# Coverage for pygeodesy/vector3d.py : 94%

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1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 | # -*- coding: utf-8 -*- u'''Generic 3-D vector base class L{Vector3d}, class L{VectorError} and functions L{intersections2}, L{iscolinearWith}, L{nearestOn}, L{parse3d}, L{sumOf} and L{trilaterate3d2}. 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 _and, _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, _and_, _coincident_, _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, _NamedBase, _xnamed, \ _xother3, _xotherError from pygeodesy.namedTuples import Vector2Tuple, 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, hypot, sqrt __all__ = _ALL_LAZY.vector3d __version__ = '21.04.15' _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}. ''' 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 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 abs(q) < EPS0: 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 (3d) 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: No intersection, colinear or near-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>} 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, 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): 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 FloatingPointError(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 _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.