# Coverage for pygeodesy/vector3d.py : 98%

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Function L{intersection3d3}, L{intersections2}, L{parse3d}, L{sumOf}, L{trilaterate2d2} and L{trilaterate3d2}. ''' from pygeodesy.basics import isnear0, isscalar, len2 from pygeodesy.errors import IntersectionError, _ValueError, VectorError, \ _xError, _xkwds, _xkwds_popitem from pygeodesy.fmath import euclid, fdot, fsum, fsum1_, hypot # from pygeodesy.fsums import fsum, fsum1_ # from .fmath # from pygeodesy.formy import _radical2 # in _intersects2 below from pygeodesy.interns import EPS, EPS0, EPS1, EPS4, INT0, MISSING, NN, \ _COMMA_, _concentric_, _datum_, _h_, _height_, \ _intersection_, _name_, _near_, _negative_, \ _no_, _too_, _z_, _0_0, _1_0 from pygeodesy.iters import Fmt, PointsIter from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY, _ALL_MODS as _MODS from pygeodesy.named import _xnamed, _xotherError from pygeodesy.namedTuples import Intersection3Tuple, NearestOn2Tuple, \ NearestOn6Tuple, Vector3Tuple # Vector4Tuple # from pygeodesy.streprs import Fmt # from .iters from pygeodesy.units import _fi_j2, Radius, Radius_ from pygeodesy.utily import atan2b, sincos2d # from pygeodesy.vector2d import .... # in .... below from pygeodesy.vector3dBase import Vector3dBase from math import sqrt __all__ = _ALL_LAZY.vector3d __version__ = '22.06.17' class Vector3d(Vector3dBase): '''Extended 3-D vector. In a geodesy context, these may be used to represent: - earth-centered, earth-fixed cartesian (ECEF) - n-vector representing a normal to a point on earth's surface - great circle normal to vector - motion vector on earth's surface - etc. ''' def bearing(self, useZ=True): '''Get the "bearing" of this vector. @kwarg useZ: If C{True}, use the Z component, otherwise consider the Y as +Z axis. @return: Bearing (compass C{degrees}), the counter-clockwise angle off the +Z axis. ''' x, y = self.x, self.y if useZ: x, y = hypot(x, y), self.z return atan2b(x, y) 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{pygeodesy.trilaterate3d2} if C{B{useZ} is True} otherwise L{pygeodesy.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 _MODS.vector2d._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{pygeodesy.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 _MODS.vector2d._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 _MODS.vector2d.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 _MODS.vector2d._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, _, _ = _MODS.vector2d._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 _MODS.vector2d.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 _nearestOn2(self, point1, point2, within=within).closest def nearestOn6(self, points, closed=False, useZ=True): # eps=EPS '''Locate the point on a path or polygon closest to this point. The closest point is either on and within the extent of a polygon edge or the nearest of that edge's end points. @arg points: The path or polygon points (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}[]). @kwarg closed: Optionally, close the path or polygon (C{bool}). @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}). @return: A L{NearestOn6Tuple}C{(closest, distance, fi, j, start, end)} with the C{closest}, the C{start} and the C{end} point each an instance of this point's (sub-)class. @raise PointsError: Insufficient number of B{C{points}} @raise TypeError: Non-cartesian B{C{points}}. @note: Distances measured with method L{Vector3d.equirectangular}. @see: Function L{nearestOn6}. ''' return nearestOn6(self, points, closed=closed, useZ=useZ) # Vector=self.classof 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 TriangleError: 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 _MODS.vector2d._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{pygeodesy.trilaterate3d2} if C{B{useZ} is True} otherwise L{pygeodesy.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 _MODS.vector2d.soddy4(self, point2, point3, eps=eps, useZ=True) def trilaterate2d2(self, radius, center2, radius2, center3, radius3, eps=EPS, z=INT0): '''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, near-concentric or -colinear 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 _MODS.vector2d._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 NumPyError: Some 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: c1 = _otherV3d(center=self, NN_OK=False) return _MODS.vector2d._trilaterate3d2(c1, 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 _intersect3d3(start1, end1, start2, end2, eps=EPS, useZ=False): # MCCABE 16 # (INTERNAL) Intersect two lines, see L{intersection3d3} below, # separated to allow callers to embellish any exceptions def _outside(t, d2, o): # -o before start#, +o after end# return -o if t < 0 else (o if t > d2 else 0) # XXX d2 + eps? def _ritri2(s1, b1, s2, useZ): # Get the C{s1'} and C{e1'}, corners of a right-angle # triangle with the hypotenuse thru C{s1} at bearing # C{b1} and the right angle at C{s2} dx, dy, d = s2.minus(s1).xyz if useZ and not isnear0(d): # not supported raise IntersectionError(useZ=d, bearing=b1) s, c = sincos2d(b1) if s and c: dx *= c / s dy *= s / c e1 = Vector3d(s2.x, s1.y + dx, s1.z) s1 = Vector3d(s1.x + dy, s2.y, s1.z) else: # orthongonal d = euclid(dx, dy) e1 = Vector3d(s1.x + s * d, s1.y + c * d, s1.z) return s1, e1 s1 = x = _otherV3d(useZ=useZ, start1=start1) s2 = _otherV3d(useZ=useZ, start2=start2) b1 = isscalar(end1) if b1: s1, e1 = _ritri2(s1, end1, s2, useZ) else: e1 = _otherV3d(useZ=useZ, end1=end1) b2 = isscalar(end2) if b2: s2, e2 = _ritri2(s2, end2, x, useZ) else: 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: # PYCHOK no cover d = d / ab.length # /= chokes PyChecker 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 = b.length2 < a.length2 if x: # make C{a} the shortest a, b = b, a s1, s2 = s2, s1 e1, e2 = e2, e1 b1, b2 = b2, b1 if b.length2 < e: # PYCHOK no cover 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 _nearestOn2(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 = 0 if b2 else _outside(t, b.length2, 1 if x else 2) return (v, o, 0) if x else (v, 0, o) raise IntersectionError(length2=ab2, txt=_no_(_intersection_)) cb = c.cross(b) t = cb.dot(ab) o1 = 0 if b1 else _outside(t, ab2, 1) v = s1.plus(a.times(t / ab2)) o2 = 0 if b2 else _outside(v.minus(s2).dot(b), b.length2, 2) return v, o1, o2 def intersection3d3(start1, end1, start2, end2, eps=EPS, useZ=True, **Vector_and_kwds): '''Compute the intersection point of two lines, each defined by two points or by a point and a bearing. @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}) or the bearing at B{C{start1}} (compass C{degrees}). @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}) or the bearing at B{C{start2}} (Ccompass C{degrees}). @kwarg eps: Tolerance for skew line distance and length (C{EPS}). @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (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. @note: The C{outside} values is C{0} for lines specified by point and bearing. @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 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 = _MODS.formy._radical2(d, r1, r2).xline y = _1_0 - (x / r1)**2 if y > EPS: y = r1 * sqrt(y) # y == a / 2 elif y < 0: # PYCHOK no cover raise IntersectionError(_negative_) else: # abutting y = _0_0 elif o < 0: # PYCHOK no cover 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, useZ=True): '''Check whether a point is colinear with two other (2- or 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}. @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}). @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}. ''' p = _otherV3d(useZ=useZ, point=point) return _MODS.vector2d._iscolinearWith(p, point1, point2, eps=eps, useZ=useZ) def nearestOn(point, point1, point2, within=True, useZ=True, Vector=None, **Vector_kwds): '''Locate the point between two points closest to a reference (2- or 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 useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (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-Line Distance<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(useZ=useZ, point =point) p1 = _otherV3d(useZ=useZ, point1=point1) p2 = _otherV3d(useZ=useZ, point2=point2) p, _ = _nearestOn2(p0, p1, p2, within=within) if Vector is not None: p = Vector(p.x, p.y, **_xkwds(Vector_kwds, z=p.z, 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, Vector_kwds.get(_z_, p.z), name=nearestOn.