ó ¡¼™\c@stddlmZmZmZddlmZddlmZm Z m Z dgZ de fd„ƒYZ d„ZdS( iÿÿÿÿ(tsympifytAddtImmutableMatrix(tunicodei(tVectorLatexPrintertVectorPrettyPrintertVectorStrPrintertDyadiccBs+eZdZd„Zd„Zd„Zd„ZeZd„Zd„Z d„Z d„Z dd „Z dd „Zd „Zd „Zd „Zdd„Zd„Zd„Zd„ZeZeZeZeZeZe Zdd„Zdd„Zd„Zd„Zd„Z d„Z!d„Z"eZ#eZ$RS(syA Dyadic object. See: https://en.wikipedia.org/wiki/Dyadic_tensor Kane, T., Levinson, D. Dynamics Theory and Applications. 1985 McGraw-Hill A more powerful way to represent a rigid body's inertia. While it is more complex, by choosing Dyadic components to be in body fixed basis vectors, the resulting matrix is equivalent to the inertia tensor. cCsÔg|_|dkrg}nxt|ƒdkr>d}xËt|jƒD]º\}}t|ddƒt|j|dƒkrIt|ddƒt|j|dƒkrI|j|d|dd|dd|ddf|j|<|j|dƒd}PqIqIW|dkr!|jj|dƒ|j|dƒq!q!Wd}xˆ|t|jƒkrÏ|j|ddk|j|ddkB|j|ddkBrÂ|jj|j|ƒ|d8}n|d7}qHWdS(s1 Just like Vector's init, you shouldn't call this unless creating a zero dyadic. zd = Dyadic(0) Stores a Dyadic as a list of lists; the inner list has the measure number and the two unit vectors; the outerlist holds each unique unit vector pair. iiiN(targstlent enumeratetstrtremovetappend(tselftinlisttaddedtitv((s:lib/python2.7/site-packages/sympy/physics/vector/dyadic.pyt__init__s.   ++# ) cCs t|ƒ}t|j|jƒS(sThe add operator for Dyadic. (t _check_dyadicRR(Rtother((s:lib/python2.7/site-packages/sympy/physics/vector/dyadic.pyt__add__<s c Csddlm}m}t|tƒr¶t|ƒ}tdƒ}xÐt|jƒD]b\}}xSt|jƒD]B\}}||d|d|d|d@|d|dB7}qiWqMWnZ||ƒ}|dƒ}x?t|jƒD].\}}||d|d|d|@7}qÞW|S(sòThe inner product operator for a Dyadic and a Dyadic or Vector. Parameters ========== other : Dyadic or Vector The other Dyadic or Vector to take the inner product with Examples ======== >>> from sympy.physics.vector import ReferenceFrame, outer >>> N = ReferenceFrame('N') >>> D1 = outer(N.x, N.y) >>> D2 = outer(N.y, N.y) >>> D1.dot(D2) (N.x|N.y) >>> D1.dot(N.y) N.x iÿÿÿÿ(tVectort _check_vectoriii(tsympy.physics.vector.vectorRRt isinstanceRRR R( RRRRtolRRti2tv2((s:lib/python2.7/site-packages/sympy/physics/vector/dyadic.pyt__and__As  A  &cCs|jd|ƒS(s0Divides the Dyadic by a sympifyable expression. i(t__mul__(RR((s:lib/python2.7/site-packages/sympy/physics/vector/dyadic.pyt__div__escCs‡|dkrtdƒ}nt|ƒ}|jgkrI|jgkrItS|jgksg|jgkrktSt|jƒt|jƒkS(s[Tests for equality. Is currently weak; needs stronger comparison testing i(RRRtTruetFalsetset(RR((s:lib/python2.7/site-packages/sympy/physics/vector/dyadic.pyt__eq__ks  cCsxg|jD] }|^q }xOt|ƒD]A\}}t|ƒ||d||d||df||>> from sympy.physics.vector import ReferenceFrame, outer >>> N = ReferenceFrame('N') >>> d = outer(N.x, N.x) >>> 5 * d 5*(N.x|N.x) iii(RR RR(RRRtnewlistR((s:lib/python2.7/site-packages/sympy/physics/vector/dyadic.pyR{s  cCs ||k S(N((RR((s:lib/python2.7/site-packages/sympy/physics/vector/dyadic.pyt__ne__•scCs|dS(Niÿÿÿÿ((R((s:lib/python2.7/site-packages/sympy/physics/vector/dyadic.pyt__neg__˜sc Csù|j}t|ƒdkr%tdƒSg}tƒ}xwt|ƒD]i\}}||ddkrŸ|jd|j||dƒd|j||dƒƒqA||ddkrñ|jd|j||dƒd|j||dƒƒqA||ddkrA|j||dƒ}t||dtƒr@d|}n|j d ƒrb|d}d}nd}|j|||j||dƒd|j||dƒƒqAqAWd j |ƒ} | j dƒrÙ| d } n| j d ƒrõ| d} n| S( Niis + s\otimes iiÿÿÿÿs - s(%s)t-tit ( RR R RR R tdoprintRRt startswithtjoin( RtprintertarRtmlpRRtarg_strt str_starttoutstr((s:lib/python2.7/site-packages/sympy/physics/vector/dyadic.pyt_latex›s:   "   @  cs,|‰dtf‡‡fd†ƒY}|ƒS(NtFakecs eZdZ‡‡fd†ZRS(ic ssˆj}ˆrˆjni}ˆr0ˆj}nddlm}|ƒ}ˆrUˆn t|ƒ}t|ƒdkr}tdƒS|r‰dnd}g} xt|ƒD]‚\} } || ddkr| j d|j || dƒ||j || dƒgƒq¢|| ddkrR| j d |j || dƒ||j || dƒgƒq¢|| ddkr¢t || dt ƒr¡|j || dƒjƒd} n|j || dƒ} | jd ƒrÚ| d} d } nd} | j | | d |j || dƒ||j || dƒgƒq¢q¢Wd j| ƒ}|jdƒrS|d }n|jdƒro|d}n|S(Niÿÿÿÿ(tpretty_use_unicodeiu⊗t|iu + iu - u-u uiR*(Rt _settingst _use_unicodet&sympy.printing.pretty.pretty_symbologyR6RR RR textendR+RRt_printtparensR,R-(RRtkwargsR/tsettingst use_unicodeR6tmpptbarRRRR1R2R3(teR.(s:lib/python2.7/site-packages/sympy/physics/vector/dyadic.pytrenderÆsR        "  (t__name__t __module__tbaselineRD((RCR.(s:lib/python2.7/site-packages/sympy/physics/vector/dyadic.pyR5Ãs(tobject(RR.R5((RCR.s:lib/python2.7/site-packages/sympy/physics/vector/dyadic.pyt_prettyÀs8cCstddlm}m}||ƒ}|dƒ}x?t|jƒD].\}}||d|d|d|@7}q>W|S(s¯The inner product operator for a Vector or Dyadic, and a Dyadic This is for: Vector dot Dyadic Parameters ========== other : Vector The vector we are dotting with Examples ======== >>> from sympy.physics.vector import ReferenceFrame, dot, outer >>> N = ReferenceFrame('N') >>> d = outer(N.x, N.x) >>> dot(N.x, d) N.x iÿÿÿÿ(RRiii(RRRR R(RRRRRRR((s:lib/python2.7/site-packages/sympy/physics/vector/dyadic.pyt__rand__ýs   &cCs d||S(Niÿÿÿÿ((RR((s:lib/python2.7/site-packages/sympy/physics/vector/dyadic.pyt__rsub__scCsnddlm}||ƒ}tdƒ}x?t|jƒD].\}}||d||dA|dB7}q8W|S(s–For a cross product in the form: Vector x Dyadic Parameters ========== other : Vector The Vector that we are crossing this Dyadic with Examples ======== >>> from sympy.physics.vector import ReferenceFrame, outer, cross >>> N = ReferenceFrame('N') >>> d = outer(N.x, N.x) >>> cross(N.y, d) - (N.z|N.x) iÿÿÿÿ(Riii(RRRR R(RRRRRR((s:lib/python2.7/site-packages/sympy/physics/vector/dyadic.pyt__rxor__s   &c Csò|j}t|ƒdkr%tdƒSg}xyt|ƒD]k\}}||ddkr”|jdt||dƒdt||dƒdƒq8||ddkrä|jdt||dƒdt||dƒdƒq8||ddkr8tƒj||dƒ}t||dtƒr6d |}n|dd krY|d}d }nd }|j||d t||dƒdt||dƒdƒq8q8Wdj |ƒ}|j d ƒrÒ|d}n|j dƒrî|d}n|S(sPrinting method. iis + (R7it)iÿÿÿÿs - (s(%s)R(s - s + s*(R)iR*( RR R R R RR+RRR-R,( RR.R/RRRR1R2R3((s:lib/python2.7/site-packages/sympy/physics/vector/dyadic.pyt__str__8s2  <<   B  cCs|j|dƒS(sThe subtraction operator. iÿÿÿÿ(R(RR((s:lib/python2.7/site-packages/sympy/physics/vector/dyadic.pyt__sub__YscCsnddlm}||ƒ}tdƒ}x?t|jƒD].