# Coverage for pygeodesy/vector3d.py : 92%

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1422 1423 1424 1425 1426 1427 1428 1429 1430 1431 1432 1433 1434 1435 1436 1437 1438 1439 1440 1441 1442 1443 1444 1445 1446 1447 1448 1449 1450 1451 1452 1453 1454 1455 1456 1457 1458 1459 1460 1461 1462 1463 1464 1465 1466 1467 1468 1469 1470 1471 1472 1473 1474 1475 1476 1477 1478 1479 1480 1481 1482 1483 1484 1485 1486 1487 1488 1489 1490 1491 1492 1493 1494 1495 1496 1497 1498 1499 1500 1501 1502 1503 1504 1505 1506 1507 1508 1509 1510 1511 1512 1513 1514 1515 1516 1517 1518 1519 1520 1521 1522 1523 1524 1525 1526 1527 1528 1529 1530 1531 1532 1533 1534 1535 1536 1537 1538 1539 | # -*- coding: utf-8 -*- u'''Extended 3-D vector class L{Vector3d} and functions. Function L{intersection3d3}, L{intersections2}, L{iscolinearWith}, L{nearestOn}, L{parse3d}, L{sumOf}, L{trilaterate2d2} and L{trilaterate3d2}. ''' from pygeodesy.basics import isnear0, len2, map1, map2, _xnumpy from pygeodesy.errors import _and, _AssertionError, IntersectionError, \ NumPyError, PointsError, _TypeError, _ValueError, \ VectorError, _xError, _xkwds, _xkwds_popitem from pygeodesy.fmath import fdot, fsum, fsum_, fsum1_, hypot, hypot2_ from pygeodesy.formy import _radical2 from pygeodesy.interns import EPS, EPS0, EPS02, EPS1, EPS4, INF, MISSING, NN, \ _EPS4e8, _and_, _center_, _coincident_, _colinear_, \ _concentric_, _COMMA_, _COMMASPACE_, _datum_, _few_, \ _h_, _height_, _intersection_, _invalid_, _name_, \ _near_, _negative_, _no_, _radius_, _s_, _SPACE_, \ _too_, _xyz_, _y_, _z_, _0_0, _0_5, _1_0, \ _1_0_T, _2_0, _4_0 from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY from pygeodesy.named import _NamedTuple, _Pass, _xnamed, _xotherError from pygeodesy.namedTuples import Intersection3Tuple, LatLon3Tuple, Vector2Tuple, \ Vector3Tuple # Vector4Tuple from pygeodesy.streprs import Fmt from pygeodesy.units import Float, Int, Meter, Radius, Radius_ from pygeodesy.vector3dBase import Vector3dBase from contextlib import contextmanager from math import sqrt __all__ = _ALL_LAZY.vector3d __version__ = '21.08.31' _cA_ = 'cA' _cB_ = 'cB' _cC_ = 'cC' _deltas_ = 'deltas' _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{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{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{trilaterate3d2} results. @note: The outer I{Soddy} radius C{outer} may be C{INF}. ''' _Names_ = (_radius_, _center_, _deltas_, _outer_) _Units_ = ( Radius, _Pass, _Pass, Radius) class Vector3d(Vector3dBase): '''Extended 3-D vector. In a geodesy context, these may be used to represent: - n-vector representing a normal to a point on earth's surface - earth-centered, earth-fixed cartesian (ECEF) - great circle normal to vector - motion vector on earth's surface - etc. ''' _numpy = None # module numpy imported once by function _numpy below def circin6(self, point2, point3, eps=EPS4): '''Return the radius and center of the I{inscribed} aka I{In- circle} of a (3-D) triangle formed by this and two other points. @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{trilaterate3d2} if C{B{useZ} is True} else L{trilaterate2d2}. @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 this (sub-)class, are co-planar with this and the two given points. @raise ImportError: Package C{numpy} not found, not installed or older than version 1.10. @raise IntersectionError: Near-coincident or colinear points or a trilateration or C{numpy} issue. @raise TypeError: Invalid B{C{point2}} or B{C{point3}}. @see: Function L{pygeodesy.circin6}, U{Incircle <https://MathWorld.Wolfram.com/Incircle.html>} and U{Contact Triangle<https://MathWorld.Wolfram.com/ContactTriangle.html>}. ''' try: return _circin6(self, point2, point3, eps=eps, useZ=True) except (AssertionError, TypeError, ValueError) as x: raise _xError(x, point=self, point2=point2, point3=point3) def circum3(self, point2, point3, circum=True, eps=EPS4): '''Return the radius and center of the