# Coverage for pygeodesy/vector2d.py : 96%

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''' from pygeodesy.basics import len2, map2, _xnumpy from pygeodesy.constants import EPS, EPS0, EPS02, EPS4, INF, INT0, \ _EPS4e8, isnear0, _0_0, _0_25, _0_5, _N_0_5, \ _1_0, _1_0_1T, _N_1_0, _2_0, _N_2_0, _4_0 from pygeodesy.errors import _and, _AssertionError, IntersectionError, NumPyError, \ PointsError, TriangleError, _xError, _xkwds from pygeodesy.fmath import fabs, fdot, hypot, hypot2_, sqrt from pygeodesy.fsums import Fsum, fsum_, fsum1_ from pygeodesy.interns import NN, _a_, _and_, _b_, _c_, _center_, _coincident_, \ _colinear_, _concentric_, _COMMASPACE_, _few_, \ _intersection_, _invalid_, _near_, _no_, _radius_, \ _rIn_, _s_, _SPACE_, _too_, _with_ # from pygeodesy.lazily import _ALL_LAZY # from .named from pygeodesy.named import _ALL_LAZY, _NamedTuple, _Pass, Property_RO from pygeodesy.namedTuples import LatLon3Tuple, Vector2Tuple # from pygeodesy.props import Property_RO # from .named from pygeodesy.streprs import Fmt, unstr from pygeodesy.units import Float, Int, Meter, Radius, Radius_ from pygeodesy.vector3d import iscolinearWith, nearestOn, _nearestOn2, _nVc, _otherV3d, \ trilaterate2d2, trilaterate3d2, Vector3d # PYCHOK unused from contextlib import contextmanager # from math import fabs, sqrt # from .fmath __all__ = _ALL_LAZY.vector2d __version__ = '22.10.12' _cA_ = 'cA' _cB_ = 'cB' _cC_ = 'cC' _deltas_ = 'deltas' _of_ = 'of' _outer_ = 'outer' _raise_ = 'raise' # PYCHOK used! _rank_ = 'rank' _residuals_ = 'residuals' _Type_ = 'Type' class Circin6Tuple(_NamedTuple): '''6-Tuple C{(radius, center, deltas, cA, cB, cC)} with the C{radius}, the trilaterated C{center} and contact points of the I{inscribed} aka I{In- circle} of a triangle. The C{center} is I{un}ambiguous if C{deltas} is C{None}, otherwise C{center} is the mean and C{deltas} the differences of the L{pygeodesy.trilaterate3d2} results. Contact points C{cA}, C{cB} and C{cC} are the points of tangency, aka the corners of the U{Contact Triangle <https://MathWorld.Wolfram.com/ContactTriangle.html>}. ''' _Names_ = (_radius_, _center_, _deltas_, _cA_, _cB_, _cC_) _Units_ = ( Radius, _Pass, _Pass, _Pass, _Pass, _Pass) class Circum3Tuple(_NamedTuple): # in .latlonBase '''3-Tuple C{(radius, center, deltas)} with the C{circumradius} and trilaterated C{circumcenter} of the C{circumcircle} through 3 points (aka {Meeus}' Type II circle) or the C{radius} and C{center} of the smallest I{Meeus}' Type I circle. The C{center} is I{un}ambiguous if C{deltas} is C{None}, otherwise C{center} is the mean and C{deltas} the differences of the L{pygeodesy.trilaterate3d2} results. ''' _Names_ = (_radius_, _center_, _deltas_) _Units_ = ( Radius, _Pass, _Pass) class Circum4Tuple(_NamedTuple): '''4-Tuple C{(radius, center, rank, residuals)} with C{radius} and C{center} of a sphere I{least-squares} fitted through given points and the C{rank} and C{residuals} -if any- from U{numpy.linalg.lstsq <https://NumPy.org/doc/stable/reference/generated/numpy.linalg.lstsq.html>}. ''' _Names_ = (_radius_, _center_, _rank_, _residuals_) _Units_ = ( Radius, _Pass, Int, _Pass) class Meeus2Tuple(_NamedTuple): '''2-Tuple C{(radius, Type)} with C{radius} and