o o h@s8ddlmZddlmZddlmZddlmZmZm Z ddl m Z ddl m Z ddl mZddlmZdd lmZdd lmZdd lmZdd lmZdd lmZddlmZmZmZm Z ddl!m"Z"ddl mZddl#m$Z$ddl%m&Z&ddl'Z(ddl)m*Z*m+Z+m,Z,m-Z-m.Z.GdddeZ/ddZ0ddl1m2Z2dS))Callable)Basic)cacheit)SDummyLambda)Str)symbols)ImmutableDenseMatrix) MatrixBase)solve) BaseScalar)Tuple)diff)sqrt)acosatan2cossin)eyesimplifytrigsimpN)Orienter AxisOrienter BodyOrienter SpaceOrienterQuaternionOrientercsjeZdZdZ  d?fdd ZddZddZed d Zed d Z d dZ eddZ eddZ ddZ eddZeddZeddZddZddZdd Zd!d"Zd#d$Zd%d&Zd'd(Zed)d*Zd+d,Z  d@d-d.Z  dAd/d0Z  dAd1d2Z  dAd3d4Z  dAd5d6Z!  dAd7d8Z"d@d9d:Z#    dBd;d<Z$eje$_ed=d>Z%Z&S)C CoordSys3Dz6 Represents a coordinate system in 3-D space. Ncs8ttjj}tjj} ttstddur}|dus!|dur%tdttt t frGtdt r=d}d}n@t ddn6tt rbtdtd\} } } t | | | f| | | nttrltnttt frtn tdt|durttd }n t|t std |}|durt|tstd |dur|j}nt||std |jD] } t| trtd q|jd|}n |j}| d}durt||ttrtdd|}\jtd}fdd}nVttr2j}t |}|dur'|!t"j#t"j#t"j#fkr'tdt|}t$|}n'tt rLt%sBtd}t&|}d}n fdd}t}d}|durtt rigd}n%ttrjdkrzgd}njdkrgd}n gd}ngd}|durgd}|durt'(|t|}n t'(|t}|_)t*d|t |}fdd|D}fdd|D}||_+t,d||d|d}t,d||d|d}t,d ||d |d }|||f|_-t*d!|t |}fd"d|D}fd#d|D}||_.||_+td||d|d} td||d|d} td ||d |d } | | | f|_/|_0||_1|| | | |_2||_3t4||d| t4||d| t4||d | t4||d|t4||d|t4||d |||_5|j5dur|j5j6|_6n||_6||_7||_8|S)$a} The orientation/location parameters are necessary if this system is being defined at a certain orientation or location wrt another. Parameters ========== name : str The name of the new CoordSys3D instance. transformation : Lambda, Tuple, str Transformation defined by transformation equations or chosen from predefined ones. location : Vector The position vector of the new system's origin wrt the parent instance. rotation_matrix : SymPy ImmutableMatrix The rotation matrix of the new coordinate system with respect to the parent. In other words, the output of new_system.rotation_matrix(parent). parent : CoordSys3D The coordinate system wrt which the orientation/location (or both) is being defined. vector_names, variable_names : iterable(optional) Iterables of 3 strings each, with custom names for base vectors and base scalars of the new system respectively. Used for simple str printing. zname should be a stringNz?specify either `transformation` or `location`/`rotation_matrix`rzx1 x2 x3clsztransformation: wrong type {}z5rotation_matrix should be an ImmutableMatrix instancez"parent should be a CoordSys3D/Nonezlocation should be a Vectorz'location should not contain BaseScalarsz.origin cartesiancs.t|d|d|dgS)Nrr )invMatrixxyz)lrm/var/www/html/construction_image-detection-poc/venv/lib/python3.10/site-packages/sympy/vector/coordsysrect.pys z$CoordSys3D.__new__..z=Parent for pre-defined coordinate system should be Cartesian.zHThe transformation equation does not create orthogonal coordinate systemcs |||SNr.r()transformationr.r/r0s x1x2x3 sphericalr-thetaphi cylindrical)r-r9r+r()ijk vector_namescg|]}d|fqS)z\mathbf{\hat{%s}_{%s}}r..0r)namer.r/ z&CoordSys3D.__new__..cr@z%s_%sr.rArCr.r/rEr%variable_namescr@)z\mathbf{{%s}_{%s}}r.rArCr.r/rErFcr@rGr.rArCr.r/rErH)9strsympyvectorVectorPoint isinstance TypeError ValueErrorrtuplelistr rrr rrformattyper r as_immutablerzero free_symbolsr origin locate_new!