# Coverage for pygeodesy/vector3d.py : 92%

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Function L{intersection3d3}, L{intersections2}, L{iscolinearWith}, L{nearestOn}, L{parse3d}, L{sumOf}, L{trilaterate2d2} and L{trilaterate3d2}. ''' from pygeodesy.basics import isnear0, len2, map2, _xnumpy from pygeodesy.errors import _and, _AssertionError, IntersectionError, \ NumPyError, _TypeError, _ValueError, \ VectorError, _xError, _xkwds_popitem from pygeodesy.fmath import fdot, fsum, fsum_, hypot2_ from pygeodesy.formy import _radical2 from pygeodesy.interns import EPS, EPS0, EPS1, MISSING, NN, _EPSqrt, \ _and_, _colinear_, _COMMA_, _COMMASPACE_, \ _datum_, _h_, _height_, _intersection_, \ _invalid_, _name_, _near_concentric_, \ _no_, _SPACE_, _too_, _xyz_, _y_, _z_, \ _0_0, _1_0, _2_0 from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY from pygeodesy.named import modulename, _xnamed, _xotherError from pygeodesy.namedTuples import Intersection3Tuple, Vector2Tuple, \ Vector3Tuple # Vector4Tuple from pygeodesy.streprs import Fmt from pygeodesy.units import Float, Radius, Radius_ from pygeodesy.vector3dBase import Vector3dBase from math import hypot, sqrt __all__ = _ALL_LAZY.vector3d __version__ = '21.06.10' _raise_ = 'raise' class Vector3d(Vector3dBase): '''Extended 3-D vector. In a geodesy context, these may be used to represent: - n-vector representing a normal to point on earth's surface - earth-centered, earth-fixed cartesian (= spherical n-vector) - great circle normal to vector - motion vector on earth's surface - etc. ''' _numpy = None # module numpy iff imported by trilaterate3d2 below def iscolinearWith(self, point1, point2, eps=EPS): '''Check whether this and two other points are colinear. @arg point1: One point (L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg point2: Another 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 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(this=self) return _iscolinearWith(v, point1, point2, eps=eps) def nearestOn(self, other1, other2, within=True): '''Locate the point between two points closest to this point. @arg other1: Start point (L{Vector3d}). @arg other2: End point (L{Vector3d}). @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 (L{Vector3d}). @raise TypeError: If B{C{other1}} or B{C{other2}} is not L{Vector3d}. @see: Method L{sphericalTrigonometry.LatLon.nearestOn3} and U{3-D Point-Line distance<https://MathWorld.Wolfram.com/ Point-LineDistance3-Dimensional.html>}. ''' return _nearestOn(self, _otherV3d(other1=other1), _otherV3d(other2=other2), 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 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 (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 (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 L{Vector3d} (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 1.15. @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 1.15 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(self if self.name else _otherV3d(center=self), Radius_(radius, low=eps), center2, radius2, center3, radius3, eps=eps, Vector=self.classof) except (AssertionError, TypeError, ValueError) as x: raise _xError(x, center=self, radius=radius, center2=center2, radius2=radius2, center3=center3, radius3=radius3) def intersection3d3(start1, end1, start2, end2, eps=EPS, useZ=True, Vector=None, **Vector_kwds): '''Compute the intersection point of two lines, each defined by or through a start and end point. @arg start1: Start point of the first line (L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg end1: End point of the first line (L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg start2: Start point of the second line (L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg end2: End point of the second line (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 component, if C{False} force C{z=0} (C{bool}). @kwarg Vector: Class to return intersections (L{Vector3d} or C{Vector3Tuple}) or C{None} for L{Vector3d}. @kwarg Vector_kwds: Optional, additional B{C{Vector}} keyword arguments, ignored if C{B{Vector}=None}. @return: An L{Intersection3Tuple}C{(point, outside1, outside2)} with C{point} a C{Vector3d} or B{C{Vector}}. @raise IntersectionError: Invalid, skew, non-coplanar 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 = _V_n(v, intersection3d3.