smallest circle I{through} or I{containing} this and two other (3-D) points. @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 passed to function L{trilaterate3d2}. @return: A L{Circum3Tuple}C{(radius, center, deltas)}. The C{center}, an instance of this (sub-)class, is co-planar with this and the two given points. @raise ImportError: Package C{numpy} not found, not installed or older than version 1.10. @raise IntersectionError: Near-concentric, coincident or colinear points or a trilateration or C{numpy} issue. @raise TypeError: Invalid B{C{point2}} or B{C{point3}}. @see: Function L{pygeodesy.circum3} and methods L{circum4_} and L{meeus2}. ''' try: return _circum3(self, point2, point3, circum=circum, eps=eps, useZ=True, clas=self.classof) except (AssertionError, TypeError, ValueError) as x: raise _xError(x, point=self, point2=point2, point3=point3, circum=circum) def circum4_(self, *points): '''Best-fit a sphere through this and two or more other (3-D) points. @arg points: Other points (each a C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @return: L{Circum4Tuple}C{(radius, center, rank, residuals)} with C{center} an instance if this (sub-)class. @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}} invalid. @see: Function L{pygeodesy.circum4_} and methods L{circum3} and L{meeus2}. ''' return circum4_(self, *points, Vector=self.classof) def iscolinearWith(self, point1, point2, eps=EPS): '''Check whether this and two other (3-D) points are colinear. @arg point1: One point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg point2: An other point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @kwarg eps: Tolerance (C{scalar}), same units as C{x}, C{y}, and C{z}. @return: C{True} if this point is colinear with B{C{point1}} and B{C{point2}}, C{False} otherwise. @raise TypeError: Invalid B{C{point1}} or B{C{point2}}. @see: Method L{nearestOn}. ''' v = self if self.name else _otherV3d(NN_OK=False, this=self) return _iscolinearWith(v, point1, point2, eps=eps) def meeus2(self, point2, point3, circum=False): '''Return the radius and I{Meeus}' Type of the smallest circle I{through} or I{containing} this and two other (3-D) points. @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}). @return: L{Meeus2Tuple}C{(radius, Type)}. @raise IntersectionError: Coincident or colinear points, iff C{B{circum}=True}. @raise TypeError: Invalid B{C{point2}} or B{C{point3}}. @see: Function L{pygeodesy.meeus2} and methods L{circum3} and L{circum4_}. ''' try: r, t, _, _ = _meeus4(self, point2, point3, circum=circum, clas=self.classof) except (TypeError, ValueError) as x: raise _xError(x, point=self, point2=point2, point3=point3, circum=circum) return Meeus2Tuple(r, t) def nearestOn(self, point1, point2, within=True): '''Locate the point between two points closest to this point. @arg point1: Start point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg point2: End point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @kwarg within: If C{True} return the closest point between the given points, otherwise the closest point on the extended line through both points (C{bool}). @return: Closest point, either B{C{point1}} or B{C{point2}} or an instance of this (sub-)class. @raise TypeError: Invalid B{C{point1}} or B{C{point2}}. @see: Method L{sphericalTrigonometry.LatLon.nearestOn3} and U{3-D Point-Line distance<https://MathWorld.Wolfram.com/ Point-LineDistance3-Dimensional.html>}. ''' return nearestOn(self, point1, point2, within=within) def parse(self, str3d, sep=_COMMA_, name=NN): '''Parse an C{"x, y, z"} string to a L{Vector3d} instance. @arg str3d: X, y and z string (C{str}), see function L{parse3d}. @kwarg sep: Optional separator (C{str}). @kwarg name: Optional instance name (C{str}), overriding this name. @return: The instance (L{Vector3d}). @raise VectorError: Invalid B{C{str3d}}. ''' return parse3d(str3d, sep=sep, Vector=self.classof, name=name or self.name) def radii11(self, point2, point3): '''Return the