U -ek@sddlmZmZmZmZmZmZmZmZ m Z ddl m Z ddl mZddlmZddlmZddlmZdgZGdddeeZGd d d eZd d Zd S)) SsympifyexpandsqrtAddzerosacosImmutableMatrix_simplify_matrix)trigsimp) Printable) filldedent) EvalfMixin) prec_to_dpsVectorc@sfeZdZdZdZdZddZeddZddZ d d Z d d Z d dZ ddZ ddZddZddZddZddZddZddZdMd d!Zd"d#Zd$d%Ze Ze ZeZd&d'Zd(d)Ze je_d*d+Zeje_d,d-Zeje_dNd.d/ZdOd0d1Z d2d3Z!d4d5Z"d6d7Z#d8d9Z$d:d;Z%dd?Z'd@dAZ(dBdCZ)dDdEZ*dFdGZ+dHdIZ,dJdKZ-dLS)PraThe class used to define vectors. It along with ReferenceFrame are the building blocks of describing a classical mechanics system in PyDy and sympy.physics.vector. Attributes ========== simp : Boolean Let certain methods use trigsimp on their outputs FcCsg|_|dkrg}t|tr"|}nDi}|D]:}|d|krT||d|d7<q*|d||d<q*|D]*\}}|tdddgkrn|j||fqndS)aThis is the constructor for the Vector class. You should not be calling this, it should only be used by other functions. You should be treating Vectors like you would with if you were doing the math by hand, and getting the first 3 from the standard basis vectors from a ReferenceFrame. The only exception is to create a zero vector: zv = Vector(0) rN)args isinstancedictitemsMatrixappend)selfinlistdZinpkvr\/var/www/html/Darija-Ai-Train/env/lib/python3.8/site-packages/sympy/physics/vector/vector.py__init__s   zVector.__init__cCstS)zReturns the class Vector. )rrrrrfunc<sz Vector.funccCstt|jSN)hashtuplerr rrr__hash__AszVector.__hash__cCs$|dkr |St|}t|j|jS)zThe add operator for Vector. r) _check_vectorrrrotherrrr__add__DszVector.__add__cCsddlm}t||rtSt|}tj}t|jD]H\}}t|jD]4\}}||dj |d |d|dd7}qDq2t j rt |ddS|SdS)aGDot product of two vectors. Returns a scalar, the dot product of the two Vectors Parameters ========== other : Vector The Vector which we are dotting with Examples ======== >>> from sympy.physics.vector import ReferenceFrame, dot >>> from sympy import symbols >>> q1 = symbols('q1') >>> N = ReferenceFrame('N') >>> dot(N.x, N.x) 1 >>> dot(N.x, N.y) 0 >>> A = N.orientnew('A', 'Axis', [q1, N.x]) >>> dot(N.y, A.y) cos(q1) rDyadicrT) recursiveN)sympy.physics.vector.dyadicr+rNotImplementedr&rZZero enumeraterTdcmrsimpr )rr(r+outiZv1jv2rrr__and__Ks"     zVector.__and__cCs|tj|S)z6This uses mul and inputs self and 1 divided by other. )__mul__rZOner'rrr __truediv__vszVector.__truediv__cCs|dkrtd}z t|}Wntk r2YdSX|jgkrL|jgkrLdS|jgks`|jgkrddS|jdd}|D]}t|||@dkrvdSqvdS)atTests for equality. It is very import to note that this is only as good as the SymPy equality test; False does not always mean they are not equivalent Vectors. If other is 0, and self is empty, returns True. If other is 0 and self is not empty, returns False. If none of the above, only accepts other as a Vector. rFTr)rr& TypeErrorrr)rr(framerrrr__eq__zs  z Vector.