha1, alpha2 or alpha3.
  • TypeError - Invalid point1, point2 or point3.

See Also: Pierlot's C function triangulationPierlot2 and function pierlot, cassini, collins5 and tienstra7.

snellius3 (a, b, degC, alpha, beta)

 

Snellius' surveying using Snellius Pothenot.

Arguments:
  • a - Length of the triangle side between corners B and C and opposite of triangle corner A (scalar, non-negative meter, conventionally).
  • b - Length of the triangle side between corners C and A and opposite of triangle corner B (scalar, non-negative meter, conventionally).
  • degC - Angle at triangle corner C, opposite triangle side c (non-negative degrees).
  • alpha - Angle subtended by triangle side b (non-negative degrees).
  • beta - Angle subtended by triangle side a (non-negative degrees).
Returns:
Survey3Tuple(PA, PB, PC) with distance from survey point P to each of the triangle corners A, B and C, same units as triangle sides a, b and c.
Raises:
  • TriangleError - Invalid a, b or degC or negative alpha or beta.

See Also: Function wildberger3.

tienstra7 (pointA, pointB, pointC, alpha, beta=None, gamma=None, useZ=False, **Clas_and_kwds)

 

3-Point resection using Tienstra's formula.

Arguments:
  • pointA - First point (Cartesian, Vector3d, Vector3Tuple, Vector4Tuple or Vector2Tuple if useZ=False).
  • pointB - Second point (Cartesian, Vector3d, Vector3Tuple, Vector4Tuple or Vector2Tuple if useZ=False).
  • pointC - Third point (Cartesian, Vector3d, Vector3Tuple, Vector4Tuple or Vector2Tuple if useZ=False).
  • alpha - Angle subtended by triangle side a from pointB to pointC (degrees, non-negative).
  • beta - Angle subtended by triangle side b from pointA to pointC (degrees, non-negative) or None if gamma is not None.
  • gamma - Angle subtended by triangle side c from pointA to pointB (degrees, non-negative) or None if beta is not None.
  • useZ - If True, use and interpolate the Z component, otherwise force z=INT0 (bool).
  • Clas_and_kwds - Optional class Clas=pointA.classof to return the survey point with optionally other Clas keyword arguments to instantiate the survey point.
Returns:
Tienstra7Tuple(pointP, A, B, C, a, b, c) with survey pointP, an instance of Clas or pointA's (sub-)class, with triangle angles A at pointA, B at pointB and C at pointC in degrees and with triangle sides a, b and c in meter, conventionally.
Raises:
  • ResectionError - Near-coincident, -colinear or -concyclic points or sum of alpha, beta and gamma not 360 or negative alpha, beta or gamma.
  • TypeError - Invalid pointA, pointB or pointC.

Note: Points pointA, pointB and pointC are ordered clockwise.

See Also: 3-Point Resection Solver, V. Pierlot, M. Van Droogenbroeck, "A New Three Object Triangulation...", 18 Triangulation Algorithms... and functions cassini, collins5, pierlot and pierlotx.

triAngle (a, b, c)

 

Compute one angle of a triangle.

Arguments:
  • a - Adjacent triangle side length (scalar, non-negative meter, conventionally).
  • b - Adjacent triangle side length (scalar, non-negative meter, conventionally).
  • c - Opposite triangle side length (scalar, non-negative meter, conventionally).
Returns:
Angle in radians at triangle corner C, opposite triangle side c.
Raises:

See Also: Functions triAngle5 and triSide.

triAngle5 (a, b, c)

 

Compute the angles of a triangle.

Arguments:
  • a - Length of the triangle side opposite of triangle corner A (scalar, non-negative meter, conventionally).
  • b - Length of the triangle side opposite of triangle corner B (scalar, non-negative meter, conventionally).
  • c - Length of the triangle side opposite of triangle corner C (scalar, non-negative meter, conventionally).
Returns:
TriAngle5Tuple(radA, radB, radC, rIn, area) with angles radA, radB and radC at triangle corners A, B and C, all in radians, the InCircle radius rIn aka inradius, same units as triangle sides a, b and c and the triangle area in those same units squared.
Raises:

See Also: Functions triAngle and triArea.

triArea (a, b, c)

 

Compute the area of a triangle using Heron's stable formula.

Arguments:
  • a - Length of the triangle side opposite of triangle corner A (scalar, non-negative meter, conventionally).
  • b - Length of the triangle side opposite of triangle corner B (scalar, non-negative meter, conventionally).
  • c - Length of the triangle side opposite of triangle corner C (scalar, non-negative meter, conventionally).
Returns:
The triangle area (float, conventionally meter or same units as a, b and c squared).
Raises:

triSide (a, b, radC)

 

Compute one side of a triangle.

Arguments:
  • a - Adjacent triangle side length (scalar, non-negative meter, conventionally).
  • b - Adjacent triangle side length (scalar, non-negative meter, conventionally).
  • radC - Angle included by sides a and b, opposite triangle side c (radians).
Returns:
Length of triangle side c, opposite triangle corner C and angle radC, same units as a and b.
Raises:

See Also: Functions sqrt_a, triAngle, triSide2 and triSide4.

triSide2 (b, c, radB)

 

Compute a side and its opposite angle of a triangle.

Arguments:
  • b - Adjacent triangle side length (scalar, non-negative meter, conventionally).
  • c - Adjacent triangle side length (scalar, non-negative meter, conventionally).
  • radB - Angle included by sides a and c, opposite triangle side b (radians).
Returns:
TriSide2Tuple(a, radA) with triangle angle radA in radians and length of the opposite triangle side a, same units as b and c.
Raises:

See Also: Functions sqrt_a, triSide and triSide4.

triSide4 (radA, radB, c)

 

Compute two sides and the height of a triangle.

Arguments:
  • radA - Angle at triangle corner A, opposite triangle side a (non-negative radians).
  • radB - Angle at triangle corner B, opposite triangle side b (non-negative radians).
  • c - Length of triangle side between triangle corners A and B, (scalar, non-negative meter, conventionally).
Returns:
TriSide4Tuple(a, b, radC, d) with triangle sides a and b and triangle height d perpendicular to triangle side c, all in the same units as c and interior angle radC in radians at triangle corner C, opposite triangle side c.
Raises:

See Also: Triangulation, Surveying and functions sqrt_a, triSide and triSide2.

wildberger3 (a, b, c, alpha, beta, R3=<built-in function min>)

 

Snellius' surveying using Rational Trigonometry.

Arguments:
  • a - Length of the triangle side between corners B and C and opposite of triangle corner A (scalar, non-negative meter, conventionally).
  • b - Length of the triangle side between corners C and A and opposite of triangle corner B (scalar, non-negative meter, conventionally).
  • c - Length of the triangle side between corners A and B and opposite of triangle corner C (scalar, non-negative meter, conventionally).
  • alpha - Angle subtended by triangle side b (degrees, non-negative).
  • beta - Angle subtended by triangle side a (degrees, non-negative).
  • R3 - Callable to determine R3 from (R3 - C)**2 = D, typically standard Python function min or max, invoked with 2 arguments.
Returns:
Survey3Tuple(PA, PB, PC) with distance from survey point P to each of the triangle corners A, B and C, same units as a, b and c.
Raises:
  • TriangleError - Invalid a, b or c or negative alpha or beta or R3 not callable.

See Also: Wildberger, Norman J., Devine Proportions, page 252 and function snellius3.


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