_compose_rotation_and_translation _projections_get_lame_coeffrD_get_transformation_lambdaslame_coefficientsrOne_set_inv_trans_equations_check_orthogonality_calculate_lame_coeffsuper__new___name_check_strings _vector_names BaseVector _base_vectors_variable_names _base_scalars_transformation_transformation_lambda_lame_coefficients"_transformation_from_parent_lambdasetattr_parent_root_parent_rotation_matrix_origin)r"rDr2parentlocationrotation_matrixr?rIrMrNr4r5r6r)rYlambda_transformation lambda_lamelambda_inversetrnameobj latex_vects pretty_vectsv1v2v3 latex_scalarspretty_scalars __class__)r,rDr-r2r/res$                                                 zCoordSys3D.__new__cC|jSr1)rf)selfprinterr.r.r/ _sympystrzCoordSys3D._sympystrcCs t|Sr1)iter base_vectorsrr.r.r/__iter__s zCoordSys3D.__iter__cCstdtd\}}}||||}tt|d|t|d|t|d|g}tt|d|t|d|t|d|g}tt|d|t|d|t|d|g}tdd|||fDr_dSt||dkr|t||dkr|t||dkr|d SdS) a Helper method for _connect_to_cartesian. It checks if set of transformation equations create orthogonal curvilinear coordinate system Parameters ========== equations : Lambda Lambda of transformation equations x1, x2, x3r!rr r%css0|]}t|d|d|ddkVqdS)rr r%NrrBr<r.r.r/ .z2CoordSys3D._check_orthogonality..FT)r rr'ranyrdot) equationsr4r5r6rrrr.r.r/rbs" $zCoordSys3D._check_orthogonalitycCs8|dkrddS|dkrddS|dkrddStd) z Store information about inverse transformation equations for pre-defined coordinate systems. Parameters ========== curv_coord_name : str Name of coordinate system r$cS |||fSr1r.r(r.r.r/r05 z5CoordSys3D._set_inv_trans_equations..r7cSsHt|d|d|dt|t|d|d|dt||fSNr%)rrrr(r.r.r/r08s"r;cSs t|d|dt|||fSr)rrr(r.r.r/r0>s...)rRr3solvedr)r*r+r3r/r0Vs z;CoordSys3D._calculate_inv_trans_equations..N)r rrmr rp)rr4r5r6rr.rr/_calculate_inv_trans_equationsFs    z)CoordSys3D._calculate_inv_trans_equationscCsLt|tr!|dkr ddS|dkrddS|dkrddStdt|S) z Store information about Lame coefficients for pre-defined coordinate systems. Parameters ========== curv_coord_name : str Name of coordinate system r$cSstjtjtjfSr1rr`r(r.r.r/r0gz,CoordSys3D._get_lame_coeff..r7cSstj||t|fSr1)rr`rr8r.r.r/r0isr;cSstj|tjfSr1rr-r9hr.r.r/r0kszAWrong set of parameters. Type of coordinate system is not defined)rOrJrQr_calculate_lame_coefficientsrr.r.r/r]Xs  zCoordSys3D._get_lame_coeffcs fddS)z It calculates Lame coefficients for given transformations equations. Parameters ========== equations : Lambda Lambda of transformation equations. c stt|||d|dt|||d|dt|||d|dtt|||d|dt|||d|dt|||d|dtt|||d|dt|||d|dt|||d|dfS)Nrr%r )rrr3rr.r/r0}s z2CoordSys3D._calculate_lame_coeff..r.rr.rr/rcps z CoordSys3D._calculate_lame_coeffcCst|jdS)z2 Returns inverse rotation matrix. )rrtrr.r.r/_inverse_rotation_matrixsz#CoordSys3D._inverse_rotation_matrixcCsFt|tr!|dkr ddS|dkrddS|dkrddStdd S) z Store information about transformation equations for pre-defined coordinate systems. Parameters ========== curv_coord_name : str Name of coordinate system r$cSrr1r.r(r.r.r/r0rz8CoordSys3D._get_transformation_lambdas..r7cSs2|t|t||t|t||t|fSr1)rrr8r.r.r/r0s r;cSs|t||t||fSr1)rrrr.r.r/r0s  rN)rOrJrQrr.r.r/r^s z&CoordSys3D._get_transformation_lambdascCst|t|S)z Returns the transformation equations obtained from rotation matrix. Parameters ========== matrix : Matrix Rotation matrix equations : tuple Transformation equations )rRr')r"matrixrr.r.r/_rotation_trans_equationssz$CoordSys3D._rotation_trans_equationscCrr1)rurr.r.r/rYszCoordSys3D.origincCrr1)rjrr.r.r/rrzCoordSys3D.base_vectorscCrr1)rlrr.r.r/ base_scalarsrzCoordSys3D.base_scalarscCrr1)rorr.r.r/r_rzCoordSys3D.lame_coefficientscCs|j|Sr1)rnrrr.r.r/transformation_to_parentsz#CoordSys3D.transformation_to_parentcCs"|jdur td|j|jS)NzHno parent coordinate system, use `transformation_from_parent_function()`)rrrQrprrr.r.r/transformation_from_parents z%CoordSys3D.transformation_from_parentcCrr1)rprr.r.r/#transformation_from_parent_functionrz.CoordSys3D.transformation_from_parent_functioncCsddlm}t|tstt|d||krtdS||jkr#|jS|j|kr,|jj S|||\}}td}d}t |D] }|||j9}q=|d7}|t |krc|||jj 9}|d7}|t |ksQ|S)ar Returns the direction cosine matrix(DCM), also known as the 'rotation matrix' of this coordinate system with respect to another system. If v_a is a vector defined in system 'A' (in matrix format) and v_b is the same vector defined in system 'B', then v_a = A.rotation_matrix(B) * v_b. A SymPy Matrix is returned. Parameters ========== other : CoordSys3D The system which the DCM is generated to. Examples ======== >>> from sympy.vector import CoordSys3D >>> from sympy import symbols >>> q1 = symbols('q1') >>> N = CoordSys3D('N') >>> A = N.orient_new_axis('A', q1, N.i) >>> N.rotation_matrix(A) Matrix([ [1, 0, 0], [0, cos(q1), -sin(q1)], [0, sin(q1), cos(q1)]]) r)_pathz is not a CoordSys3Dr#rr%r ) sympy.vector.functionsrrOrrPrJrrrrtTrangelen)rotherr rootindexpathresultr<r.r.r/rxs, !      zCoordSys3D.rotation_matrixcCs |j|S)ab Returns the position vector of the origin of this coordinate system with respect to another Point/CoordSys3D. Parameters ========== other : Point/CoordSys3D If other is a Point, the position of this system's origin wrt it is returned. If its an instance of CoordSyRect, the position wrt its origin is returned. Examples ======== >>> from sympy.vector import CoordSys3D >>> N = CoordSys3D('N') >>> N1 = N.locate_new('N1', 10 * N.i) >>> N.position_wrt(N1) (-10)*N.i )rY position_wrt)rrr.r.r/rs zCoordSys3D.position_wrtcsZt|||fddt|D}||t|fddt|DS)as Returns a dictionary which expresses the coordinate variables (base scalars) of this frame in terms of the variables of otherframe. Parameters ========== otherframe : CoordSys3D The other system to map the variables to. Examples ======== >>> from sympy.vector import CoordSys3D >>> from sympy import Symbol >>> A = CoordSys3D('A') >>> q = Symbol('q') >>> B = A.orient_new_axis('B', q, A.k) >>> A.scalar_map(B) {A.x: B.x*cos(q) - B.y*sin(q), A.y: B.x*sin(q) + B.y*cos(q), A.z: B.z} csg|] \}}||qSr.r.rBr<r)) origin_coordsr.r/rEEsz)CoordSys3D.scalar_map..csi|] \}}|t|qSr.rr) vars_matrixr.r/ Jsz)CoordSys3D.scalar_map..)rRr to_matrix enumeraterrxr')rrrelocated_scalarsr.)rrr/ scalar_map+s    zCoordSys3D.scalar_mapcCs.