__name__, Vector, Vector_kwds) 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=None, **Vector_kwds): '''Compute the intersection of two spheres or circles, each defined by a center point and a radius. @arg center1: Center of the first sphere or circle (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 (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: Class to return intersections (L{Vector3d} or C{Vector3Tuple}) or C{None} for L{Vector3d}. @kwarg Vector_kwds: Optional, additional B{C{Vector}} keyword arguments, ignored if C{B{Vector}=None}. @return: If B{C{sphere}} is C{True}, a 2-Tuple of the C{center} and C{radius} of the intersection of the spheres. The C{radius} is C{0.0} for abutting spheres. If B{C{sphere}} is C{False}, a 2-tuple of the intersection points of two circles. For abutting circles, both points are the same B{C{Vector}} instance. @raise IntersectionError: Concentric, invalid or non-intersecting spheres or circles. @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>} intersections. ''' try: return _intersects2(center1, Radius_(radius1=radius1), center2, Radius_(radius2=radius2), sphere=sphere, Vector=Vector, **Vector_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 .ellipsoidalBaseDI._intersections2 Vector=None, **Vector_kwds): # (INTERNAL) Intersect two spheres or circles, see L{intersections2} # above, separated to allow callers to embellish any exceptions def _V3(x, y, z): v = Vector3d(x, y, z) n = intersections2.__name__ return _V_n(v, n, Vector, Vector_kwds) def _xV3(c1, u, x, y): xy1 = x, y, _1_0 # transform to original space return _V3(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 = fsum_(-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(_invalid_) 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 = _V3(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 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{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 nearestOn(point, point1, point2, within=True, Vector=None, **Vector_kwds): '''Locate the point between two points closest to a reference. @arg point: Reference point (L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg point1: Start point (L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg point2: End point (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 (L{Vector3d} or C{Vector3Tuple}) or C{None} for L{Vector3d}. @kwarg Vector_kwds: Optional, additional B{C{Vector}} keyword arguments, ignored if C{B{Vector}=None}. @return: Closest point (L{Vector3d} or C{Vector}). @raise TypeError: Invalid B{C{point}}, B{C{point1}} or B{C{point2}}. @see: Methods L{sphericalTrigonometry.LatLon.nearestOn3} and L{sphericalTrigonometry.LatLon.nearestOn3} U{3-D Point-Line distance<https://MathWorld.Wolfram.com/ Point-LineDistance3-Dimensional.html>}. ''' v = _nearestOn(_otherV3d(point =point), _otherV3d(point1=point1), _otherV3d(point2=point2), within=within) return _V_n(v, nearestOn.__name__, Vector, Vector_kwds) 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 p2 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=modulename(_null_space2, True)) # 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=modulename(_null_space2, True)) e = float(numpy.max(numpy.abs(numpy.dot(A, n)))) if e > t: # residual not near-zero raise _AssertionError(eps=e, txt=modulename(_null_space2, True)) else: # coincident, colinear, concentric centers, ambiguous, etc. n = None # del A, s, vh # release numpy return n, r def _otherV3d(useZ=True, **name_v): # check B{C{name#}} vector instance, return Vector3d if not name_v: raise _AssertionError(name_v=MISSING) name, v = _xkwds_popitem(name_v) if useZ and isinstance(v, Vector3dBase): return v try: return Vector3d(v.x, v.y, v.z if useZ else _0_0, name=name) except AttributeError: # no _x_ or _y_ attr pass raise _xotherError(Vector3d(0, 0, 0), v, name=name, up=2) def parse3d(str3d, sep=_COMMA_, name=NN, 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 name: Optional instance name (C{str}). @kwarg Vector: Optional class (L{Vector3d}). @kwarg Vector_kwds: Optional B{C{Vector}} keyword arguments, ignored if C{B{Vector}=None}. @return: New B{C{Vector}} or if B{C{Vector}} is C{None}, a 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 _V_n(Vector3Tuple(*v), name, Vector, Vector_kwds) 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}=None}. @return: Vectorial sum as B{C{Vector}} or if B{C{Vector}} is C{None}, a 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 _V_n(v, sumOf.