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PyGeodesy
Package pygeodesy :: Module vector3d
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Module vector3d

Extended 3-D vector class Vector3d and functions.

Function intersection3d3, intersections2, iscolinearWith, nearestOn, parse3d, sumOf, trilaterate2d2 and trilaterate3d2.

Version: 21.08.31

Classes
  Vector3dBase
(INTERNAL) Generic 3-D vector base class.
  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 and 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.
  Vector3d
Extended 3-D vector.
Functions
 
circin6(point1, point2, point3, eps=8.881784197e-16, useZ=True)
Return the radius and center of the inscribed aka In- circle of a (2- or 3-D) triangle.
 
circum3(point1, point2, point3, circum=True, eps=8.881784197e-16, useZ=True)
Return the radius and center of the smallest circle through or containing three (2- or 3-D) points.
 
circum4_(*points, **Vector_and_kwds)
Best-fit a sphere through three or more (3-D) points.
 
intersection3d3(start1, end1, start2, end2, eps=2.22044604925e-16, useZ=True, **Vector_and_kwds)
Compute the intersection point of two lines, each defined by or through a start and end point (3-D).
 
intersections2(center1, radius1, center2, radius2, sphere=True, **Vector_and_kwds)
Compute the intersection of two spheres or circles, each defined by a (3-D) center point and a radius.
 
iscolinearWith(point, point1, point2, eps=2.22044604925e-16)
Check whether a point is colinear with two other (3-D) points.
 
meeus2(point1, point2, point3, circum=False, useZ=True)
Return the radius and Meeus' Type of the smallest circle through or containing three (3-D) points.
 
parse3d(str3d, sep=',', Vector=<class 'pygeodesy.vector3d.Vector3d'>, **Vector_kwds)
Parse an "x, y, z" string.
 
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=8.881784197e-16, useZ=True)
Return the radius and center of the inner Soddy circle of a (2- or 3-D) triangle.
 
sumOf(vectors, Vector=<class 'pygeodesy.vector3d.Vector3d'>, **Vector_kwds)
Compute the vectorial sum of several vectors.
 
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.
 
trilaterate3d2(center1, radius1, center2, radius2, center3, radius3, eps=2.22044604925e-16, **Vector_and_kwds)
Trilaterate three spheres, each given as a (3-D) center and a radius.
Variables
  __all__ = _ALL_LAZY.vector3d
Function Details

circin6 (point1, point2, point3, eps=8.881784197e-16, useZ=True)

 

Return the radius and center of the inscribed aka In- 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 trilaterate3d2 if useZ is True else trilaterate2d2.
  • useZ - If True, use the Z components, otherwise force z=0 (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, Incircle and Contact Triangle.

circum3 (point1, point2, point3, circum=True, eps=8.881784197e-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 trilaterate3d2 if useZ is True else trilaterate2d2.
  • useZ - If True, use the Z components, otherwise force z=0 (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: Jean Meeus, "Astronomical Algorithms", 2nd Ed. 1998, page 127ff, circumradius, circumcircle and functions pygeodesy.circum4_ and pygeodesy.meeus2.

circum4_ (*points, **Vector_and_kwds)

 

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

Arguments:
  • points - The points (each a Cartesian, Vector3d, Vector3Tuple, or Vector4Tuple).
  • Vector_and_kwds - Optional class Vector=None to return the center and optional, additional Vector keyword arguments, otherwise the first 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: Charles F. Jekel, "Least Squares Sphere Fit", Sep 13, 2015 and Appendix A, numpy.linalg.lstsq, Eberly 6 and functions pygeodesy.circum3 and pygeodesy.meeus2.

intersection3d3 (start1, end1, start2, end2, eps=2.22044604925e-16, useZ=True, **Vector_and_kwds)

 

Compute the intersection point of two lines, each defined by or through a start and end point (3-D).

Arguments:
  • start1 - Start point of the first line (Cartesian, Vector3d, Vector3Tuple or Vector4Tuple).
  • end1 - End point of the first line (Cartesian, Vector3d, Vector3Tuple or Vector4Tuple).
  • start2 - Start point of the second line (Cartesian, Vector3d, Vector3Tuple or Vector4Tuple).
  • end2 - End point of the second line (Cartesian, Vector3d, Vector3Tuple or Vector4Tuple).
  • eps - Tolerance for skew line distance and length (EPS).
  • useZ - If True, use the Z components, otherwise force z=0 (bool).
  • Vector_and_kwds - Optional class Vector=None to return the intersection points and optional, additional Vector keyword arguments, otherwise start1's (sub-)class.
Returns:
An Intersection3Tuple(point, outside1, outside2) with point an instance of Vector or start1's (sub-)class.
Raises:
  • IntersectionError - Invalid, skew, non-co-planar or otherwise non-intersecting lines.

See Also: Line-line intersection and line-line distance, skew lines and point-line distance.

intersections2 (center1, radius1, center2, radius2, sphere=True, **Vector_and_kwds)

 

Compute the intersection of two spheres or circles, each defined by a (3-D) center point and a radius.

Arguments:
  • center1 - Center of the first sphere or circle (Cartesian, Vector3d, Vector3Tuple or Vector4Tuple).
  • radius1 - Radius of the first sphere or circle (same units as the center1 coordinates).
  • center2 - Center of the second sphere or circle (Cartesian, Vector3d, Vector3Tuple or Vector4Tuple).
  • radius2 - Radius of the second sphere or circle (same units as the center1 and center2 coordinates).
  • sphere - If True compute the center and radius of the intersection of two spheres. If False, ignore the z-component and compute the intersection of two circles (bool).
  • Vector_and_kwds - Optional class Vector=None to return the intersection points and optional, additional Vector keyword arguments, otherwise center1's (sub-)class.
Returns:
If sphere is True, a 2-tuple of the center and radius of the intersection of the spheres. The radius is 0.0 for abutting spheres (and the center is aka radical center).

