ó ¡¼™\c@sýddlmZddlmZmZmZmZmZmZm Z m Z m Z ddl mZddlZdefd„ƒYZdefd„ƒYZd efd „ƒYZd efd „ƒYZd efd„ƒYZdefd„ƒYZd„ZdS(iÿÿÿÿ(tBasic( tsympifyteyetsintcost rot_axis1t rot_axis2t rot_axis3tImmutableMatrixtSymbol(tcacheitNtOrientercBseZdZd„ZRS(s/ Super-class for all orienter classes. cCs|jS(sV The rotation matrix corresponding to this orienter instance. (t_parent_orient(tself((s5lib/python2.7/site-packages/sympy/vector/orienters.pytrotation_matrix s(t__name__t __module__t__doc__R(((s5lib/python2.7/site-packages/sympy/vector/orienters.pyR st AxisOrientercBsMeZdZd„Zd„Zed„ƒZed„ƒZed„ƒZ RS(s+ Class to denote an axis orienter. cCsdt|tjjƒs$tdƒ‚nt|ƒ}tt|ƒj|||ƒ}||_ ||_ |S(Nsaxis should be a Vector( t isinstancetsympytvectortVectort TypeErrorRtsuperRt__new__t_anglet_axis(tclstangletaxistobj((s5lib/python2.7/site-packages/sympy/vector/orienters.pyRs    cCsdS(sá 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 ========== 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 Examples ======== >>> from sympy.vector import CoordSys3D >>> from sympy import symbols >>> q1 = symbols('q1') >>> N = CoordSys3D('N') >>> from sympy.vector import AxisOrienter >>> orienter = AxisOrienter(q1, N.i + 2 * N.j) >>> B = N.orient_new('B', (orienter, )) N((R RR((s5lib/python2.7/site-packages/sympy/vector/orienters.pyt__init__&scCsÂtjj|j|ƒjƒ}|j|ƒ}|j}tdƒ||jt |ƒt d|d |dg|dd|d g|d |ddggƒt |ƒ||j}|j}|S(sø The rotation matrix corresponding to this orienter instance. Parameters ========== system : CoordSys3D The coordinate system wrt which the rotation matrix is to be computed iiii( RRtexpressRt normalizet to_matrixRRtTRtMatrixR(R tsystemRtthetat parent_orient((s5lib/python2.7/site-packages/sympy/vector/orienters.pyRDs q cCs|jS(N(R(R ((s5lib/python2.7/site-packages/sympy/vector/orienters.pyR]scCs|jS(N(R(R ((s5lib/python2.7/site-packages/sympy/vector/orienters.pyRas( RRRRR R RtpropertyRR(((s5lib/python2.7/site-packages/sympy/vector/orienters.pyRs  tThreeAngleOrientercBsSeZdZd„Zed„ƒZed„ƒZed„ƒZed„ƒZRS(s3 Super-class for Body and Space orienters. c Csîd}|}t|ƒjƒ}t|ƒdks?tdƒ‚ng|D]}|jddƒ^qF}g|D]}|jddƒ^qk}g|D]}|jddƒ^q}d j|ƒ}||krØtdƒ‚nt|dