# Coverage for pygeodesy/vector3d.py : 94%

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Pure Python implementation of vector-based functions by I{(C) Chris Veness 2011-2015} published under the same MIT Licence**, see U{Vector-based geodesy <https://www.Movable-Type.co.UK/scripts/latlong-vectors.html>}. @newfield example: Example, Examples @newfield JSname: JS name, JS names ''' from pygeodesy.basics import copysign, len2, map1, map2, neg, _xnumpy from pygeodesy.errors import _AssertionError, CrossError, IntersectionError, \ _TypeError, _ValueError, _xkwds_popitem from pygeodesy.fmath import euclid_, fdot, fsum, fsum_, hypot_, hypot2_ from pygeodesy.formy import n_xyz2latlon, n_xyz2philam, _radical2, sincos2 # utily from pygeodesy.interns import EPS, EPS0, EPS1, MISSING, NN, PI, PI2, \ _EPSqrt, _coincident_, _colinear_, _COMMA_, \ _COMMASPACE_, _datum_, _h_, _height_, _invalid_, \ _intersection_, _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, _NamedBase, _xnamed, _xother3, \ _xotherError from pygeodesy.namedTuples import Vector3Tuple # Vector4Tuple from pygeodesy.props import deprecated_method, Property_RO, property_doc_ from pygeodesy.streprs import Fmt, strs from pygeodesy.units import Float, Radius, Radius_, Scalar from math import atan2, sqrt __all__ = _ALL_LAZY.vector3d __version__ = '21.03.29' _raise_ = 'raise' 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 class VectorError(_ValueError): '''L{Vector3d} or C{*Nvector} issue. ''' pass class Vector3d(_NamedBase): # XXX or _NamedTuple or Vector3Tuple? '''Generic 3-D vector manipulation. 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 vector (= n-vector for spherical model) - great circle normal to vector - motion vector on earth's surface - etc. ''' _crosserrors = True # un/set by .errors.crosserrors _fromll = None # original latlon, '_fromll' _numpy = None # module numpy iff imported by trilaterate3d2 below _x = 0 # X component _y = 0 # Y component _z = 0 # Z component def __init__(self, x, y, z, ll=None, name=NN): '''New 3-D L{Vector3d}. The vector may be normalised or use x/y/z values for height relative to the surface of the sphere or ellipsoid, distance from earth centre, etc. @arg x: X component of vector (C{scalar}). @arg y: Y component of vector (C{scalar}). @arg z: Z component of vector (C{scalar}). @kwarg ll: Optional latlon reference (C{LatLon}). @kwarg name: Optional name (C{str}). ''' self._x = x self._y = y self._z = z if ll: self._fromll = ll if name: self.name = name def __abs__(self): '''Return the norm of this vector. @return: Norm, unit length (C{float}); ''' return self.length def __add__(self, other): '''Add this to an other vector (L{Vector3d}). @return: Vectorial sum (L{Vector3d}). @raise TypeError: Incompatible B{C{other}} C{type}. ''' return self.plus(other) # __iadd__ = __add__ __radd__ = __add__ def __cmp__(self, other): # Python 2- '''Compare this and an other vector @arg other: The other vector (L{Vector3d}). @return: -1, 0 or +1 (C{int}). @raise TypeError: Incompatible B{C{other}} C{type}. ''' self.others(other) return -1 if self.length < other.length else ( +1 if self.length > other.length else 0) cmp = __cmp__ def __div__(self, scalar): '''Divide this vector by a scalar. @arg scalar: The divisor (C{scalar}). @return: Quotient (L{Vector3d}). @raise TypeError: Non-scalar B{C{scalar}}. ''' return self.dividedBy(scalar) # __itruediv__ = __div__ __truediv__ = __div__ def __eq__(self, other): '''Is this vector equal to an other vector? @arg other: The other vector (L{Vector3d}). @return: C{True} if equal, C{False} otherwise. @raise TypeError: Incompatible B{C{other}} C{type}. ''' self.others(other) return self.isequalTo(other) def __ge__(self, other): '''Is this vector longer than or equal to an other vector? @arg other: The other vector (L{Vector3d}). @return: C{True} if so, C{False} otherwise. @raise TypeError: Incompatible