If sphere is False, a 2-tuple with the two intersection points of the circles. For abutting circles, both points are the same instance, aka radical center.

Raises:
  • IntersectionError - Concentric, invalid or non-intersecting spheres or circles.
  • TypeError - Invalid center1 or center2.
  • UnitError - Invalid radius1 or radius2.

See Also: Sphere-Sphere and Circle-Circle Intersection.

iscolinearWith (point, point1, point2, eps=2.22044604925e-16)

 

Check whether a point is colinear with two other (3-D) points.

Arguments:
  • point - The point (Vector3d, Vector3Tuple or Vector4Tuple).
  • point1 - First point (Vector3d, Vector3Tuple or Vector4Tuple).
  • point2 - Second point (Vector3d, Vector3Tuple or Vector4Tuple).
  • eps - Tolerance (scalar), same units as x, y and z.
Returns:
True if point is colinear point1 and point2, False otherwise.
Raises:
  • TypeError - Invalid point, point1 or point2.

See Also: Function vector3d.nearestOn.

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

 

Return the radius and Meeus' Type of the smallest circle through or containing three (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 non-zero circumradius always, ignoring the Meeus' Type I case (bool).
  • useZ - If True, use the Z components, otherwise force z=0 (bool).
Returns:
Meeus2Tuple(radius, Type).
Raises:
  • IntersectionError - Near-coincident or colinear points, iff circum=True.
  • TypeError - Invalid point1, point2 or point3.

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

parse3d (str3d, sep=',', Vector=<class 'pygeodesy.vector3d.Vector3d'>, **Vector_kwds)

 

Parse an "x, y, z" string.

Arguments:
  • str3d - X, y and z values (str).
  • sep - Optional separator (str).
  • Vector - Optional class (Vector3d).
  • Vector_kwds - Optional Vector keyword arguments, ignored if Vector is None.
Returns:
New Vector or if Vector is None, a named Vector3Tuple(x, y, z).
Raises:

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=0 (bool).
Returns:
Radii11Tuple(rA, rB, rC, cR, rIn, riS, roS, a, b, c, s).
Raises:
  • IntersectionError - 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=8.881784197e-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 trilaterate3d2 if useZ is True else trilaterate2d2.
  • useZ - If True, use the Z components, otherwise force z=0 (bool).
Returns:
Soddy4Tuple(radius, center, deltas, outer). 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 radii11 and circum3.

sumOf (vectors, Vector=<class 'pygeodesy.vector3d.Vector3d'>, **Vector_kwds)

 

Compute the vectorial sum of several vectors.

Arguments:
  • vectors - Vectors to be added (Vector3d[]).
  • Vector - Optional class for the vectorial sum (Vector3d).
  • Vector_kwds - Optional Vector keyword arguments, ignored if Vector is None.
Returns:
Vectorial sum as Vector or if Vector is None, a named Vector3Tuple(x, y, z).
Raises:

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 - Check the trilaterated point delta on all 3 circles (scalar) or None.
  • Vector_and_kwds - Optional class Vector=None to return the trilateration and optional, additional Vector keyword arguments, otherwise (Vector3d).
Returns:
Trilaterated point as Vector(x, y, **Vector_kwds) or Vector2Tuple(x, y) if Vector is None..
Raises:
  • IntersectionError - No intersection, colinear or near-concentric centers, trilateration failed some other way or the trilaterated point is off one circle by more than eps.
  • UnitError - Invalid radius1, radius2 or radius3.

See Also: Issue #49, Find X location using 3 known (X,Y) location using trilateration and trilaterate3d2.

trilaterate3d2 (center1, radius1, center2, radius2, center3, radius3, eps=2.22044604925e-16, **Vector_and_kwds)

 

Trilaterate three spheres, each given as a (3-D) center and a radius.

Arguments:
  • center1 - Center of the 1st sphere (Cartesian, Vector3d, Vector3Tuple or Vector4Tuple).
  • radius1 - Radius of the 1st sphere (same units as x, y and z).
  • center2 - Center of the 2nd sphere (Cartesian, Vector3d, Vector3Tuple or Vector4Tuple).
  • radius2 - Radius of this sphere (same units as x, y and z).
  • center3 - Center of the 3rd sphere (Cartesian, Vector3d, Vector3Tuple or Vector4Tuple).
  • radius3 - Radius of the 3rd sphere (same units as x, y and z).
  • eps - Tolerance (scalar), same units as x, y, and z.
  • Vector_and_kwds - Optional class Vector=None to return the trilateration and optional, additional Vector keyword arguments, otherwise center1's (sub-)class.
Returns:
2-Tuple with two trilaterated points, each a Vector instance. Both points are the same instance if all three spheres abut/intersect in a single point.
Raises:
  • ImportError - Package numpy not found, not installed or older than version 1.10.
  • IntersectionError - Near-concentric, colinear, too distant or non-intersecting spheres.
  • NumPyError - Some numpy issue.
  • TypeError - Invalid center1, center2 or center3.
  • UnitError - Invalid radius1, radius2 or radius3.

See Also: Norrdine, A. An Algebraic Solution to the Multilateration Problem and implementation and trilaterate2d2.


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