|dur|j}|dur|j}t|||||dS)a+ Returns a CoordSys3D with its origin located at the given position wrt this coordinate system's origin. Parameters ========== name : str The name of the new CoordSys3D instance. position : Vector The position vector of the new system's origin wrt this one. vector_names, variable_names : iterable(optional) Iterables of 3 strings each, with custom names for base vectors and base scalars of the new system respectively. Used for simple str printing. Examples ======== >>> from sympy.vector import CoordSys3D >>> A = CoordSys3D('A') >>> B = A.locate_new('B', 10 * A.i) >>> B.origin.position_wrt(A.origin) 10*A.i N)rwr?rIrv)rkrhr)rrDpositionr?rIr.r.r/rZMszCoordSys3D.locate_newcCs|dur|j}|dur|j}t|tr't|tr||}n|}t|}nttd}|D]}t|tr>|||9}q/||9}q/t ||||||dS)at Creates a new CoordSys3D oriented in the user-specified way with respect to this system. Please refer to the documentation of the orienter classes for more information about the orientation procedure. Parameters ========== name : str The name of the new CoordSys3D instance. orienters : iterable/Orienter An Orienter or an iterable of Orienters for orienting the new coordinate system. If an Orienter is provided, it is applied to get the new system. If an iterable is provided, the orienters will be applied in the order in which they appear in the iterable. location : Vector(optional) The location of the new coordinate system's origin wrt this system's origin. If not specified, the origins are taken to be coincident. vector_names, variable_names : iterable(optional) Iterables of 3 strings each, with custom names for base vectors and base scalars of the new system respectively. Used for simple str printing. Examples ======== >>> from sympy.vector import CoordSys3D >>> from sympy import symbols >>> q0, q1, q2, q3 = symbols('q0 q1 q2 q3') >>> N = CoordSys3D('N') Using an AxisOrienter >>> from sympy.vector import AxisOrienter >>> axis_orienter = AxisOrienter(q1, N.i + 2 * N.j) >>> A = N.orient_new('A', (axis_orienter, )) Using a BodyOrienter >>> from sympy.vector import BodyOrienter >>> body_orienter = BodyOrienter(q1, q2, q3, '123') >>> B = N.orient_new('B', (body_orienter, )) Using a SpaceOrienter >>> from sympy.vector import SpaceOrienter >>> space_orienter = SpaceOrienter(q1, q2, q3, '312') >>> C = N.orient_new('C', (space_orienter, )) Using a QuaternionOrienter >>> from sympy.vector import QuaternionOrienter >>> q_orienter = QuaternionOrienter(q0, q1, q2, q3) >>> D = N.orient_new('D', (q_orienter, )) Nr#)rxr?rIrwrv) rkrhrOrrrxrr'rr)rrD orientersrwr?rI final_matrixorienterr.r.r/ orient_newvs(A      zCoordSys3D.orient_newcCs:|dur|j}|dur|j}t||}|j|||||dS)a Axis rotation is a rotation about an arbitrary axis by some angle. The angle is supplied as a SymPy expr scalar, and the axis is supplied as a Vector. Parameters ========== name : string The name of the new coordinate system angle : Expr The angle by which the new system is to be rotated axis : Vector The axis around which the rotation has to be performed location : Vector(optional) The location of the new coordinate system's origin wrt this system's origin. If not specified, the origins are taken to be coincident. vector_names, variable_names : iterable(optional) Iterables of 3 strings each, with custom names for base vectors and base scalars of the new system respectively. Used for simple str printing. Examples ======== >>> from sympy.vector import CoordSys3D >>> from sympy import symbols >>> q1 = symbols('q1') >>> N = CoordSys3D('N') >>> B = N.orient_new_axis('B', q1, N.i + 2 * N.j) Nrwr?rI)rkrhrr)rrDangleaxisrwr?rIrr.r.r/orient_new_axiss' zCoordSys3D.orient_new_axisc