__eq__cCsLt|j}t|}t|D](\}}|||d||df||<qt|S)aMultiplies the Vector by a sympifyable expression. Parameters ========== other : Sympifyable The scalar to multiply this Vector with Examples ======== >>> from sympy.physics.vector import ReferenceFrame >>> from sympy import Symbol >>> N = ReferenceFrame('N') >>> b = Symbol('b') >>> V = 10 * b * N.x >>> print(V) 10*b*N.x rr)listrrr/r)rr(Znewlistr4rrrrr8s  "zVector.__mul__cCs|dSNrr rrr__neg__szVector.__neg__cCs ddlm}t|}|d}t|jD]\}}t|jD]\}}|||dd|dd|dj|djfg7}|||dd|dd|dj|djfg7}|||dd|dd|dj|djfg7}|||dd|dd|dj|djfg7}|||dd|dd|dj|djfg7}|||dd|dd|dj|djfg7}|||dd|dd|dj|djfg7}|||dd|dd|dj|djfg7}|||dd|dd|dj|djfg7}q:q&|SaOuter product between two Vectors. A rank increasing operation, which returns a Dyadic from two Vectors Parameters ========== other : Vector The Vector to take the outer product with Examples ======== >>> from sympy.physics.vector import ReferenceFrame, outer >>> N = ReferenceFrame('N') >>> outer(N.x, N.x) (N.x|N.x) rr*rr-r+r&r/rxyzrr(r+olr4ri2r6rrr__or__s 444444448z Vector.__or__c Csl|j}t|dkrtdSg}t|D]\}}dD]}||d|dkrj|d||dj|q4||d|dkr|d||dj|q4||d|dkr4|||d|}t||d|trd|}|ddkr|dd }d}nd}|||||dj|q4q&d |} | drP| d d } n| d rh| dd } | S) zLatex Printing method. rrrrBr + r? - (%s)-N ) rlenstrr/rZ latex_vecs_printrrjoin startswith) rprinterarrHr4rr5arg_str str_startoutstrrrr_latexs2  $    z Vector._latexcs,ddlm|Gfddd}|S)zPretty Printing method. r) prettyFormcseZdZfddZdS)zVector._pretty..Fakec sj}t|dkrtdSg}t|D]\}}dD] }||d|dkrh||dj|}n||d|dkr||dj|}|d}j} |d| i}n~||d|dkr4||d|}t||d|t r| } | d| d}| d||dj|}nq4| |q4q&j |}|d|d<|d |d <|j||} d d | d D} d | S) NrrKrr?rMZbindingrRZ wrap_lineZ num_columnscSsg|] }|qSr)rstrip).0linerrr *sz7Vector._pretty..Fake.render.. )rrSrTr/rUZ pretty_vecsleftZNEGrrparensrightrr)getrendersplitrV) rrkwargsrYZpformsr4rr5ZpformbintmpZout_strZmlineser^rXrrrhs<    z#Vector._pretty..Fake.renderN)__name__ __module__ __qualname__rhrrmrrFakesrr)Z sympy.printing.pretty.stringpictr^)rrXrrrrmr_prettys (zVector._prettycCs ddlm}t|}|d}t|jD]\}}t|jD]\}}|||dd|dd|dj|djfg7}|||dd|dd|dj|djfg7}|||dd|dd|dj|djfg7}|||dd|dd|dj|djfg7}|||dd|dd|dj|djfg7}|||dd|dd|dj|djfg7}|||dd|dd|dj|djfg7}|||dd|dd|dj|djfg7}|||dd|dd|dj|djfg7}q:q&|SrArCrGrrr__ror__/s 444444448zVector.__ror__cCs d||Sr>rr'rrr__rsub__WszVector.__rsub__TcCs|rt|jdkrt|j}n\t|jdkr6|dSdd|jD}t|ddd}g}|D]}||||fqbg}t|D]\}} dD]} ||d| dkr|d ||dj| q||d| d kr|d ||dj| q||d| dkr|||d| } t ||d| t rFd | } | dd krf| dd} d } nd } || | d||dj| qqd |} | d r| dd} n| dr| dd} | S)zPrinting method. rrcSsi|]}|d|dqS)rrr)r`rrrr asz$Vector._sympystr..cSs|jSr")indexrDrrrbz"Vector._sympystr..)keyrKrLr?rMrNrON*rPrQrR) rSrr=rUsortedkeysrr/Zstr_vecsrrrVrW)rrXorderrYrr~r{rHr4rr5rZr[r\rrr _sympystrZs>   (    zVector._sympystrcCs||dS)zThe subtraction operator. r?)r)r'rrr__sub__szVector.