I{Meeus}' C{Type} of the smallest circle I{containing} 3 points. C{Type} is C{None} for a I{Meeus}' Type II C{circumcircle} passing through all 3 points. Otherwise C{Type} is the center of a I{Meeus}' Type I circle with 2 points on (a diameter of) and 1 point inside the circle. ''' _Names_ = (_radius_, _Type_) _Units_ = ( Radius, _Pass) class Radii11Tuple(_NamedTuple): '''11-Tuple C{(rA, rB, rC, cR, rIn, riS, roS, a, b, c, s)} with the C{Tangent} circle radii C{rA}, C{rB} and C{rC}, the C{circumradius} C{cR}, the C{Incircle} radius C{rIn} aka C{inradius}, the inner and outer I{Soddy} circle radii C{riS} and C{roS}, the sides C{a}, C{b} and C{c} and semi-perimeter C{s} of a triangle, all in C{meter} conventionally. @note: C{Circumradius} C{cR} and outer I{Soddy} radius C{roS} may be C{INF}. ''' _Names_ = ('rA', 'rB', 'rC', 'cR', _rIn_, 'riS', 'roS', _a_, _b_, _c_, _s_) _Units_ = ( Meter,) * len(_Names_) class Soddy4Tuple(_NamedTuple): '''4-Tuple C{(radius, center, deltas, outer)} with C{radius} and (trilaterated) C{center} of the I{inner} I{Soddy} circle and the radius of the C{outer} I{Soddy} circle. The C{center} is I{un}ambiguous if C{deltas} is C{None}, otherwise C{center} is the mean and C{deltas} the differences of the L{pygeodesy.trilaterate3d2} results. @note: The outer I{Soddy} radius C{outer} may be C{INF}. ''' _Names_ = (_radius_, _center_, _deltas_, _outer_) _Units_ = ( Radius, _Pass, _Pass, Radius) def circin6(point1, point2, point3, eps=EPS4, useZ=True): '''Return the radius and center of the I{inscribed} aka I{Incircle} of a (2- or 3-D) triangle. @arg point1: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}). @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}). @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}). @kwarg eps: Tolerance for function L{pygeodesy.trilaterate3d2} if C{B{useZ} is True} else L{pygeodesy.trilaterate2d2}. @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}). @return: L{Circin6Tuple}C{(radius, center, deltas, cA, cB, cC)}. The C{center} and contact points C{cA}, C{cB} and C{cC}, each an instance of B{C{point1}}'s (sub-)class, are co-planar with the three given points. @raise ImportError: Package C{numpy} not found, not installed or older than version 1.10 and C{B{useZ} is True}. @raise IntersectionError: Near-coincident or -colinear points or a trilateration or C{numpy} issue. @raise TypeError: Invalid B{C{point1}}, B{C{point2}} or B{C{point3}}. @see: Functions L{radii11} and L{circum3}, U{Contact Triangle <https://MathWorld.Wolfram.com/ContactTriangle.html>} and U{Incircle<https://MathWorld.Wolfram.com/Incircle.html>}. ''' try: return _circin6(point1, point2, point3, eps=eps, useZ=useZ) except (AssertionError, TypeError, ValueError) as x: raise _xError(x, point1=point1, point2=point2, point3=point3) def _circin6(point1, point2, point3, eps=EPS4, useZ=True, dLL3=False, **Vector_kwds): # (INTERNAL) Radius, center, deltas, 3 contact points def _fraction(r, a): return (r / a) if a > EPS0 else _0_5 def _contact2(a, p2, r2, p3, r3, V, V_kwds): c = p2.intermediateTo(p3, _fraction(r2, a)) if r2 > r3 else \ p3.intermediateTo(p2, _fraction(r3, a)) C = V(c.x, c.y, c.z, **V_kwds) return c, C t, p1, p2, p3 = _radii11ABC(point1, point2, point3, useZ=useZ) V, r1, r2, r3 = point1.classof, t.rA, t.rB, t.rC c1, cA = _contact2(t.a, p2, r2, p3, r3, V, _xkwds(Vector_kwds, name=_cA_)) c2, cB = _contact2(t.b, p3, r3, p1, r1, V, _xkwds(Vector_kwds, name=_cB_)) c3, cC = _contact2(t.c, p1, r1, p2, r2, V, _xkwds(Vector_kwds, name=_cC_)) r = t.rIn c, d = _tricenter3d2(c1, r, c2, r, c3, r, eps=eps, useZ=useZ, dLL3=dLL3, **_xkwds(Vector_kwds, Vector=V, name=circin6.