__name__) return p def _nearestOn2(p0, p1, p2, within=True, eps=EPS): # (INTERNAL) Closest point and fraction, 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 # ~= p2 t = 0 else: # see comments in .points.nearestOn5 t = p0.minus(p1).dot(p21) / d2 if within and t < eps: p = p1 t = 0 elif within and t > (_1_0 - eps): p = p2 t = 1 else: p = p1.plus(p21.times(t)) return NearestOn2Tuple(p, t) def nearestOn6(point, points, closed=False, useZ=True, **Vector_and_kwds): # eps=EPS '''Locate the point on a path or polygon closest to a reference point. The closest point is either on and within the extent of a polygon edge or the nearest of that edge's end points. @arg point: Reference point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg points: The path or polygon points (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}[]). @kwarg closed: Optionally, close the path or polygon (C{bool}). @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}). @kwarg Vector_and_kwds: Optional class C{B{Vector}=None} to return the closest point and optional, additional B{C{Vector}} keyword arguments, otherwise B{C{point}}'s (sub-)class. @return: A L{NearestOn6Tuple}C{(closest, distance, fi, j, start, end)} with the C{closest}, the C{start} and the C{end} point each an instance of the B{C{Vector}} keyword argument of if {B{Vector}=None} or not specified, an instance of the B{C{point}}'s (sub-)class. @raise PointsError: Insufficient number of B{C{points}} @raise TypeError: Non-cartesian B{C{point}} and B{C{points}}. @note: Distances measured with method L{Vector3d.equirectangular}. @see: Method C{LatLon.nearestOn6} or function L{nearestOn5} for geodetic points. ''' r = _otherV3d(useZ=useZ, point=point) D2 = r.equirectangular # distance squared Ps = PointsIter(points, loop=1, name=nearestOn6.__name__) p1 = c = s = e = _otherV3d(useZ=useZ, i=0, points=Ps[0]) c2 = D2(c) # == r.minus(c).length2 f = i = 0 # p1..p2 == points[i]..[j] for j, p2 in Ps.enumerate(closed=closed): p2 = _otherV3d(useZ=useZ, i=j, points=p2) p, t = _nearestOn2(r, p1, p2) # within=True, eps=EPS d2 = D2(p) # == r.minus(p).length2 if d2 < c2: c2, c, s, e, f = d2, p, p1, p2, (i + t) p1, i = p2, j f, j = _fi_j2(f, len(Ps)) # like .ellipsoidalBaseDI._nearestOn2_ kwds = _xkwds(Vector_and_kwds, clas=point.classof, name=Ps.name) v = _nVc(c, **kwds) s = _nVc(s, **kwds) if s is not c else v e = _nVc(e, **kwds) if e is not c else v return NearestOn6Tuple(v, sqrt(c2), f, j, s, e) def _nVc(v, clas=None, name=NN, Vector=None, **Vector_kwds): # in .vector2d # 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, # check named vector instance, return Vector3d .Ellipsoid.height4, .formy.hartzell, .vector2d 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 INT0), name=_name_i(name, i)) except AttributeError: # no .x, .y or .z 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: A B{C{Vector}} instance 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 # *v Vector(*v, **Vector_kwds)), parse3d.__name__) 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 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, near-concentric or -colinear 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 function L{pygeodesy.trilaterate3d2}. ''' return _MODS.vector2d._trilaterate2d2(x1, y1, radius1, x2, y2, radius2, x3, y3, radius3, eps=eps, **Vector_and_kwds) 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>}, the U{I{implementation}<https://www.ResearchGate.net/publication/ 288825016_Trilateration_Matlab_Code>} and function L{pygeodesy.trilaterate2d2}. ''' try: return _MODS.vector2d._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, TypeError, ValueError) as x: raise _xError(x, center1=center1, radius1=radius1, center2=center2, radius2=radius2, center3=center3, radius3=radius3) def _xyzhdn3(xyz, height, datum, ll): # in .cartesianBase, .nvectorBase '''(INTERNAL) Get an C{(h, d, name)} 3-tuple. ''' 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 h, d, getattr(xyz, _name_, NN) __all__ += _ALL_DOCS(intersections2, sumOf, Vector3dBase) # **) MIT License # # Copyright (C) 2016-2022 -- 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.