C"t||||} |j|| |||dS)an Body orientation takes this coordinate system through three successive simple rotations. Body fixed rotations include both Euler Angles and Tait-Bryan Angles, see https://en.wikipedia.org/wiki/Euler_angles. Parameters ========== name : string The name of the new coordinate system angle1, angle2, angle3 : Expr Three successive angles to rotate the coordinate system by rotation_order : string String defining the order of axes for rotation location : Vector(optional) The location of the new coordinate system's origin wrt this system's origin. If not specified, the origins are taken to be coincident. vector_names, variable_names : iterable(optional) Iterables of 3 strings each, with custom names for base vectors and base scalars of the new system respectively. Used for simple str printing. Examples ======== >>> from sympy.vector import CoordSys3D >>> from sympy import symbols >>> q1, q2, q3 = symbols('q1 q2 q3') >>> N = CoordSys3D('N') A 'Body' fixed rotation is described by three angles and three body-fixed rotation axes. To orient a coordinate system D with respect to N, each sequential rotation is always about the orthogonal unit vectors fixed to D. For example, a '123' rotation will specify rotations about N.i, then D.j, then D.k. (Initially, D.i is same as N.i) Therefore, >>> D = N.orient_new_body('D', q1, q2, q3, '123') is same as >>> D = N.orient_new_axis('D', q1, N.i) >>> D = D.orient_new_axis('D', q2, D.j) >>> D = D.orient_new_axis('D', q3, D.k) Acceptable rotation orders are of length 3, expressed in XYZ or 123, and cannot have a rotation about about an axis twice in a row. >>> B = N.orient_new_body('B', q1, q2, q3, '123') >>> B = N.orient_new_body('B', q1, q2, 0, 'ZXZ') >>> B = N.orient_new_body('B', 0, 0, 0, 'XYX') r)rr rrDangle1angle2angle3rotation_orderrwr?rIrr.r.r/orient_new_bodys AzCoordSys3D.orient_new_bodyc Cr)a Space rotation is similar to Body rotation, but the rotations are applied in the opposite order. Parameters ========== name : string The name of the new coordinate system angle1, angle2, angle3 : Expr Three successive angles to rotate the coordinate system by rotation_order : string String defining the order of axes for rotation location : Vector(optional) The location of the new coordinate system's origin wrt this system's origin. If not specified, the origins are taken to be coincident. vector_names, variable_names : iterable(optional) Iterables of 3 strings each, with custom names for base vectors and base scalars of the new system respectively. Used for simple str printing. See Also ======== CoordSys3D.orient_new_body : method to orient via Euler angles Examples ======== >>> from sympy.vector import CoordSys3D >>> from sympy import symbols >>> q1, q2, q3 = symbols('q1 q2 q3') >>> N = CoordSys3D('N') To orient a coordinate system D with respect to N, each sequential rotation is always about N's orthogonal unit vectors. For example, a '123' rotation will specify rotations about N.i, then N.j, then N.k. Therefore, >>> D = N.orient_new_space('D', q1, q2, q3, '312') is same as >>> B = N.orient_new_axis('B', q1, N.i) >>> C = B.orient_new_axis('C', q2, N.j) >>> D = C.orient_new_axis('D', q3, N.k) r)rrrr.r.r/orient_new_spaceNs ;zCoordSys3D.orient_new_spacec Cr)aF Quaternion orientation orients the new CoordSys3D with Quaternions, defined as a finite rotation about lambda, a unit vector, by some amount theta. This orientation is described by four parameters: q0 = cos(theta/2) q1 = lambda_x sin(theta/2) q2 = lambda_y sin(theta/2) q3 = lambda_z sin(theta/2) Quaternion does not take in a rotation order. Parameters ========== name : string The name of the new coordinate system q0, q1, q2, q3 : Expr The quaternions to rotate the coordinate system by location : Vector(optional) The location of the new coordinate system's origin wrt this system's origin. If not specified, the origins are taken to be coincident. vector_names, variable_names : iterable(optional) Iterables of 3 strings each, with custom names for base vectors and base scalars of the new system respectively. Used for simple str printing. Examples ======== >>> from sympy.vector import CoordSys3D >>> from sympy import symbols >>> q0, q1, q2, q3 = symbols('q0 q1 q2 q3') >>> N = CoordSys3D('N') >>> B = N.orient_new_quaternion('B', q0, q1, q2, q3) r)rr) rrDq0q1q2q3rwr?rIrr.r.r/orient_new_quaternions 1z CoordSys3D.orient_new_quaternioncCst|||||dS)a  Returns a CoordSys3D which is connected to self by transformation. Parameters ========== name : str The name of the new CoordSys3D instance. transformation : Lambda, Tuple, str Transformation defined by transformation equations or chosen from predefined ones. vector_names, variable_names : iterable(optional) Iterables of 3 strings each, with custom names for base vectors and base scalars of the new system respectively. Used for simple str printing. Examples ======== >>> from sympy.vector import CoordSys3D >>> a = CoordSys3D('a') >>> b = a.create_new('b', transformation='spherical') >>> b.transformation_to_parent() (b.r*sin(b.theta)*cos(b.phi), b.r*sin(b.phi)*sin(b.theta), b.r*cos(b.theta)) >>> b.transformation_from_parent() (sqrt(a.x**2 + a.y**2 + a.z**2), acos(a.z/sqrt(a.x**2 + a.y**2 + a.z**2)), atan2(a.y, a.x)) )rvr2rIr?)r)rrDr2rIr?r.r.r/ create_newszCoordSys3D.create_newc CsdSr1r.) rrDrwrxrvr?rIr~rrrr2r.r.r/__init__szCoordSys3D.__init__csRfdd|dur Sfdd|D\fddfddS)Ncst|||fSr1)rrr()rotr.r/r0sz>CoordSys3D._compose_rotation_and_translation..csg|]}|qSr.)rr) translationr.r/rEsz@CoordSys3D._compose_rotation_and_translation..cs|||fSr1r.r()dxdydzr.r/r0scs|||Sr1r.r()r-tr.r/r0r)r)rrrvr.)rrrr-rrrr/r[s z,CoordSys3D._compose_rotation_and_translation)NNNNNN)NN)NNN) NNNNNNNNNN)'__name__ __module__ __qualname____doc__rerr staticmethodrbrarr]rcrr^ classmethodrpropertyrYrrr_rrrrxrrrrZrrrrrrrr[ __classcell__r.r.rr/rsza "      8 " ) _ 3 H A 7" rcCs<|d}t|dkrt||D] }t|tst|qdS)Nz& must be an iterable of 3 string-typesr#)rrQrOrJrP)arg_nameargerrorstrsr.r.r/rgs  rg)ri)3collections.abcrsympy.core.basicrsympy.core.cacher sympy.corerrrsympy.core.symbolrr sympy.matrices.immutabler r'sympy.matrices.matrixbaser sympy.solversr sympy.vector.scalarr sympy.core.containersrsympy.core.functionr(sympy.functions.elementary.miscellaneousr(sympy.functions.elementary.trigonometricrrrrsympy.matrices.densersympy.simplify.simplifyrsympy.simplify.trigsimpr sympy.vectorrKsympy.vector.orientersrrrrrrrgsympy.vector.vectorrir.r.r.r/s<               n