__name__)) return Circin6Tuple(r, c, d, cA, cB, cC) def circum3(point1, point2, point3, circum=True, eps=EPS4, useZ=True): '''Return the radius and center of the smallest circle I{through} or I{containing} three (2- or 3-D) points. @arg point1: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @kwarg circum: If C{True} return the C{circumradius} and C{circumcenter} always, ignoring the I{Meeus}' Type I case (C{bool}). @kwarg eps: Tolerance for function L{pygeodesy.trilaterate3d2} if C{B{useZ} is True} else L{pygeodesy.trilaterate2d2}. @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}). @return: A L{Circum3Tuple}C{(radius, center, deltas)}. The C{center}, an instance of B{C{point1}}'s (sub-)class, is co-planar with the three given points. @raise ImportError: Package C{numpy} not found, not installed or older than version 1.10 and C{B{useZ} is True}. @raise IntersectionError: Near-coincident or -colinear points or a trilateration or C{numpy} issue. @raise TypeError: Invalid B{C{point1}}, B{C{point2}} or B{C{point3}}. @see: Functions L{pygeodesy.circum4_} and L{pygeodesy.meeus2} and Meeus, J. U{I{Astronomical Algorithms}<http://www.Agopax.IT/Libri_astronomia/pdf/ Astronomical%20Algorithms.pdf>}, 2nd Ed. 1998, page 127ff, U{circumradius <https://MathWorld.Wolfram.com/Circumradius.html>} and U{circumcircle <https://MathWorld.Wolfram.com/Circumcircle.html>}. ''' try: p1 = _otherV3d(useZ=useZ, point1=point1) return _circum3(p1, point2, point3, circum=circum, eps=eps, useZ=useZ, clas=point1.classof) except (AssertionError, TypeError, ValueError) as x: raise _xError(x, point1=point1, point2=point2, point3=point3, circum=circum) def _circum3(p1, point2, point3, circum=True, eps=EPS4, useZ=True, dLL3=False, clas=Vector3d, **clas_kwds): # in .latlonBase # (INTERNAL) Radius, center, deltas r, d, p2, p3 = _meeus4(p1, point2, point3, circum=circum, useZ=useZ, clas=clas, **clas_kwds) if d is None: # Meeus' Type II or circum=True kwds = _xkwds(clas_kwds, eps=eps, Vector=clas, name=circum3.__name__) c, d = _tricenter3d2(p1, r, p2, r, p3, r, useZ=useZ, dLL3=dLL3, **kwds) else: # Meeus' Type I c, d = d, None return Circum3Tuple(r, c, d) def circum4_(*points, **useZ_Vector_and_kwds): '''Best-fit a sphere through three or more (3-D) points. @arg points: The points (each a C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, or C{Vector4Tuple}). @kwarg useZ_Vector_and_kwds: Keyword arguments C{B{useZ}=True} (C{bool}) to use the Z components, otherwise force all C{z=INT0}, class C{B{Vector}=None} to return the center point with optionally, additional nB{C{Vector}} keyword arguments, otherwise the first B{C{points}}' (sub-)class is used. @return: L{Circum4Tuple}C{(radius, center, rank, residuals)} with C{center} an instance of C{B{points}[0]}' (sub-)class or B{C{Vector}} if specified. @raise