Package pygeodesy :: Module vector2d

Module vector2d

2- or 3-D vectorial functions circin6, circum3, circum4_, iscolinearWith, meeus2, radii11, soddy4, triaxum5 and trilaterate2d2.

Note: Functions circin6, circum3, circum4_, soddy4 and triaxum5 require numpyhttps://PyPI.org/project/numpy> version 1.10 or newer to be installed.

Version: 25.05.26

Classes
	Circin6Tuple 6-Tuple `(radius, center, deltas, cA, cB, cC)` with the `radius`, the trilaterated `center` and contact points of the inscribed aka In- circle of a triangle.
	Circum3Tuple 3-Tuple `(radius, center, deltas)` with the `circumradius` and trilaterated `circumcenter` of the `circumcircle` through 3 points (aka {Meeus}' Type II circle) or the `radius` and `center` of the smallest Meeus' Type I circle.
	Circum4Tuple 4-Tuple `(radius, center, rank, residuals)` with `radius` and `center` of a sphere least-squares fitted through given points and the `rank` and `residuals` -if any- from numpy.linalg.lstsq.
	Meeus2Tuple 2-Tuple `(radius, Type)` with `radius` and Meeus' `Type` of the smallest circle containing 3 points.
	Radii11Tuple 11-Tuple `(rA, rB, rC, cR, rIn, riS, roS, a, b, c, s)` with the `Tangent` circle radii `rA`, `rB` and `rC`, the `circumradius` `cR`, the `Incircle` radius `rIn` aka `inradius`, the inner and outer Soddy circle radii `riS` and `roS`, the sides `a`, `b` and `c` and semi-perimeter `s` of a triangle, all in `meter` conventionally.
	Soddy4Tuple 4-Tuple `(radius, center, deltas, outer)` with `radius` and (trilaterated) `center` of the inner Soddy circle and the radius of the `outer` Soddy circle.
	Triaxum5Tuple 5-Tuple `(a, b, c, rank, residuals)` with the (unordered) triaxial radii `a`, `b` and `c` of an ellipsoid least-squares fitted through given points and the `rank` and `residuals` -if any- from numpy.linalg.lstsq.

Functions

circin6(point1, point2, point3, eps=8.881784197001252e-16, useZ=True)
Return the radius and center of the inscribed aka Incircle of a (2- or 3-D) triangle.

circum3(point1, point2, point3, circum=True, eps=8.881784197001252e-16, useZ=True)
Return the radius and center of the smallest circle through or containing three (2- or 3-D) points.

circum4(points, useZ=True, **Vector_and_kwds)
Best-fit a sphere through three or more (3-D) points.

circum4_(*points, **useZ_Vector_and_kwds)
Best-fit a sphere through three or more (3-D) positional points.

meeus2(point1, point2, point3, circum=False, useZ=True)
Return the radius and Meeus' Type of the smallest circle through or containing three (2- or 3-D) points.

radii11(point1, point2, point3, useZ=True)
Return the radii of the In-, Soddy and Tangent circles of a (2- or 3-D) triangle.

soddy4(point1, point2, point3, eps=8.881784197001252e-16, useZ=True)
Return the radius and center of the inner Soddy circle of a (2- or 3-D) triangle.

triaxum5(points, useZ=True)
Best-fit a triaxial ellipsoid through three or more (3-D) points.

trilaterate2d2(x1, y1, radius1, x2, y2, radius2, x3, y3, radius3, eps=None, **Vector_and_kwds)
Trilaterate three circles, each given as a (2-D) center and a radius.

Variables
	__all__ = `_ALL_LAZY.vector2d`

Function Details

circin6 (point1, point2, point3, eps=8.881784197001252e-16, useZ=True)

Return the radius and center of the inscribed aka Incircle of a (2- or 3-D) triangle.

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).
eps - Tolerance for function pygeodesy.trilaterate3d2 if useZ is True else pygeodesy.trilaterate2d2.
useZ - If True, use the Z components, otherwise force z=INT0 (bool).

Returns:

Circin6Tuple

(radius, center, deltas, cA, 
          cB, cC)

. The center and contact points cA, cB and cC, each an instance of point1's (sub-)class, are co-planar with the three given points.

Raises:

ImportError - Package numpy not found, not installed or older than version 1.10 and useZ is True.
IntersectionError - Near-coincident or -colinear points or a trilateration or numpy issue.
TypeError - Invalid point1, point2 or point3.

See Also: Functions radii11 and circum3, Contact Triangle and Incircle.

circum3 (point1, point2, point3, circum=True, eps=8.881784197001252e-16, useZ=True)

Return the radius and center of the smallest circle through or containing three (2- or 3-D) points.

Arguments:

point1 - First point (Cartesian, Vector3d, Vector3Tuple or Vector4Tuple).
point2 - Second point (Cartesian, Vector3d, Vector3Tuple or Vector4Tuple).
point3 - Third point (Cartesian, Vector3d, Vector3Tuple or Vector4Tuple).
circum - If True, return the circumradius and circumcenter always, ignoring the Meeus' Type I case (bool).
eps - Tolerance for function pygeodesy.trilaterate3d2 if useZ is True else pygeodesy.trilaterate2d2.
useZ - If True, use the Z components, otherwise force z=INT0 (bool).

Returns:

A Circum3Tuple

(radius, center, 
          deltas)

. The center, an instance of point1's (sub-)class, is co-planar with the three given points.

Raises:

ImportError - Package numpy not found, not installed or older than version 1.10 and useZ is True.
IntersectionError - Near-coincident or -colinear points or a trilateration or numpy issue.
TypeError - Invalid point1, point2 or point3.