B{C{other}} C{type}. ''' self.others(other) return self.length >= other.length def __gt__(self, other): '''Is this vector longer than an other vector? @arg other: The other vector (L{Vector3d}). @return: C{True} if so, C{False} otherwise. @raise TypeError: Incompatible B{C{other}} C{type}. ''' self.others(other) return self.length > other.length def __le__(self, other): # Python 3+ '''Is this vector shorter than or equal to an other vector? @arg other: The other vector (L{Vector3d}). @return: C{True} if so, C{False} otherwise. @raise TypeError: Incompatible B{C{other}} C{type}. ''' self.others(other) return self.length <= other.length def __lt__(self, other): # Python 3+ '''Is this vector shorter than an other vector? @arg other: The other vector (L{Vector3d}). @return: C{True} if so, C{False} otherwise. @raise TypeError: Incompatible B{C{other}} C{type}. ''' self.others(other) return self.length < other.length # Luciano Ramalho, "Fluent Python", page 397, O'Reilly 2016 def __matmul__(self, other): # PYCHOK Python 3.5+ ... c = a @ b '''Compute the cross product of this and an other vector. @arg other: The other vector (L{Vector3d}). @return: Cross product (L{Vector3d}). @raise TypeError: Incompatible B{C{other}} C{type}. ''' return self.cross(other) # __imatmul__ = __matmul__ def __mul__(self, scalar): '''Multiply this vector by a scalar @arg scalar: Factor (C{scalar}). @return: Product (L{Vector3d}). ''' return self.times(scalar) # __imul__ = __mul__ # __rmul__ = __mul__ def __ne__(self, other): '''Is this vector not equal to an other vector? @arg other: The other vector (L{Vector3d}). @return: C{True} if so, C{False} otherwise. @raise TypeError: Incompatible B{C{other}} C{type}. ''' self.others(other) return not self.isequalTo(other) def __neg__(self): '''Negate this vector. @return: Negative (L{Vector3d}) ''' return self.negate() def __pos__(self): '''Copy this vector. @return: Positive (L{Vector3d}) ''' return self.copy() # Luciano Ramalho, "Fluent Python", page 397, O'Reilly 2016 def __rmatmul__(self, other): # PYCHOK Python 3.5+ ... c = a @ b '''Compute the cross product of an other and this vector. @arg other: The other vector (L{Vector3d}). @return: Cross product (L{Vector3d}). @raise TypeError: Incompatible B{C{other}} C{type}. ''' self.others(other) return other.cross(self) def __rsub__(self, other): '''Subtract this vector from an other vector. @arg other: The other vector (L{Vector3d}). @return: Difference (L{Vector3d}). @raise TypeError: Incompatible B{C{other}} C{type}. ''' self.others(other) return other.minus(self) def __sub__(self, other): '''Subtract an other vector from this vector. @arg other: The other vector (L{Vector3d}). @return: Difference (L{Vector3d}). @raise TypeError: Incompatible B{C{other}} C{type}. ''' return self.minus(other) # __isub__ = __sub__ def angleTo(self, other, vSign=None, wrap=False): '''Compute the angle between this and an other vector. @arg other: The other vector (L{Vector3d}). @kwarg vSign: Optional vector, if supplied (and out of the plane of this and the other), angle is signed positive if this->other is clockwise looking along vSign or negative in opposite direction, otherwise angle is unsigned. @kwarg warp: Wrap/unroll the angle to +/-PI (c{bool}). @return: Angle (C{radians}). @raise TypeError: If B{C{other}} or B{C{vSign}} not a L{Vector3d}. ''' x = self.cross(other) s = x.length if s < EPS0: return _0_0 # use vSign as reference to get sign of s if vSign and x.dot(vSign) < 0: s = neg(s) a = atan2(s, self.dot(other)) if wrap and abs(a) > PI: a -= copysign(PI2, a) return a def cross(self, other, raiser=None): # raiser=NN '''Compute the cross product of this and an other vector. @arg other: The other vector (L{Vector3d}). @kwarg raiser: Optional, L{CrossError} label if raised (C{str}). @return: Cross product (L{Vector3d}). @raise CrossError: Zero or near-zero cross product and both