__name__, Vector, Vector_kwds) def trilaterate2d2(x1, y1, radius1, x2, y2, radius2, x3, y3, radius3, eps=None): '''Trilaterate three circles, each given as a (2d) 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}. @return: Trilaterated point as L{Vector2Tuple}C{(x, y)}. @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} ''' def _abct4(x1, y1, r1, x2, y2, r2): a = x2 - x1 b = y2 - y1 t = _tri_r2h(r1, r2, hypot(a, b)) 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) 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) 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) if t: raise IntersectionError(_and(_astr(x3=x3, y3=y3, radius3=r3), _astr(x1=x1, y1=y1, radius1=r1)), txt=t) q = _2_0 * (e * a - b * d) 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_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()) raise IntersectionError(t, txt=Fmt.exceeds_eps(eps)) return t def trilaterate3d2(center1, radius1, center2, radius2, center3, radius3, eps=EPS, Vector=None, **Vector_kwds): '''Trilaterate three spheres, each given as a (3-D) center and a radius. @arg center1: Center of the 1st sphere (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 (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 (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: Class to return intersections (L{Vector3d} or C{Vector3Tuple}) or C{None} for L{Vector3d}. @kwarg Vector_kwds: Optional, additional B{C{Vector}} keyword arguments, ignored if C{B{Vector}=None}. @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 1.15. @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}}. @note: Package U{numpy<https://pypi.org/project/numpy>} is required, version 1.15 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>} and L{trilaterate2d2}. ''' try: return _trilaterate3d2(_otherV3d(center1=center1), Radius_(radius1=radius1, low=eps), center2, radius2, center3, radius3, eps=eps, Vector=Vector, **Vector_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, Vector=None, **Vector_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 _0f3d(F): # map numpy 4-vector to floats and split F0, x, y, z = map(float, F) return F0, Vector3d(x, y, z) def _N3(t01, x, z): # compute x, y and z and return as Vector v = x.plus(z.times(t01)) n = trilaterate3d2.__name__ return _V_n(v, n, Vector, Vector_kwds) def _real_roots(numpy, *coeffs): # non-complex roots of a polynomial rs = numpy.polynomial.polynomial.polyroots(coeffs) return tuple(float(r) for r in rs if not numpy.iscomplex(r)) np = Vector3d._numpy if np is None: # get numpy, once or ImportError Vector3d._numpy = np = _xnumpy(trilaterate3d2, 1, 10) # macOS' Python 2.7 numpy 1.8 OK c2 = _otherV3d(center2=c2) c3 = _otherV3d(center3=c3) R = [r1, Radius_(radius2=r2, low=eps), Radius_(radius3=r3, low=eps)] # get null_space Z and pseudo-inverse A, once t = -_2_0 A = [(_1_0, t * c.x, t * c.y, t * c.z) for c in (c1, c2, c3)] # 3 x 4 try: # <https://NumPy.org/doc/stable/reference/generated/numpy.seterr.html> e = np.seterr(all=_raise_) # throw FloatingPointError for numpy errors Z, _ = _null_space2(np, A, eps) A = np.linalg.pinv(A) # Moore-Penrose pseudo-inverse except Exception as X: # mostly FloatingPointError? raise NumPyError(X.__class__.__name__, txt=str(X)) finally: # restore numpy error handling np.seterr(**e) if Z is None: # coincident, colinear, concentric, etc. raise _trilaterror(c1, r1, c2, r2, c3, r3, eps) B = [c.length2 for c in (c1, c2, c3)] # perturbe radii to handle corner cases like this # <https://GitHub.com/mrJean1/PyGeodesy/issues/49> if eps > _0_0: p1 = max(eps, EPS) p2 = max(eps, _EPSqrt) if p2 > p1: ps = _0_0, p1, -p1, p2, -p2 else: ps = _0_0, p1, -p1 else: ps = _0_0, for p in ps: b = [((r + p)**2 - b) for r, b in zip(R, B)] # 1 x 3 or 3 x 1 # t = () try: # <https://NumPy.org/doc/stable/reference/generated/numpy.seterr.html> e = np.seterr(all=_raise_) # throw FloatingPointError for numpy errors X = np.dot(A, b) X0, x = _0f3d(X) Z0, z = _0f3d(Z) # quadratic polynomial coefficients, ordered (^0, ^1, ^2) t = _real_roots(np, x.length2 - X0, # fdot(X, -_1_0, *x.xyz) z.dot(x) * _2_0 - Z0, # fdot(Z, -_0_5, *x.xyz) * 2 z.length2) # fdot(Z, _0_0, *z.xyz) except Exception as X: # mostly FloatingPointError? raise NumPyError(X.__class__.__name__, txt=str(X)) finally: # restore numpy error handling np.seterr(**e) if t: break else: # coincident, concentric, colinear, too distant, no intersection, etc. raise _trilaterror(c1, r1, c2, r2, c3, r3, eps) 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 < v else (v, u) return t def _trilaterror(c1, r1, c2, r2, c3, r3, eps): # return FloatingPointError with the cause of the error def _txt(c1, r1, c2, r2): t = _tri_r2h(r1, r2, c1.minus(c2).length) 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) def _tri_r2h(r1, r2, h): # check for near-concentric or too distant spheres/circles return _too_(Fmt.distant(h)) if h > (r1 + r2) else ( _near_concentric_ if h < abs(r1 - r2) else NN) def _V_n(v, name, Vector, Vector_kwds): # return a named Vector instance if Vector is not None: v = Vector(v.x, v.y, v.z, **Vector_kwds) return _xnamed(v, name) 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.