See Also: Functions pygeodesy.circum4_ and pygeodesy.meeus2 and Meeus, J. Astronomical Algorithms, 2nd Ed. 1998, page 127ff, circumradius and circumcircle.

circum4 (points, useZ=True, **Vector_and_kwds)

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

Arguments:

points - Iterable of points (each a Cartesian, Vector3d, Vector3Tuple or Vector4Tuple).
useZ - If True, use the points' Z component, otherwise force z=INT0 (bool).
Vector_and_kwds - Optional class Vector=None to return the center point and optionally, additional Vector keyword arguments, otherwise the points' (sub-)class.

Returns:

Circum4Tuple

(radius, center, rank, 
          residuals)

with center an instance of points[0]' (sub-)class or Vector if specified.

Raises:

ImportError - Package numpy not found, not installed or older than version 1.10.
NumPyError - Some numpy issue.
PointsError - Too few points.
TypeError - One of the points is invalid.

See Also: Functions pygeodesy.circum3 and pygeodesy.meeus2, Charles Jekel's "Least Squares Sphere Fit", Appendix A, numpy.linalg.lstsq and Eberly 6.

circum4_ (*points, **useZ_Vector_and_kwds)

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

Arguments:

points - The points (each a Cartesian, Vector3d, Vector3Tuple or Vector4Tuple), all positional.
useZ_Vector_and_kwds - Keyword arguments useZ=True and Vector=None, see function circum4.

See Also: Function circum4 for further details.

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

Return the radius and Meeus' Type of the smallest circle through or containing three (2- or 3-D) points.

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).
circum - If True, return the circumradius and circumcenter always, overriding Meeus' Type II case (bool).
useZ - If True, use the Z components, otherwise force z=INT0 (bool).

Returns:

Meeus2Tuple(radius, Type), with Type the circumcenter iff circum=True.

Raises:

IntersectionError - Near-coincident or -colinear points, iff circum=True.
TypeError - Invalid point1, point2 or point3.

See Also: Functions pygeodesy.circum3 and pygeodesy.circum4_ and Meeus, J. Astronomical Algorithms, 2nd Ed. 1998, page 127ff, circumradius and circumcircle.

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

Return the radii of the In-, Soddy and Tangent circles of a (2- or 3-D) triangle.

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).
useZ - If True, use the Z components, otherwise force z=INT0 (bool).

Returns:

Radii11Tuple

(rA, rB, rC, cR, rIn, riS, 
          roS, a, b, c, s)

Raises:

TriangleError - Near-coincident or -colinear points.
TypeError - Invalid point1, point2 or point3.

See Also: Circumradius, Incircle, Soddy Circles and Tangent Circles.

soddy4 (point1, point2, point3, eps=8.881784197001252e-16, useZ=True)

Return the radius and center of the inner Soddy circle of a (2- or 3-D) triangle.

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).
eps - Tolerance for function pygeodesy.trilaterate3d2 if useZ is True otherwise pygeodesy.trilaterate2d2.
useZ - If True, use the Z components, otherwise force z=INT0 (bool).

Returns:

Soddy4Tuple

(radius, center, deltas, 
          outer)

. The center, an instance of point1's (sub-)class, is co-planar with the three given points. The outer Soddy radius may be INF.

Raises:

ImportError - Package numpy not found, not installed or older than version 1.10 and useZ is True.
IntersectionError - Near-coincident or -colinear points or a trilateration or numpy issue.
TypeError - Invalid point1, point2 or point3.

See Also: Functions radii11 and circum3 and Soddy Circles.

triaxum5 (points, useZ=True)

Best-fit a triaxial ellipsoid through three or more (3-D) points.

Arguments:

points - Iterable of points (each a Cartesian, Vector3d, Vector3Tuple or Vector4Tuple).
useZ - If True, use the points' Z component, otherwise force z=INT0 (bool).

Returns:

Triaxum5Tuple

(a, b, c, rank, 
          residuals)

with the unordered triaxial radii a, b and c in meter, same units as the points' coordinates.

Raises:

ImportError - Package numpy not found, not installed or older than version 1.10.
NumPyError - Some numpy issue.
PointsError - Too few points.
TypeError - One of the points is invalid.

See Also: Charles Jekel's "Least Squares Ellipsoid Fit" and numpy.linalg.lstsq.

trilaterate2d2 (x1, y1, radius1, x2, y2, radius2, x3, y3, radius3, eps=None, **Vector_and_kwds)

Trilaterate three circles, each given as a (2-D) center and a radius.

Arguments:

x1 - Center x coordinate of the 1st circle (scalar).
y1 - Center y coordinate of the 1st circle (scalar).
radius1 - Radius of the 1st circle (scalar).
x2 - Center x coordinate of the 2nd circle (scalar).
y2 - Center y coordinate of the 2nd circle (scalar).
radius2 - Radius of the 2nd circle (scalar).
x3 - Center x coordinate of the 3rd circle (scalar).
y3 - Center y coordinate of the 3rd circle (scalar).
radius3 - Radius of the 3rd circle (scalar).
eps - Tolerance to check the trilaterated point delta on all 3 circles (scalar) or None for no checking.
Vector_and_kwds - Optional class Vector=None to return the trilateration and optionally, additional Vector keyword arguments).

Returns:

Trilaterated point as

Vector(x, y, 
          **Vector_kwds)

or Vector2Tuple(x, y) if Vector is None.

Raises:

IntersectionError - No intersection, near-concentric or -colinear centers, trilateration failed some other way or the trilaterated point is off one circle by more than eps.
UnitError - Invalid radius1, radius2 or radius3.