B{C{raiser}} and L{crosserrors} set. @raise TypeError: Incompatible B{C{other}} C{type}. ''' self.others(other) x = self.y * other.z - self.z * other.y y = self.z * other.x - self.x * other.z z = self.x * other.y - self.y * other.x if raiser and self.crosserrors and max(map1(abs, x, y, z)) < EPS: t = _coincident_ if self.isequalTo(other) else _colinear_ r = getattr(other, '_fromll', None) or other raise CrossError(raiser, r, txt=t) return self.classof(x, y, z) @property_doc_('''raise or ignore L{CrossError} exceptions (C{bool}).''') def crosserrors(self): '''Get L{CrossError} exceptions (C{bool}). ''' return self._crosserrors @crosserrors.setter # PYCHOK setter! def crosserrors(self, raiser): '''Raise L{CrossError} exceptions (C{bool}). ''' self._crosserrors = bool(raiser) def dividedBy(self, factor): '''Divide this vector by a scalar. @arg factor: The divisor (C{scalar}). @return: New, scaled vector (L{Vector3d}). @raise TypeError: Non-scalar B{C{factor}}. @raise VectorError: Invalid or zero B{C{factor}}. ''' f = Scalar(factor=factor) try: return self.times(_1_0 / f) except (ValueError, ZeroDivisionError) as x: raise VectorError(factor=factor, txt=str(x)) def dot(self, other): '''Compute the dot (scalar) product of this and an other vector. @arg other: The other vector (L{Vector3d}). @return: Dot product (C{float}). @raise TypeError: Incompatible B{C{other}} C{type}. ''' if other is self: d = self.length2 else: self.others(other) d = fdot(self.xyz, *other.xyz) return d @deprecated_method def equals(self, other, units=False): # PYCHOK no cover '''DEPRECATED, use method C{isequalTo}. ''' return self.isequalTo(other, units=units) @Property_RO def euclid(self): '''Approximate the length (norm, magnitude) of this vector. @see: Function L{euclid_} and properties C{length} and C{length2}. ''' return Float(euclid=euclid_(self.x, self.y, self.z)) 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 isequalTo(self, other, units=False, eps=EPS): '''Check if this and an other vector are equal or equivalent. @arg other: The other vector (L{Vector3d}). @kwarg units: Optionally, compare the normalized, unit version of both vectors. @kwarg eps: Tolerance (C{scalar}), same units as C{x}, C{y}, and C{z}. @return: C{True} if vectors are identical, C{False} otherwise. @raise TypeError: Incompatible B{C{other}} C{type}. @example: >>> v1 = Vector3d(52.205, 0.119) >>> v2 = Vector3d(52.205, 0.119) >>> e = v1.isequalTo(v2) # True ''' self.others(other) if units: d = self.unit().minus(other.unit()) else: d = self.minus(other) return max(map(abs, d.xyz)) < eps @Property_RO def length(self): # __dict__ value overwritten by Property_RO C{_united} '''Get the length (norm, magnitude) of this vector (C{float}). @see: Properties L{length2} and L{euclid}. ''' return Float(length=hypot_(self.x, self.y, self.z)) @Property_RO def length2(self): # __dict__ value overwritten by Property_RO C{_united} '''Get the length I{squared} of this vector (C{float}). @see: Property L{length}. ''' return Float(length2=hypot2_(self.x, self.y, self.z)) def minus(self, other): '''Subtract an other vector from this vector. @arg other: The other vector (L{Vector3d}). @return: New vector difference (L{Vector3d}). @raise TypeError: Incompatible B{C{other}} C{type}. ''' self.others(other) return self.classof(self.x - other.x, self.y - other.y, self.z - other.z) 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, _V3d(other1) or _otherV3d(other1=other1), _V3d(other2) or _otherV3d(other2=other2), within=within) def negate(self): '''Return this vector in opposite direction. @return: New, opposite vector (L{Vector3d}). ''' return self.classof(-self.x, -self.y, -self.z) @Property_RO def _N_vector(self): '''(INTERNAL) Get the (C{nvectorBase._N_vector_}) ''' from pygeodesy.nvectorBase import _N_vector_ return _N_vector_(*self.xyz, name=self.name) def others(self, *other, **name_other_up): '''Refined