ImportError: Package C{numpy} not found, not installed or older than version 1.10. @raise NumPyError: Some C{numpy} issue. @raise PointsError: Too few B{C{points}}. @raise TypeError: One of the B{C{points}} is invalid. @see: Functions L{pygeodesy.circum3} and L{pygeodesy.meeus2}, Jekel, Charles F. U{I{Least Squares Sphere Fit}<https://Jekel.me/2015/Least-Squares-Sphere-Fit/>} Sep 13, 2015, U{Appendix A<https://hdl.handle.net/10019.1/98627>}, U{numpy.linalg.lstsq<https:// NumPy.org/doc/stable/reference/generated/numpy.linalg.lstsq.html>} and U{Eberly 6 <https://www.sci.Utah.EDU/~balling/FEtools/doc_files/LeastSquaresFitting.pdf>}. ''' def _useZ_kwds(useZ=True, **kwds): return useZ, kwds n, ps = len2(points) if n < 3: raise PointsError(points=n, txt=_too_(_few_)) useZ, kwds = _useZ_kwds(**useZ_Vector_and_kwds) A, b = [], [] for i, p in enumerate(ps): v = _otherV3d(useZ=useZ, i=i, points=p) A.append(v.times(_2_0).xyz + _1_0_1T) b.append(v.length2) with _numpy(circum4_, n=n) as _np: A = _np.array(A).reshape((n, 4)) b = _np.array(b).reshape((n, 1)) C, R, rk, _ = _np.least_squares4(A, b, rcond=None) # to silence warning C = map2(float, C) R = map2(float, R) # empty if rk < 4 or n <= 4 n = circum4_.__name__ c = Vector3d(*C[:3], name=n) r = Radius(sqrt(fsum_(C[3], *c.x2y2z2)), name=n) c = _nVc(c, **_xkwds(kwds, clas=ps[0].classof, name=n)) return Circum4Tuple(r, c, rk, R) def _iscolinearWith(p, point1, point2, eps=EPS, useZ=True): # (INTERNAL) Check colinear, see L{iscolinearWith} above, # separated to allow callers to embellish any exceptions p1 = _otherV3d(useZ=useZ, point1=point1) p2 = _otherV3d(useZ=useZ, point2=point2) n, _ = _nearestOn2(p, p1, p2, within=False, eps=eps) return n is p1 or n.minus(p).length2 < eps def meeus2(point1, point2, point3, circum=False, useZ=True): '''Return the radius and I{Meeus}' Type of the smallest circle I{through} or I{containing} three (2- or 3-D) points. @arg point1: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}). @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}). @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}). @kwarg circum: If C{True} return the C{circumradius} and C{circumcenter} always, overriding I{Meeus}' Type II case (C{bool}). @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}). @return: L{Meeus2Tuple}C{(radius, Type)}, with C{Type} the C{circumcenter} iff C{B{circum}=True}. @raise IntersectionError: Near-coincident or -colinear points, iff C{B{circum}=True}. @raise TypeError: Invalid B{C{point1}}, B{C{point2}} or B{C{point3}}. @see: Functions L{pygeodesy.circum3} and L{pygeodesy.circum4_} and Meeus, J. U{I{Astronomical Algorithms}<http://www.Agopax.IT/Libri_astronomia/pdf/ Astronomical%20Algorithms.pdf>}, 2nd Ed. 1998, page 127ff, U{circumradius <https://MathWorld.Wolfram.com/Circumradius.html>} and U{circumcircle <https://MathWorld.Wolfram.com/Circumcircle.html>}. ''' try: A = _otherV3d(useZ=useZ, point1=point1) return _meeus2(A, point2, point3, circum, useZ=useZ, clas=point1.classof) except (TypeError, ValueError) as x: raise _xError(x, point1=point1, point2=point2, point3=point3, circum=circum) def _meeus2(A, point2, point3, circum, useZ=True, **clas_and_kwds): # in .vector3d # (INTERNAL) Radius and center or Meeus' Type f = _circum3 if circum else _meeus4 t = f(A, point2, point3, circum=circum, useZ=useZ, **clas_and_kwds)[:2] return Meeus2Tuple(t) def _meeus4(A, point2, point3, circum=False, useZ=True, clas=None, **clas_kwds): # (INTERNAL) Radius and Meeus' Type B = p2 = _otherV3d(useZ=useZ, point2=point2) C = p3 = _otherV3d(useZ=useZ, point3=point3) a = B.minus(C).length2 b = C.minus(A).length2 c = A.minus(B).length2 if a < b: a, b, A, B = b, a, B, A if a < c: a, c, A, C = c, a, C, A if a > EPS02 and (circum or a < (b + c)): # circumradius b = sqrt(b / a) c = sqrt(c / a) R = Fsum(_1_0, b, c) * Fsum(_1_0, b, -c) * Fsum(_1_0, -b, c) * Fsum(_N_1_0, b, c) r = R.fover(a) if r < EPS02: raise IntersectionError(_coincident_ if b < EPS0 or c < EPS0 else ( _colinear_ if _iscolinearWith(A, B, C) else _invalid_)) r = b * c / sqrt(r) t = None # Meeus' Type II else: # obtuse or right angle at A r = sqrt(a * _0_25) if a > EPS02 else INT0 t = B.plus(C).times(_0_5) # Meeus' Type I if clas is not None: t = clas(t.x, t.y, t.z, **_xkwds(clas_kwds, name=meeus2.__name__)) return r, t, p2, p3 class _numpy(object): '''(INTERNAL) Partial C{NumPy} wrapper. ''' @contextmanager # <https://www.python.org/dev/peps/pep-0343/> Examples def __call__(self, where, *args, **kwds): '''(INTERNAL) Yield self with any errors raised as L{NumPyError}. ''' np = self.np try: # <https://NumPy.org/doc/stable/reference/generated/numpy.seterr.html> e = np.seterr(all=_raise_) # throw FloatingPointError for numpy errors yield self except Exception as x: # mostly FloatingPointError? t = unstr(where, *args, **kwds) raise NumPyError(t, cause=x) # _xError2? finally: # restore numpy error handling np.seterr(**e) @Property_RO def array(self): return self.np.array @Property_RO def least_squares4(self): '''Linear least-squares function. ''' return self.np.linalg.lstsq @Property_RO def np(self): '''Import numpy 1.10+ once. ''' return _xnumpy(self.__class__, 1, 10) def null_space2(self, A, rcond=None): '''Return the C{null_space} and C{rank} of matrix B{C{A}}. @see: U{Source<https://docs.SciPy.org/doc/scipy/reference/generated/scipy.linalg.null_space.html>} U{SciPY Cookbook<https://SciPy-Cookbook.ReadTheDocs.io/items/RankNullspace.html>}, U{here <https://NumPy.org/doc/stable/reference/generated/numpy.linalg.svd.html>}, U{here <https://StackOverflow.com/questions/19820921>}, U{here <https://StackOverflow.com/questions/2992947>} and U{here <https://StackOverflow.com/questions/5889142>}. ''' def _Error(txt=self.null_space2.__name__, **kwds): return _AssertionError(txt=txt, **kwds) np = self.np A = np.array(A) m = max(A.shape) if m != 4: # for this usage raise _Error(shape=m) # if needed, square A, pad with zeros A = np.resize(A, m * m).reshape(m, m) # try: # no np.linalg.null_space <https://docs.SciPy.org/doc/> # Z = scipy.linalg.null_space(A) # XXX no scipy.linalg? # return Z, ... # except AttributeError: # pass U, S, V = np.linalg.svd(A) s = max(EPS, rcond) if rcond else (EPS * max(U.shape[0], V.shape[1])) t = max(EPS, float(np.max(S) * s)) # abs_tol, rel_tol * largest singular r = int(np.sum(S > t)) # rank if r == 3: # get null_space Z = np.transpose(V[r:]) s = map2(int, Z.shape) if s != (m, 1): # bad null_space shape raise _Error(shape=s, m=m) D = A.dot(Z) # near-zeros-vector n = float(np.linalg.norm(D, INF)) # INF = max(fabs(D)), 2 = hypot_(*D) if n > t: # largest exceed tol raise _Error(dot=tuple(D.ravel()), norm=n, tol=t) else: # coincident, colinear, concentric centers, ambiguous, etc. Z = None # del A, S, U, V # release numpy return Z, r @Property_RO def pseudo_inverse(self): '''Moore-Penrose pseudo-inverse function. ''' return self.np.linalg.pinv def real_roots(self, *coeffs): '''Compute the real, non-complex roots of a polynomial. ''' np = self.np rs = np.polynomial.polynomial.polyroots(coeffs) return tuple(float(r) for r in rs if not np.iscomplex(r)) _numpy = _numpy() # PYCHOK singleton def radii11(point1, point2, point3, useZ=True): '''Return the radii of the C{In-}, I{Soddy} and C{Tangent} circles of a (2- or 3-D) triangle. @arg point1: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}). @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}). @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}). @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}). @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{point1}}, B{C{point2}} or B{C{point3}}. @see: U{Circumradius<https://MathWorld.Wolfram.com/Circumradius.html>}, U{Incircle<https://MathWorld.Wolfram.com/Incircle.html>}, U{Soddy Circles<https://MathWorld.Wolfram.com/SoddyCircles.html>} and U{Tangent Circles<https://MathWorld.Wolfram.com/TangentCircles.html>}. ''' try: return _radii11ABC(point1, point2, point3, useZ=useZ)[0] except (TypeError, ValueError) as x: raise _xError(x, point1=point1, point2=point2, point3=point3) def _radii11ABC(point1, point2, point3, useZ=True): # (INTERNAL) Tangent, Circum, Incircle, Soddy radii, sides and semi-perimeter A = _otherV3d(useZ=useZ, point1=point1, NN_OK=False) B = _otherV3d(useZ=useZ, point2=point2, NN_OK=False) C = _otherV3d(useZ=useZ, point3=point3, NN_OK=False) a = B.minus(C).length b = C.minus(A).length c = A.minus(B).length S = Fsum(a, b, c) * _0_5 s = float(S) # semi-perimeter if s > EPS0: rs = float(S - a), float(S - b), float(S - c) r3, r2, r1 = sorted(rs) # r3 <= r2 <= r1 if r3 > EPS0: # and r2 > EPS0 and r1 > EPS0 r3_r1 = r3 / r1 r3_r2 = r3 / r2 # t = r1 * r2 * r3 * (r1 + r2 + r3) # = r1**2 * r2 * r3 * (1 + r2 / r1 + r3 / r1) # = (r1 * r2)**2 * (r3 / r2) * (1 + r2 / r1 + r3 / r1) t = r3_r2 * fsum1_(_1_0, r2 / r1, r3_r1) # * (r1 * r2)**2 if t > EPS02: t = sqrt(t) * _2_0 # * r1 * r2 # d = r1 * r2 + r2 * r3 + r3 * r1 # = r1 * (r2 + r2 * r3 / r1 + r3) # = r1 * r2 * (1 + r3 / r1 + r3 / r2) d = fsum1_(_1_0, r3_r1, r3_r2) # * r1 * r2 # si/o = r1 * r2 * r3 / (r1 * r2 * (d +/- t)) # = r3 / (d +/- t) si = r3 / (d + t) so = (r3 / (d - t)) if d > t else INF # ci = sqrt(r1 * r2 * r3 / s) # = r1 * sqrt(r2 * r3 / r1 / s) ci = r1 * sqrt(r2 * r3_r1 / s) # co = a * b * c / (4 * ci * s) t = ci * s * _4_0 co = (a * b * c / t) if t > EPS0 else INF r1, r2, r3 = rs # original order t = Radii11Tuple(r1, r2, r3, co, ci, si, so, a, b, c, s) return t, A, B, C raise TriangleError(_near_(_coincident_) if min(a, b, c) < EPS0 else ( _colinear_ if _iscolinearWith(A, B, C) else _invalid_)) def soddy4(point1, point2, point3, eps=EPS4, useZ=True): '''Return the radius and center of the C{inner} I{Soddy} circle of a (2- or 3-D) triangle. @arg point1: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}). @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}). @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}). @kwarg eps: Tolerance for function L{pygeodesy.trilaterate3d2} if C{B{useZ} is True} otherwise L{pygeodesy.trilaterate2d2}. @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}). @return: L{Soddy4Tuple}C{(radius, center, deltas, outer)}. The C{center}, an instance of B{C{point1}}'s (sub-)class, is co-planar with the three given points. The C{outer} I{Soddy} radius may be C{INF}. @raise ImportError: Package C{numpy} not found, not installed or older than version 1.10 and C{B{useZ} is True}. @raise IntersectionError: Near-coincident or -colinear points or a trilateration or C{numpy} issue. @raise TypeError: Invalid B{C{point1}}, B{C{point2}} or B{C{point3}}. @see: Functions L{radii11} and L{circum3} and U{Soddy Circles <https://MathWorld.Wolfram.com/SoddyCircles.html>}. ''' t, p1, p2, p3 = _radii11ABC(point1, point2, point3, useZ=useZ) r = t.riS c, d = _tricenter3d2(p1, t.rA + r, p2, t.rB + r, p3, t.rC + r, eps=eps, useZ=useZ, Vector=point1.classof, name=soddy4.__name__) return Soddy4Tuple(r, c, d, t.roS) def _tricenter3d2(p1, r1, p2, r2, p3, r3, eps=EPS4, useZ=True, dLL3=False, **kwds): # (INTERNAL) Trilaterate and disambiguate the 3-D center d, kwds = None, _xkwds(kwds, eps=eps, coin=True) if useZ and p1.z != p2.z != p3.z: # ignore z if all match a, b = _trilaterate3d2(p1, r1, p2, r2, p3, r3, **kwds) if a is b: # no unambiguity c = a # == b else: c = a.plus(b).times(_0_5) # mean if not a.isconjugateTo(b, minum=0, eps=eps): if dLL3: # deltas as (lat, lon, height) a = a.toLatLon() b = b.toLatLon() d = LatLon3Tuple(b.lat - a.lat, b.lon - a.lon, b.height - a.height, name=_deltas_) else: d = b.minus(a) # vectorial deltas else: if useZ: # pass z to Vector if given kwds = _xkwds(kwds, z=p1.z) c = _trilaterate2d2(p1.x, p1.y, r1, p2.x, p2.y, r2, p3.x, p3.y, r3, **kwds) return c, d def _trilaterate2d2(x1, y1, radius1, x2, y2, radius2, x3, y3, radius3, coin=False, eps=None, Vector=None, **Vector_kwds): # (INTERNAL) Trilaterate three circles, see L{pygeodesy.trilaterate2d2} def _abct4(x1, y1, r1, x2, y2, r2): a = x2 - x1 b = y2 - y1 t = _tri3near2far(r1, r2, hypot(a, b), coin) c = _0_0 if t else (hypot2_(r1, x2, y2) - hypot2_(r2, x1, y1)) return a, b, c, t def _astr(**kwds): # kwds as (name=value, ...) strings return Fmt.PAREN(_COMMASPACE_(*(Fmt.EQUAL(*t) for t in kwds.items()))) r1 = Radius_(radius1=radius1) r2 = Radius_(radius2=radius2) r3 = Radius_(radius3=radius3) a, b, c, t = _abct4(x1, y1, r1, x2, y2, r2) 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 = (a * e - b * d) * _2_0 if isnear0(q): t = _no_(_intersection_) raise IntersectionError(_and(_astr(x1=x1, y1=y1, radius1=r1), _astr(x2=x2, y2=y2, radius2=r2), _astr(x3=x3, y3=y3, radius3=r3)), txt=t) t = Vector2Tuple((c * e - b * f) / q, (a * f - c * d) / q, name=trilaterate2d2.