radii of the C{Circum-}, C{In-}, I{Soddy} and C{Tangent} circles of a (3-D) triangle. @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}). @return: L{Radii11Tuple}C{(rA, rB, rC, cR, rIn, riS, roS, a, b, c, s)}. @raise IntersectionError: Near-coincident or colinear points. @raise TypeError: Invalid B{C{point2}} or B{C{point3}}. @see: Function L{pygeodesy.radii11}, 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(self, point2, point3, useZ=True)[0] except (TypeError, ValueError) as x: raise _xError(x, point=self, point2=point2, point3=point3) def soddy4(self, point2, point3, eps=EPS4): '''Return the radius and center of the C{inner} I{Soddy} circle of a (3-D) triangle. @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{trilaterate3d2} if C{B{useZ} is True} else L{trilaterate2d2}. @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. @raise IntersectionError: Near-coincident or colinear points or a trilateration or C{numpy} issue. @raise TypeError: Invalid B{C{point2}} or B{C{point3}}. @see: Function L{pygeodesy.soddy4}. ''' return soddy4(self, point2, point3, eps=eps, useZ=True) def trilaterate2d2(self, radius, center2, radius2, center3, radius3, eps=EPS, z=0): '''Trilaterate this and two other circles, each given as a (2-D) center and a radius. @arg radius: Radius of this circle (same C{units} as this C{x} and C{y}. @arg center2: Center of the 2nd circle (C{Cartesian}, L{Vector3d}, C{Vector2Tuple}, C{Vector3Tuple} or C{Vector4Tuple}). @arg radius2: Radius of this circle (same C{units} as this C{x} and C{y}. @arg center3: Center of the 3rd circle (C{Cartesian}, L{Vector3d}, C{Vector2Tuple}, C{Vector3Tuple} or C{Vector4Tuple}). @arg radius3: Radius of the 3rd circle (same C{units} as this C{x} and C{y}. @kwarg eps: Check the trilaterated point I{delta} on all 3 circles (C{scalar}) or C{None}. @kwarg z: Optional Z component of the trilaterated point (C{scalar}). @return: Trilaterated point, an instance of this (sub-)class with C{z=B{z}}. @raise IntersectionError: No intersection, colinear or near-concentric centers, trilateration failed some other way or the trilaterated point is off one circle by more than B{C{eps}}. @raise TypeError: Invalid B{C{center2}} or B{C{center3}}. @raise UnitError: Invalid B{C{radius1}}, B{C{radius2}} or B{C{radius3}}. @see: Function L{pygeodesy.trilaterate2d2}. ''' def _xyr3(r, **name_v): v = _otherV3d(useZ=False, **name_v) return v.x, v.y, r try: return _trilaterate2d2(*(_xyr3(radius, center=self) + _xyr3(radius2, center2=center2) + _xyr3(radius3, center3=center3)), eps=eps, Vector=self.classof, z=z) except (AssertionError, TypeError, ValueError) as x: raise _xError(x, center=self, radius=radius, center2=center2, radius2=radius2, center3=center3, radius3=radius3) def trilaterate3d2(self, radius, center2, radius2, center3, radius3, eps=EPS): '''Trilaterate this and two other spheres, each given as a (3-D) center and a radius. @arg radius: Radius of this sphere (same C{units} as this C{x}, C{y} and C{z}). @arg center2: Center of the 2nd sphere (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg radius2: Radius of this sphere (same C{units} as this C{x}, C{y} and C{z}). @arg center3: Center of the 3rd sphere (C{Cartesian}, , L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg radius3: Radius of the 3rd sphere (same C{units} as this C{x}, C{y} and C{z}). @kwarg eps: Tolerance (C{scalar}), same units as C{x}, C{y}, and C{z}. @return: 2-Tuple with two trilaterated points, each an instance of this (sub-)class. Both points are the same instance if all three spheres intersect or abut in a single point. @raise ImportError: Package C{numpy} not found, not installed or older than version 1.10. @raise IntersectionError: Near-concentric, colinear, too distant or non-intersecting spheres or C{numpy} issue. @raise TypeError: Invalid B{C{center2}} or B{C{center3}}. @raise UnitError: Invalid B{C{radius}}, B{C{radius2}} or B{C{radius3}}. @note: Package U{numpy<https://pypi.org/project/numpy>} is