class comparison. @arg other: The other vector (L{Vector3d}). @kwarg name_other_up: Overriding C{name=other} and C{up=1} keyword arguments. @return: The B{C{other}} if compatible. @raise TypeError: Incompatible B{C{other}} C{type}. ''' other, name, up = _xother3(self, other, **name_other_up) if not isinstance(other, Vector3d): _NamedBase.others(self, other, name=name, up=up + 1) return other def parse(self, str3d, sep=_COMMA_, name=NN): '''Parse an C{"x, y, z"} string to a similar 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 similar instance (L{Vector3d}). @raise VectorError: Invalid B{C{str3d}}. ''' return parse3d(str3d, sep=sep, Vector=self.classof, name=name or self.name) def plus(self, other): '''Add this vector and an other vector. @arg other: The other vector (L{Vector3d}). @return: Vectorial sum (L{Vector3d}). @raise TypeError: Incompatible B{C{other}} C{type}. ''' self.others(other) return self.classof(self.x + other.x, self.y + other.y, self.z + other.z) sum = plus # alternate name def rotate(self, axis, theta): '''Rotate this vector around an axis by a specified angle. See U{Rotation matrix from axis and angle <https://WikiPedia.org/wiki/Rotation_matrix#Rotation_matrix_from_axis_and_angle>} and U{Quaternion-derived rotation matrix <https://WikiPedia.org/wiki/Quaternions_and_spatial_rotation#Quaternion-derived_rotation_matrix>}. @arg axis: The axis being rotated around (L{Vector3d}). @arg theta: The angle of rotation (C{radians}). @return: New, rotated vector (L{Vector3d}). @JSname: I{rotateAround}. ''' s, c = sincos2(theta) a = self.others(axis=axis).unit() # axis being rotated around b = a.times(_1_0 - c) s = a.times(s) p = self.unit().xyz # point being rotated # multiply p by a quaternion-derived rotation matrix ax, ay, az = a.x, a.y, a.z bx, by, bz = b.x, b.y, b.z sx, sy, sz = s.x, s.y, s.z return self.classof(fdot(p, ax * bx + c, ax * by - sz, ax * bz + sy), fdot(p, ay * bx + sz, ay * by + c, ay * bz - sx), fdot(p, az * bx - sy, az * by + sx, az * bz + c)) @deprecated_method def rotateAround(self, axis, theta): # PYCHOK no cover '''DEPRECATED, use method C{rotate}.''' return self.rotate(axis, theta) def times(self, factor): '''Multiply this vector by a scalar. @arg factor: Scale factor (C{scalar}). @return: New, scaled vector (L{Vector3d}). @raise TypeError: Non-scalar B{C{factor}}. ''' f = Scalar(factor=factor) return self.classof(self.x * f, self.y * f, self.z * f) @deprecated_method def to2ab(self): # PYCHOK no cover '''DEPRECATED, use property C{Nvector.philam}. @return: A L{PhiLam2Tuple}C{(phi, lam)}. ''' return n_xyz2philam(self.x, self.y, self.z) @deprecated_method def to2ll(self): # PYCHOK no cover '''DEPRECATED, use property C{Nvector.latlon}. @return: A L{LatLon2Tuple}C{(lat, lon)}. ''' return n_xyz2latlon(self.x, self.y, self.z) @deprecated_method def to3xyz(self): # PYCHOK no cover '''DEPRECATED, use property L{xyz}. ''' return self.xyz def toStr(self, prec=5, fmt=Fmt.PAREN, sep=_COMMASPACE_): # PYCHOK expected '''Return a string representation of this vector. @kwarg prec: Optional number of decimal places (C{int}). @kwarg fmt: Optional, enclosing format to use (C{str}). @kwarg sep: Optional separator between components (C{str}). @return: Vector as "(x, y, z)" (C{str}). ''' t = sep.join(strs(self.xyz, prec=prec)) return (fmt % (t,)) if fmt else t def trilaterate3d2(self, radius, center2, radius2, center3, radius3, eps=EPS): '''Trilaterate this and two other spheres, each given as a (3d) 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: No intersection, colinear or near concentric centers or trilateration failed some other way. @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, FloatingPointError) as x: raise IntersectionError(center=self, radius=radius, center2=center2, radius2=radius2, center3=center3, radius3=radius3, txt=str(x)) def unit(self, ll=None): '''Normalize this vector to