__sub__c Csddlm}t||rtSt|}|jgkr4tdSdd}g}|j}t|D]\}}|dj}|dj } |dj } || | g||@|| @|| @gt||g|@t||g| @t||g| @gg} ||| j7}qNt|S)aThe cross product operator for two Vectors. Returns a Vector, expressed in the same ReferenceFrames as self. Parameters ========== other : Vector The Vector which we are crossing with Examples ======== >>> from sympy import symbols >>> from sympy.physics.vector import ReferenceFrame, cross >>> q1 = symbols('q1') >>> N = ReferenceFrame('N') >>> cross(N.x, N.y) N.z >>> A = ReferenceFrame('A') >>> A.orient_axis(N, q1, N.x) >>> cross(A.x, N.y) N.z >>> cross(N.y, A.x) - sin(q1)*A.y - cos(q1)*A.z rr*cSs|dd|dd|dd|dd|dd|dd|dd|dd|dd|dd|dd|dd|dd|dd|ddS)aThis is needed as a little method for to find the determinant of a list in python; needs to work for a 3x3 list. SymPy's Matrix will not take in Vector, so need a custom function. You should not be calling this. rrrBr)matrrr_dets:*  zVector.__xor__.._detr) r-r+rr.r&rrr/rDrErF) rr(r+rZoutlistrYr4rZtempxZtempyZtempzZtempmrrr__xor__s*        zVector.__xor__cCs&i}|jD]}t|g||d<q |S)a The constituents of this vector in different reference frames, as per its definition. Returns a dict mapping each ReferenceFrame to the corresponding constituent Vector. Examples ======== >>> from sympy.physics.vector import ReferenceFrame >>> R1 = ReferenceFrame('R1') >>> R2 = ReferenceFrame('R2') >>> v = R1.x + R2.x >>> v.separate() == {R1: R1.x, R2: R2.x} True r)rr)r componentsrDrrrseparates zVector.separatecCs||@Sr"rr'rrrdotsz Vector.dotcCs||ASr"rr'rrrcrosssz Vector.crosscCs||BSr"rr'rrroutersz Vector.outerc Csddlm}||t|}g}|jD]}|d}|d}||krX||||fg7}q&|rv|||tddkr||||fg7}q&t|g|} | jdd|} |t| |fgj7}q&t|S)a2Returns the partial derivative of the vector with respect to a variable in the provided reference frame. Parameters ========== var : Symbol What the partial derivative is taken with respect to. frame : ReferenceFrame The reference frame that the partial derivative is taken in. var_in_dcm : boolean If true, the differentiation algorithm assumes that the variable may be present in any of the direction cosine matrices that relate the frame to the frames of any component of the vector. But if it is known that the variable is not present in the direction cosine matrices, false can be set to skip full reexpression in the desired frame. Examples ======== >>> from sympy import Symbol >>> from sympy.physics.vector import dynamicsymbols, ReferenceFrame >>> from sympy.physics.vector import Vector >>> from sympy.physics.vector import init_vprinting >>> init_vprinting(pretty_print=False) >>> Vector.simp = True >>> t = Symbol('t') >>> q1 = dynamicsymbols('q1') >>> N = ReferenceFrame('N') >>> A = N.orientnew('A', 'Axis', [q1, N.y]) >>> A.x.diff(t, N) - sin(q1)*q1'*N.x - cos(q1)*q1'*N.z >>> A.x.diff(t, N).express(A) - q1'*A.z >>> B = ReferenceFrame('B') >>> u1, u2 = dynamicsymbols('u1, u2') >>> v = u1 * A.x + u2 * B.y >>> v.diff(u2, N, var_in_dcm=False) B.y r) _check_framerrQ) Zsympy.physics.vector.framerrrdiffr1rrexpress) rvarr;Z var_in_dcmrrZvector_componentZmeasure_numberZcomponent_frameZreexp_vec_compZderivrrrrs"+  z Vector.diffcCsddlm}||||dS)a, Returns a Vector equivalent to this one, expressed in otherframe. Uses