required, version 1.10 or later. @see: Norrdine, A. U{I{An Algebraic Solution to the Multilateration Problem}<https://www.ResearchGate.net/publication/ 275027725_An_Algebraic_Solution_to_the_Multilateration_Problem>} and U{I{implementation}<https://www.ResearchGate.net/publication/ 288825016_Trilateration_Matlab_Code>}. ''' try: return _trilaterate3d2(_otherV3d(center=self, NN_OK=False), Radius_(radius, low=eps), center2, radius2, center3, radius3, eps=eps, clas=self.classof) except (AssertionError, TypeError, ValueError) as x: raise _xError(x, center=self, radius=radius, center2=center2, radius2=radius2, center3=center3, radius3=radius3) 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{trilaterate3d2} if C{B{useZ} is True} else L{trilaterate2d2}. @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=0} (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, Vector=V, name=circin6.__name__, **Vector_kwds) 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{trilaterate3d2} if C{B{useZ} is True} else L{trilaterate2d2}. @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=0} (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_T) 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 intersection3d3(start1, end1, start2, end2, eps=EPS, useZ=True, **Vector_and_kwds): '''Compute the intersection point of two lines, each defined by or through a start and end point (3-D). @arg start1: Start point of the first line (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg end1: End point of the first line (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg start2: Start point of the second line (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg end2: End point of the second line (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @kwarg eps: Tolerance for skew line distance and length (C{EPS}). @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=0} (C{bool}). @kwarg Vector_and_kwds: Optional class C{B{Vector}=None} to return the intersection points and optional, additional B{C{Vector}} keyword arguments, otherwise B{C{start1}}'s (sub-)class. @return: An L{Intersection3Tuple}C{(point, outside1, outside2)} with C{point} an instance of B{C{Vector}} or B{C{start1}}'s (sub-)class. @raise IntersectionError: Invalid, skew, non-co-planar or otherwise non-intersecting lines. @see: U{Line-line intersection<https://MathWorld.Wolfram.com/Line-LineIntersection.html>} and U{line-line distance<https://MathWorld.Wolfram.com/Line-LineDistance.html>}, U{skew lines<https://MathWorld.Wolfram.com/SkewLines.html>} and U{point-line distance<https://MathWorld.Wolfram.com/Point-LineDistance3-Dimensional.html>}. ''' try: v, o1, o2 = _intersect3d3(start1, end1, start2, end2, eps=eps, useZ=useZ) except (TypeError, ValueError) as x: raise _xError(x, start1=start1, end1=end1, start2=start2, end2=end2) v = _nVc(v, **_xkwds(Vector_and_kwds, clas=start1.classof, name=intersection3d3.__name__)) return Intersection3Tuple(v, o1, o2) def _intersect3d3(start1, end1, start2, end2, eps=EPS, useZ=False): # (INTERNAL) Intersect two lines, see L{intersection} above, # separated to allow callers to embellish any exceptions def _outside(t, d2): # -1 before start#, +1 after end# return -1 if t < 0 else (+1 if t > d2 else 0) # XXX d2 + eps? s1 = _otherV3d(useZ=useZ, start1=start1) e1 = _otherV3d(useZ=useZ, end1=end1) s2 = _otherV3d(useZ=useZ, start2=start2) e2 = _otherV3d(useZ=useZ, end2=end2) a = e1.minus(s1) b = e2.minus(s2) c = s2.minus(s1) ab = a.cross(b) d = abs(c.dot(ab)) e = max(EPS0, eps or _0_0) if d > EPS0 and ab.length > e: d = d / ab.length if d > e: # argonic, skew lines distance raise IntersectionError(skew_d=d, txt=_no_(_intersection_)) # co-planar, non-skew lines ab2 = ab.length2 if ab2 < e: # colinear, parallel or null line(s) x = a.length2 > b.length2 if x: # make a shortest a, b = b, a s1, s2 = s2, s1 e1, e2 = e2, e1 if b.length2 < e: # both null if c.length < e: return s1, 0, 0 elif e2.minus(e1).length < e: return e1, 0, 0 elif a.length2 < e: # null (s1, e1), non-null (s2, e2) # like _nearestOn(s1, s2, e2, within=False, eps=e) t = s1.minus(s2).dot(b) v = s2.plus(b.times(t / b.length2)) if s1.minus(v).length < e: o = _outside(t, b.length2) return (v, o, 0) if x else (v, 0, (o * 2)) raise IntersectionError(length2=ab2, txt=_no_(_intersection_)) cb = c.cross(b) t = cb.dot(ab) o1 = _outside(t, ab2) v = s1.plus(a.times(t / ab2)) o2 = _outside(v.minus(s2).dot(b), b.length2) * 2 return v, o1, o2 def 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. @arg center1: Center of the first sphere or circle (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg radius1: Radius of the first sphere or circle (same units as the B{C{center1}} coordinates). @arg center2: Center of the second sphere or circle (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg radius2: Radius of the second sphere or circle (same units as the B{C{center1}} and B{C{center2}} coordinates). @kwarg sphere: If C{True} compute the center and radius of the intersection of two spheres. If C{False}, ignore the C{z}-component and compute the intersection of two circles (C{bool}). @kwarg Vector_and_kwds: Optional class C{B{Vector}=None} to return the intersection points and optional, additional B{C{Vector}} keyword arguments, otherwise B{C{center1}}'s (sub-)class. @return: If B{C{sphere}} is C{True}, a 2-tuple of the C{center} and C{radius} of the intersection of the I{spheres}. The C{radius} is C{0.0} for abutting spheres (and the C{center} is aka I{radical center}). If B{C{sphere}} is C{False}, a 2-tuple with the two intersection points of the I{circles}. For abutting circles, both points are the same instance, aka I{radical center}. @raise IntersectionError: Concentric, invalid or non-intersecting spheres or circles. @raise TypeError: Invalid B{C{center1}} or B{C{center2}}. @raise UnitError: Invalid B{C{radius1}} or B{C{radius2}}. @see: U{Sphere-Sphere<https://MathWorld.Wolfram.com/Sphere-SphereIntersection.html>} and U{Circle-Circle<https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>} Intersection. ''' try: return _intersects2(center1, Radius_(radius1=radius1), center2, Radius_(radius2=radius2), sphere=sphere, clas=center1.classof, **Vector_and_kwds) except (TypeError, ValueError) as x: raise _xError(x, center1=center1, radius1=radius1, center2=center2, radius2=radius2) def _intersects2(center1, r1, center2, r2, sphere=True, too_d=None, # in CartesianEllipsoidalBase.intersections2, **clas_Vector_and_kwds): # .ellipsoidalBaseDI._intersections2 # (INTERNAL) Intersect two spheres or circles, see L{intersections2} # above, separated to allow callers to embellish any exceptions def _nV3(x, y, z): v = Vector3d(x, y, z) n = intersections2.__name__ return _nVc(v, **_xkwds(clas_Vector_and_kwds, name=n)) def _xV3(c1, u, x, y): xy1 = x, y, _1_0 # transform to original space return _nV3(fdot(xy1, u.x, -u.y, c1.x), fdot(xy1, u.y, u.x, c1.y), _0_0) c1 = _otherV3d(useZ=sphere, center1=center1) c2 = _otherV3d(useZ=sphere, center2=center2) if r1 < r2: # r1, r2 == R, r c1, c2 = c2, c1 r1, r2 = r2, r1 m = c2.minus(c1) d = m.length if d < max(r2 - r1, EPS): raise IntersectionError(_near_(_concentric_)) o = fsum1_(-d, r1, r2) # overlap == -(d - (r1 + r2)) # compute intersections with c1 at (0, 0) and c2 at (d, 0), like # <https://MathWorld.Wolfram.com/Circle-CircleIntersection.html> if o > EPS: # overlapping, r1, r2 == R, r x = _radical2(d, r1, r2).xline y = _1_0 - (x / r1)**2 if y > EPS: y = r1 * sqrt(y) # y == a / 2 elif y < 0: raise IntersectionError(_negative_) else: # abutting y = _0_0 elif o < 0: t = d if too_d is None else too_d raise IntersectionError(_too_(Fmt.distant(t))) else: # abutting x, y = r1, _0_0 u = m.unit() if sphere: # sphere center and radius c = c1 if x < EPS else ( c2 if x > EPS1 else c1.plus(u.times(x))) t = _nV3(c.x, c.y, c.z), Radius(y) elif y > 0: # intersecting circles t = _xV3(c1, u, x, y), _xV3(c1, u, x, -y) else: # abutting circles