unit length. @kwarg ll: Optional, original location (C{LatLon}). @return: Normalized vector (L{Vector3d}). ''' u = self._united if ll: u._fromll = ll return u @Property_RO def _united(self): # __dict__ value overwritten below '''(INTERNAL) Get normalized vector (L{Vector3d}). ''' n = self.length if n > EPS0 and abs(n - _1_0) > EPS0: u = self._xnamed(self.dividedBy(n)) u._overwrite(length=_1_0, length2=_1_0) else: u = self.copy() u._overwrite(_united=u) if self._fromll: u._fromll = self._fromll return u @Property_RO def x(self): '''Get the X component (C{float}). ''' return self._x @Property_RO def xyz(self): '''Get the X, Y and Z components (L{Vector3Tuple}C{(x, y, z)}). ''' return Vector3Tuple(self.x, self.y, self.z, name=self.name) @Property_RO def y(self): '''Get the Y component (C{float}). ''' return self._y @Property_RO def z(self): '''Get the Z component (C{float}). ''' return self._z 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 IntersectionError(center1=center1, radius1=radius1, center2=center2, radius2=radius2, txt=str(x)) def _intersects2(center1, r1, center2, r2, sphere=True, too_d=None, # in .ellipsoidalBase._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(sphere=sphere, center1=center1) c2 = _otherV3d(sphere=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 ValueError(_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 ValueError(_invalid_) else: # abutting y = _0_0 elif o < 0: t = d if too_d is None else too_d raise ValueError(_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(_V3d(point) or _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 = _V3d(p1) or _otherV3d(point1=p1) v2 = _V3d(p2) or _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(_V3d(point) or _otherV3d(point =point), _V3d(point1) or _otherV3d(point1=point1), _V3d(point2) or _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(sphere=True, **name_v): # check B{C{name#}} vector instance, return Vector3d name, v = _xkwds_popitem(name_v) try: return Vector3d(v.x, v.y, v.z if sphere 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 trilaterate3d2(center1, radius1, center2, radius2, center3, radius3, eps=EPS, Vector=None, **Vector_kwds): '''Trilaterate three spheres, each given as a (3d) center point 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: No intersection, colinear or concentric centers or trilateration failed some other way. @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>}. ''' try: return _trilaterate3d2(_otherV3d(center1=center1), Radius_(radius1=radius1, low=eps), center2, radius2, center3, radius3, eps=eps, Vector=Vector, **Vector_kwds) except (AssertionError, FloatingPointError) as x: raise IntersectionError(center1=center1, radius1=radius1, center2=center2, radius2=radius2, center3=center3, radius3=radius3, txt=str(x)) def _trilaterate3d2(c1, r1, c2, r2, c3, r3, eps=EPS, Vector=None, **Vector_kwds): # MCCABE 13 # (INTERNAL) Intersect three spheres or circles, see L{trilaterate3d2} # above, separated to allow callers to embellish any exceptions 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 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 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) finally: # restore numpy error handling np.seterr(**e) if t: break else: # coincident, 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): # check for concentric or too distant spheres d = c1.minus(c2).length if d < abs(r1 - r2): t = _near_concentric_ elif d > (r1 + r2): t = _too_(Fmt.distant(d)) else: return NN return _SPACE_(c1.name, 'and', c2.name, 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 FloatingPointError(t) def _V3d(v3d): # return v3d if it's a L{Vector3d} instance return v3d if isinstance(v3d, Vector3d) else None 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) __all__ += _ALL_DOCS(intersections2, sumOf) # **) 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.