the global express method. Parameters ========== otherframe : ReferenceFrame The frame for this Vector to be described in variables : boolean If True, the coordinate symbols(if present) in this Vector are re-expressed in terms otherframe Examples ======== >>> from sympy.physics.vector import ReferenceFrame, dynamicsymbols >>> from sympy.physics.vector import init_vprinting >>> init_vprinting(pretty_print=False) >>> q1 = dynamicsymbols('q1') >>> N = ReferenceFrame('N') >>> A = N.orientnew('A', 'Axis', [q1, N.y]) >>> A.x.express(N) cos(q1)*N.x - sin(q1)*N.z r)r) variables)sympy.physics.vectorr)r otherframerrrrrr1s zVector.expresscstfdd|DddS)aReturns the matrix form of the vector with respect to the given frame. Parameters ---------- reference_frame : ReferenceFrame The reference frame that the rows of the matrix correspond to. Returns ------- matrix : ImmutableMatrix, shape(3,1) The matrix that gives the 1D vector. Examples ======== >>> from sympy import symbols >>> from sympy.physics.vector import ReferenceFrame >>> a, b, c = symbols('a, b, c') >>> N = ReferenceFrame('N') >>> vector = a * N.x + b * N.y + c * N.z >>> vector.to_matrix(N) Matrix([ [a], [b], [c]]) >>> beta = symbols('beta') >>> A = N.orientnew('A', 'Axis', (beta, N.x)) >>> vector.to_matrix(A) Matrix([ [ a], [ b*cos(beta) + c*sin(beta)], [-b*sin(beta) + c*cos(beta)]]) csg|]}|qSr)r)r`Zunit_vecr rrrbusz$Vector.to_matrix..rQr)rZreshaperreference_framerr r to_matrixPs % zVector.to_matrixc s6i}|jD]"}|dfdd||d<q t|S)z(Calls .doit() on each term in the Vectorrcs |jfSr")doitrxhintsrrry|rzzVector.doit..r)r applyfuncr)rrrrrrrrxs  z Vector.doitcCsddlm}|||S)a. Returns a Vector which is the time derivative of the self Vector, taken in frame otherframe. Calls the global time_derivative method Parameters ========== otherframe : ReferenceFrame The frame to calculate the time derivative in r)time_derivative)rr)rrrrrrdts z Vector.dtcCs,i}|jD]}t|d||d<q t|S)zReturns a simplified Vector.rr)rr r)rrrrrrsimplifys zVector.simplifycOs0i}|jD]}|dj||||d<q t|S)a+Substitution on the Vector. Examples ======== >>> from sympy.physics.vector import ReferenceFrame >>> from sympy import Symbol >>> N = ReferenceFrame('N') >>> s = Symbol('s') >>> a = N.x * s >>> a.subs({s: 2}) 2*N.x rr)rsubsr)rrrjrrrrrrs z Vector.subscCs t||@S)a,Returns the magnitude (Euclidean norm) of self. Warnings ======== Python ignores the leading negative sign so that might give wrong results. ``-A.x.magnitude()`` would be treated as ``-(A.x.magnitude())``, instead of ``(-A.x).magnitude()``. )rr rrr magnitudes zVector.magnitudecCst|jg|S)z9Returns a Vector of magnitude 1, codirectional with self.)rrrr rrr normalizeszVector.normalizecCs>t|stdi}|jD]}|d|||d<qt|S)z/Apply a function to each component of a vector.z`f` must be callable.rr)callabler:rrr)rfrrrrrrs  zVector.applyfunccCs"|}|}t||}|S)a Returns the smallest angle between Vector 'vec' and