t = _xV3(c1, u, x, 0) t = t, t return t def iscolinearWith(point, point1, point2, eps=EPS): '''Check whether a point is colinear with two other (3-D) points. @arg point: The point (L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg point1: First point (L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg point2: Second point (L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @kwarg eps: Tolerance (C{scalar}), same units as C{x}, C{y} and C{z}. @return: C{True} if B{C{point}} is colinear B{C{point1}} and B{C{point2}}, C{False} otherwise. @raise TypeError: Invalid B{C{point}}, B{C{point1}} or B{C{point2}}. @see: Function L{vector3d.nearestOn}. ''' return _iscolinearWith(_otherV3d(point=point), point1, point2, eps=eps) def _iscolinearWith(v, p1, p2, eps=EPS): # (INTERNAL) Check colinear, see L{isColinear} above, # separated to allow callers to embellish any exceptions v1 = _otherV3d(point1=p1) v2 = _otherV3d(point2=p2) n = _nearestOn(v, v1, v2, within=False, eps=eps) return n is v1 or n.minus(v).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 (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=0} (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_(_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 = 0 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 nearestOn(point, point1, point2, within=True, Vector=None, **Vector_kwds): '''Locate the point between two points closest to a reference (3-D). @arg point: Reference point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg point1: Start point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg point2: End point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @kwarg within: If C{True} return the closest point between both given points, otherwise the closest point on the extended line through both points (C{bool}). @kwarg Vector: Class to return closest point (C{Cartesian}, L{Vector3d} or C{Vector3Tuple}) or C{None}. @kwarg Vector_kwds: Optional, additional B{C{Vector}} keyword arguments, ignored if C{B{Vector} is None}. @return: Closest point, either B{C{point1}} or B{C{point2}} or an instance of the B{C{point}}'s (sub-)class or B{C{Vector}} if not C{None}. @raise TypeError: Invalid B{C{point}}, B{C{point1}} or B{C{point2}}. @see: U{3-D Point-Linedistance<https://MathWorld.Wolfram.com/Point-LineDistance3-Dimensional.html>}, C{Cartesian} and C{LatLon} methods C{nearestOn}, method L{sphericalTrigonometry.LatLon.nearestOn3} and function L{sphericalTrigonometry.nearestOn3}. ''' p0 = _otherV3d(point =point) p1 = _otherV3d(point1=point1) p2 = _otherV3d(point2=point2) p = _nearestOn(p0, p1, p2, within=within) if Vector is not None: p = Vector(p.x, p.y, p.z, **_xkwds(Vector_kwds, name=nearestOn.__name__)) elif p is p1: p = point1 elif p is p2: p = point2 else: # ignore Vector_kwds p = point.classof(p.x, p.y, p.z, name=nearestOn.__name__) return p def _nearestOn(p0, p1, p2, within=True, eps=EPS): # (INTERNAL) Get closest point, see L{nearestOn} above, # separated to allow callers to embellish any exceptions p21 = p2.minus(p1) d2 = p21.length2 if d2 < eps: # coincident p = p1 # ... or point2 else: t = p0.minus(p1).dot(p21) / d2 p = p1 if (within and t < eps) else ( p2 if (within and t > (_1_0 - eps)) else p1.plus(p21.times(t))) return p 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): # get numpy with any errors raised as NumPyError np = Vector3d._numpy if np is None: # get numpy, once or ImportError Vector3d._numpy = np = _xnumpy(where, 1, 10) # macOS' Python 2.7 numpy 1.8 OK 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)) finally: # restore numpy error handling np.seterr(**e) def _nVc(v, clas=None, name=NN, Vector=None, **Vector_kwds): # return a named C{Vector} or C{clas} instance if Vector is not None: v = Vector(v.x, v.y, v.z, **Vector_kwds) elif clas is not None: v = clas(v.x, v.y, v.z) # ignore Vector_kwds return _xnamed(v, name) if name else v def _otherV3d(useZ=True, NN_OK=True, i=None, **name_v): # in .CartesianEllipsoidalBase.intersections2, .Ellipsoid.height4, .formy.hartzell # check named vector instance, return Vector3d def _name_i(name, i): return name if i is None else Fmt.SQUARE(name, i) name, v = _xkwds_popitem(name_v) if useZ and isinstance(v, Vector3dBase): return v if NN_OK or v.name else v.copy(name=_name_i(name, i)) try: return Vector3d(v.x, v.y, (v.z if useZ else _0_0), name=_name_i(name, i)) except AttributeError: # no _x_ or _y_ attr pass raise _xotherError(Vector3d(0, 0, 0), v, name=_name_i(name, i), up=2) def parse3d(str3d, sep=_COMMA_, Vector=Vector3d, **Vector_kwds): '''Parse an C{"x, y, z"} string. @arg str3d: X, y and z values (C{str}). @kwarg sep: Optional separator (C{str}). @kwarg Vector: Optional class (L{Vector3d}). @kwarg Vector_kwds: Optional B{C{Vector}} keyword arguments, ignored if C{B{Vector} is None}. @return: New B{C{Vector}} or if B{C{Vector}} is C{None}, a named L{Vector3Tuple}C{(x, y, z)}. @raise VectorError: Invalid B{C{str3d}}. ''' try: v = [float(v.strip()) for v in str3d.split(sep)] n = len(v) if n != 3: raise _ValueError(len=n) except (TypeError, ValueError) as x: raise VectorError(str3d=str3d, txt=str(x)) return _xnamed((Vector3Tuple(*v) if Vector is None else Vector(*v, **Vector_kwds)), parse3d.__name__) 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=0} (C{bool}). @return: L{Radii11Tuple}C{(rA, rB, rC, cR, rIn, riS, roS, a, b, c, s)}. @raise IntersectionError: 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 = _4_0 * s * ci 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 IntersectionError(_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{trilaterate3d2} if C{B{useZ} is True} else L{trilaterate2d2}. @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=0} (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 sumOf(vectors, Vector=Vector3d, **Vector_kwds): '''Compute the vectorial sum of several vectors. @arg vectors: Vectors to be added (L{Vector3d}[]). @kwarg Vector: Optional class for the vectorial sum (L{Vector3d}). @kwarg Vector_kwds: Optional B{C{Vector}} keyword arguments, ignored if C{B{Vector} is None}. @return: Vectorial sum as B{C{Vector}} or if B{C{Vector}} is C{None}, a named L{Vector3Tuple}C{(x, y, z)}. @raise VectorError: No B{C{vectors}}. ''' n, vectors = len2(vectors) if n < 1: raise VectorError(vectors=n, txt=MISSING) v = Vector3Tuple(fsum(v.x for v in vectors), fsum(v.y for v in vectors), fsum(v.z for v in vectors)) return _xnamed((v if Vector is None else Vector(*v, **Vector_kwds)), sumOf.__name__) 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, eps=None, **Vector_and_kwds): '''Trilaterate three circles, each given as a (2-D) center and a radius. @arg x1: Center C{x} coordinate of the 1st circle (C{scalar}). @arg y1: Center C{y} coordinate of the 1st circle (C{scalar}). @arg radius1: Radius of the 1st circle (C{scalar}). @arg x2: Center C{x} coordinate of the 2nd circle (C{scalar}). @arg y2: Center C{y} coordinate of the 2nd circle (C{scalar}). @arg radius2: Radius of the 2nd circle (C{scalar}). @arg x3: Center C{x} coordinate of the 3rd circle (C{scalar}). @arg y3: Center C{y} coordinate of the 3rd circle (C{scalar}). @arg radius3: Radius of the 3rd circle (C{scalar}). @kwarg eps: Check the trilaterated point I{delta} on all 3 circles (C{scalar}) or C{None}. @kwarg Vector_and_kwds: Optional class C{B{Vector}=None} to return the trilateration and optional, additional B{C{Vector}} keyword arguments, otherwise (L{Vector3d}). @return: Trilaterated point as C{B{Vector}(x, y, **B{Vector_kwds})} or L{Vector2Tuple}C{(x, y)} if C{B{Vector} is None}.. @raise IntersectionError: No intersection, colinear or near-concentric centers, trilateration failed some other way or the trilaterated point is off one circle by more than B{C{eps}}. @raise UnitError: Invalid B{C{radius1}}, B{C{radius2}} or B{C{radius3}}. @see: U{Issue #49<https://GitHub.com/mrJean1/PyGeodesy/issues/49>}, U{Find X location using 3 known (X,Y) location using trilateration <https://math.StackExchange.com/questions/884807>} and L{trilaterate3d2}. ''' return _trilaterate2d2(x1, y1, radius1, x2, y2, radius2, x3, y3, radius3, eps=eps, **Vector_and_kwds) 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{trilaterate2d2} above def _abct4(x1, y1, r1, x2, y2, r2, coin): 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, coin) 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, coin) 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, coin) 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(center1, radius1, center2, radius2, center3, radius3, eps=EPS, **Vector_and_kwds): '''Trilaterate three spheres, each given as a (3-D) center and a radius. @arg center1: Center of the 1st sphere (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg radius1: Radius of the 1st sphere (same C{units} as C{x}, C{y} and C{z}). @arg center2: Center of the 2nd sphere (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg radius2: Radius of this sphere (same C{units} as C{x}, C{y} and C{z}). @arg center3: Center of the 3rd sphere (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg radius3: Radius of the 3rd sphere (same C{units} as C{x}, C{y} and C{z}). @kwarg eps: Tolerance (C{scalar}), same units as C{x}, C{y}, and C{z}. @kwarg Vector_and_kwds: Optional class C{B{Vector}=None} to return the trilateration and optional, additional B{C{Vector}} keyword arguments, otherwise B{C{center1}}'s (sub-)class. @return: 2-Tuple with two trilaterated points, each a B{C{Vector}} instance. Both points are the same instance if all three spheres abut/intersect in a single point. @raise ImportError: Package C{numpy} not found, not installed or older than version 1.10. @raise IntersectionError: Near-concentric, colinear, too distant or non-intersecting spheres. @raise NumPyError: Some C{numpy} issue. @raise TypeError: Invalid B{C{center1}}, B{C{center2}} or B{C{center3}}. @raise UnitError: Invalid B{C{radius1}}, B{C{radius2}} or B{C{radius3}}. @see: Norrdine, A. U{I{An Algebraic Solution to the Multilateration Problem}<https://www.ResearchGate.net/publication/ 275027725_An_Algebraic_Solution_to_the_Multilateration_Problem>} and U{I{implementation}<https://www.ResearchGate.net/publication/ 288825016_Trilateration_Matlab_Code>} and L{trilaterate2d2}. ''' try: return _trilaterate3d2(_otherV3d(center1=center1, NN_OK=False), Radius_(radius1=radius1, low=eps), center2, radius2, center3, radius3, eps=eps, clas=center1.classof, **Vector_and_kwds) except (AssertionError, NumPyError, TypeError, ValueError) as x: raise _xError(x, center1=center1, radius1=radius1, center2=center2, radius2=radius2, center3=center3, radius3=radius3) 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{trilaterate3d2} # above, 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 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 yield -min(q, r) 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_T + c.times(-_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, -_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) def _xyzn4(xyz, y, z, Error=_TypeError): # see: .ecef '''(INTERNAL) Get an C{(x, y, z, name)} 4-tuple. ''' try: t = xyz.x, xyz.y, xyz.z n = getattr(xyz, _name_, NN) except AttributeError: t = xyz, y, z n = NN try: x, y, z = map2(float, t) except (TypeError, ValueError) as x: d = dict(zip((_xyz_, _y_, _z_), t)) raise Error(txt=str(x), **d) return x, y, z, n def _xyzhdn6(xyz, y, z, height, datum, ll, Error=_TypeError): # by .cartesianBase, .nvectorBase '''(INTERNAL) Get an C{(x, y, z, h, d, name)} 6-tuple. ''' x, y, z, n = _xyzn4(xyz, y, z, Error=Error) h = height or getattr(xyz, _height_, None) \ or getattr(xyz, _h_, None) \ or getattr(ll, _height_, None) d = datum or getattr(xyz, _datum_, None) \ or getattr(ll, _datum_, None) return x, y, z, h, d, n __all__ += _ALL_DOCS(intersections2, sumOf, Vector3dBase) # **) MIT License # # Copyright (C) 2016-2021 -- 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.