\}}||d|d|d|AB7}q8W|S(s•For a cross product in the form: Dyadic x Vector. Parameters ========== other : Vector The Vector that we are crossing this Dyadic with Examples ======== >>> from sympy.physics.vector import ReferenceFrame, outer, cross >>> N = ReferenceFrame('N') >>> d = outer(N.x, N.x) >>> cross(d, N.y) (N.x|N.z) iÿÿÿÿ(Riii(RRRR R(RRRRRR((s:lib/python2.7/site-packages/sympy/physics/vector/dyadic.pyt__xor__]s   &cCs*ddlm}||ddƒ}d|S(s^ IPython/Jupyter LaTeX printing To change the behavior of this (e.g., pass in some settings to LaTeX), use init_printing(). init_printing() will also enable LaTeX printing for built in numeric types like ints and container types that contain SymPy objects, like lists and dictionaries of expressions. iÿÿÿÿ(tlatextmodetplains$\displaystyle %s$(tsympy.printing.latexRQ(RRQts((s:lib/python2.7/site-packages/sympy/physics/vector/dyadic.pyt _repr_latex_|s cCs ddlm}||||ƒS(s´Expresses this Dyadic in alternate frame(s) The first frame is the list side expression, the second frame is the right side; if Dyadic is in form A.x|B.y, you can express it in two different frames. If no second frame is given, the Dyadic is expressed in only one frame. Calls the global express function Parameters ========== frame1 : ReferenceFrame The frame to express the left side of the Dyadic in frame2 : ReferenceFrame If provided, the frame to express the right side of the Dyadic in Examples ======== >>> from sympy.physics.vector import ReferenceFrame, outer, dynamicsymbols >>> N = ReferenceFrame('N') >>> q = dynamicsymbols('q') >>> B = N.orientnew('B', 'Axis', [q, N.z]) >>> d = outer(N.x, N.x) >>> d.express(B, N) cos(q)*(B.x|N.x) - sin(q)*(B.y|N.x) iÿÿÿÿ(texpress(tsympy.physics.vector.functionsRW(Rtframe1tframe2RW((s:lib/python2.7/site-packages/sympy/physics/vector/dyadic.pyRW‘scCs]|dkr|}ntg|D]+}|D]}|j|ƒj|ƒ^q)qƒjddƒS(s”Returns the matrix form of the dyadic with respect to one or two reference frames. Parameters ---------- reference_frame : ReferenceFrame The reference frame that the rows and columns of the matrix correspond to. If a second reference frame is provided, this only corresponds to the rows of the matrix. second_reference_frame : ReferenceFrame, optional, default=None The reference frame that the columns of the matrix correspond to. Returns ------- matrix : ImmutableMatrix, shape(3,3) The matrix that gives the 2D tensor form. Examples ======== >>> from sympy import symbols >>> from sympy.physics.vector import ReferenceFrame, Vector >>> Vector.simp = True >>> from sympy.physics.mechanics import inertia >>> Ixx, Iyy, Izz, Ixy, Iyz, Ixz = symbols('Ixx, Iyy, Izz, Ixy, Iyz, Ixz') >>> N = ReferenceFrame('N') >>> inertia_dyadic = inertia(N, Ixx, Iyy, Izz, Ixy, Iyz, Ixz) >>> inertia_dyadic.to_matrix(N) Matrix([ [Ixx, Ixy, Ixz], [Ixy, Iyy, Iyz], [Ixz, Iyz, Izz]]) >>> beta = symbols('beta') >>> A = N.orientnew('A', 'Axis', (beta, N.x)) >>> inertia_dyadic.to_matrix(A) Matrix([ [ Ixx, Ixy*cos(beta) + Ixz*sin(beta), -Ixy*sin(beta) + Ixz*cos(beta)], [ Ixy*cos(beta) + Ixz*sin(beta), Iyy*cos(2*beta)/2 + Iyy/2 + Iyz*sin(2*beta) - Izz*cos(2*beta)/2 + Izz/2, -Iyy*sin(2*beta)/2 + Iyz*cos(2*beta) + Izz*sin(2*beta)/2], [-Ixy*sin(beta) + Ixz*cos(beta), -Iyy*sin(2*beta)/2 + Iyz*cos(2*beta) + Izz*sin(2*beta)/2, -Iyy*cos(2*beta)/2 + Iyy/2 - Iyz*sin(2*beta) + Izz*cos(2*beta)/2 + Izz/2]]) iN(tNonetMatrixtdottreshape(Rtreference_frametsecond_reference_frameRtj((s:lib/python2.7/site-packages/sympy/physics/vector/dyadic.pyt to_matrix²s,  cKsPtg|jD]3}t|dj||d|dfgƒ^q tdƒƒS(s(Calls .doit() on each term in the Dyadiciii(tsumRRtdoit(RthintsR((s:lib/python2.7/site-packages/sympy/physics/vector/dyadic.pyRdäscCsddlm}|||ƒS(sDTake the time derivative of this Dyadic in a frame. This function calls the global time_derivative method Parameters ========== frame : ReferenceFrame The frame to take the time derivative in Examples ======== >>> from sympy.physics.vector import ReferenceFrame, outer, dynamicsymbols >>> N = ReferenceFrame('N') >>> q = dynamicsymbols('q') >>> B = N.orientnew('B', 'Axis', [q, N.z]) >>> d = outer(N.x, N.x) >>> d.dt(B) - q'*(N.y|N.x) - q'*(N.x|N.y) iÿÿÿÿ(ttime_derivative(RXRf(RtframeRf((s:lib/python2.7/site-packages/sympy/physics/vector/dyadic.pytdtéscCsRtdƒ}x?|jD]4}|t|djƒ|d|dfgƒ7}qW|S(sReturns a simplified Dyadic.iii(RRtsimplify(RtoutR((s:lib/python2.7/site-packages/sympy/physics/vector/dyadic.pyRis 2cOsStg|jD]6}t|dj||Ž|d|dfgƒ^q tdƒƒS(s5Substitution on the Dyadic. Examples ======== >>> from sympy.physics.vector import ReferenceFrame >>> from sympy import Symbol >>> N = ReferenceFrame('N') >>> s = Symbol('s') >>> a = s*(N.x|N.x) >>> a.subs({s: 2}) 2*(N.x|N.x) iii(RcRRtsubs(RRR>R((s:lib/python2.7/site-packages/sympy/physics/vector/dyadic.pyRk scCs`t|ƒstdƒ‚ntdƒ}x2|jD]'\}}}|||ƒ||B7}q1W|S(s/Apply a function to each component of a Dyadic.s`f` must be callable.i(tcallablet TypeErrorRR(RtfRjtatbtc((s:lib/python2.7/site-packages/sympy/physics/vector/dyadic.pyt applyfuncs   N(%RERFt__doc__RRRR t __truediv__R$RR&R'R[R4RIRJRKRLRNRORPRVt_repr_latex_origt _sympystrt _sympyreprt__repr__t__radd__t__rmul__RWRbRdRhRiRkRrR]tcross(((s:lib/python2.7/site-packages/sympy/physics/vector/dyadic.pyR sD  &  $      % =    !    ! 2     cCs"t|tƒstdƒ‚n|S(NsA Dyadic must be supplied(RRRm(R((s:lib/python2.7/site-packages/sympy/physics/vector/dyadic.pyR+sN(tsympy.core.backendRRRR\tsympy.core.compatibilityRtprintingRRRt__all__RHRR(((s:lib/python2.7/site-packages/sympy/physics/vector/dyadic.pyts ÿÿ$