self. Parameter ========= vec : Vector The Vector between which angle is needed. Examples ======== >>> from sympy.physics.vector import ReferenceFrame >>> A = ReferenceFrame("A") >>> v1 = A.x >>> v2 = A.y >>> v1.angle_between(v2) pi/2 >>> v3 = A.x + A.y + A.z >>> v1.angle_between(v3) acos(sqrt(3)/3) Warnings ======== Python ignores the leading negative sign so that might give wrong results. ``-A.x.angle_between()`` would be treated as ``-(A.x.angle_between())``, instead of ``(-A.x).angle_between()``. )rrr)rZvecZvec1Zvec2Zanglerrr angle_betweens!zVector.angle_betweencCs ||jS)aReturns the free symbols in the measure numbers of the vector expressed in the given reference frame. Parameters ========== reference_frame : ReferenceFrame The frame with respect to which the free symbols of the given vector is to be determined. Returns ======= set of Symbol set of symbols present in the measure numbers of ``reference_frame``. )r free_symbolsrrrrrszVector.free_symbolscCsddlm}|||dS)aReturns the free dynamic symbols (functions of time ``t``) in the measure numbers of the vector expressed in the given reference frame. Parameters ========== reference_frame : ReferenceFrame The frame with respect to which the free dynamic symbols of the given vector is to be determined. Returns ======= set Set of functions of time ``t``, e.g. ``Function('f')(me.dynamicsymbols._t)``. r)find_dynamicsymbols)r)Z!sympy.physics.mechanics.functionsr)rrrrrrfree_dynamicsymbolss zVector.free_dynamicsymbolscCsD|js |Sg}t|}|jD]\}}||j|d|gqt|S)N)n)rrrZevalfr)rprecnew_argsZdpsrr;rrr _eval_evalfszVector._eval_evalfcCs4g}|jD] \}}||}|||gq t|S)anReplace occurrences of objects within the measure numbers of the vector. Parameters ========== rule : dict-like Expresses a replacement rule. Returns ======= Vector Result of the replacement. Examples ======== >>> from sympy import symbols, pi >>> from sympy.physics.vector import ReferenceFrame >>> A = ReferenceFrame('A') >>> x, y, z = symbols('x y z') >>> ((1 + x*y) * A.x).xreplace({x: pi}) (pi*y + 1)*A.x >>> ((1 + x*y) * A.x).xreplace({x: pi, y: 2}) (1 + 2*pi)*A.x Replacements occur only if an entire node in the expression tree is matched: >>> ((x*y + z) * A.x).xreplace({x*y: pi}) (z + pi)*A.x >>> ((x*y*z) * A.x).xreplace({x*y: pi}) x*y*z*A.x )rxreplacerr)rrulerrr;rrrr"s & zVector.xreplaceN)T)T)F).rorprq__doc__r2Z is_numberrpropertyr!r%r)r7r9r<r8r@rJr]rsrtrurrr__radd____rand____rmul__rrrrrrrrrrrrrrrrrrrrrrrrs\  +("/( (> D ( & cseZdZfddZZS)VectorTypeErrorcs*tdt||t|f}t|dS)Nz;Expected an instance of %s, but received object '%s' of %s.)r typesuperr)rr(Zwantmsg __class__rrrQszVectorTypeError.__init__)rorprqr __classcell__rrrrrOsrcCst|tstd|S)NzA Vector must be supplied)rrr:)r(rrrr&Ws r&N)Zsympy.core.backendrrrrrrrr rr Zsympy.simplify.trigsimpr Zsympy.printing.defaultsr Zsympy.utilities.miscr Zsympy.core.evalfrZmpmath.libmp.libmpfr__all__rr:rr&rrrrs,     G