__name__) if eps and eps > 0: # check distances to center vs radius for x, y, r in ((x1, y1, r1), (x2, y2, r2), (x3, y3, r3)): d = hypot(x - t.x, y - t.y) e = fabs(d - r) if e > eps: t = _and(Float(delta=e).toRepr(), r.toRepr(), Float(distance=d).toRepr(), t.toRepr()) raise IntersectionError(t, txt=Fmt.exceeds_eps(eps)) if Vector is not None: t = Vector(t.x, t.y, **_xkwds(Vector_kwds, name=t.name)) return t def _trilaterate3d2(c1, r1, c2, r2, c3, r3, eps=EPS, coin=False, **clas_Vector_and_kwds): # (INTERNAL) Intersect three spheres or circles, see L{pygeodesy.trilaterate3d2}, # separated to allow callers to embellish any exceptions, like # C{FloatingPointError}s from C{numpy} def _F3d2(F): # map numpy 4-vector to floats tuple and Vector3d T = map2(float, F) return T, Vector3d(*T[1:]) def _N3(t01, x, z): # compute x, y and z and return as B{C{clas}} or B{C{Vector}} v = x.plus(z.times(t01)) n = trilaterate3d2.__name__ return _nVc(v, **_xkwds(clas_Vector_and_kwds, name=n)) c2 = _otherV3d(center2=c2, NN_OK=False) c3 = _otherV3d(center3=c3, NN_OK=False) rs = (r1, Radius_(radius2=r2, low=EPS), Radius_(radius3=r3, low=EPS)) # get matrix A[3 x 4], its null_space Z and pseudo-invert A A = [(_1_0_1T + c.times(_N_2_0).xyz) for c in (c1, c2, c3)] with _numpy(trilaterate3d2, A=A, eps=eps) as _np: Z, _ = _np.null_space2(A, eps) if Z is not None: Z, z = _F3d2(Z) # [4 x 1] z2 = z.length2 A = _np.pseudo_inverse(A) # [4 x 3] bs = [c.length2 for c in (c1, c2, c3)] # perturbe radii and vector b slightly by eps and eps * 4 for p in _tri5perturb(eps, min(rs)): b = [((r + p)**2 - b) for r, b in zip(rs, bs)] # [3 x 1] X, x = _F3d2(A.dot(b)) # quadratic polynomial, coefficients ordered (^0, ^1, ^2) t = _np.real_roots(fdot(X, _N_1_0, *x.xyz), fdot(Z, _N_0_5, *x.xyz) * _2_0, z2) if t: v = _N3(t[0], x, z) if len(t) < 2: # one intersection t = v, v elif fabs(t[0] - t[1]) < eps: # abutting t = v, v else: # "lowest" intersection first (to avoid test failures) u = _N3(t[1], x, z) t = (u, v) if u.x < v.x else (v, u) return t # coincident, concentric, colinear, too distant, no intersection: # create the explanation and and throw an IntersectionError def _no_intersection(coin): t = _no_(_intersection_) if coin: def _reprs(*crs): return _and(*map(repr, crs)) r = repr(r1) if r1 == r2 == r3 else _reprs(r1, r2, r3) t = _SPACE_(t, _of_, _reprs(c1, c2, c3), _with_, _radius_, r) elif Z is None: t = _COMMASPACE_(t, _no_(_numpy.null_space2.__name__)) return t t = _tri4near2far(c1, r1, c2, r2, coin) or \ _tri4near2far(c1, r1, c3, r3, coin) or \ _tri4near2far(c2, r2, c3, r3, coin) or ( _colinear_ if _iscolinearWith(c1, c2, c3, eps=eps) else _no_intersection(coin)) raise IntersectionError(t, txt=None) def _tri3near2far(r1, r2, h, coin): # check for near-coincident/-concentric or too distant spheres/circles return _too_(Fmt.distant(h)) if h > (r1 + r2) else (_near_( _coincident_ if coin else _concentric_) if h < fabs(r1 - r2) else NN) def _tri4near2far(c1, r1, c2, r2, coin): # check for near-coincident/-concentric or too distant spheres/circles t = _tri3near2far(r1, r2, c1.minus(c2).length, coin) return _SPACE_(c1.name, _and_, c2.name, t) if t else NN def _tri5perturb(eps, r): # perturb the radii to handle this corner case # <https://GitHub.com/mrJean1/PyGeodesy/issues/49> yield _0_0 if eps and eps > 0: p = max(eps, EPS) yield p m = min(p, r) yield -m q = max(eps * _4_0, _EPS4e8) if q > p: yield q q = min(q, r) if q > m: yield -q # **) 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.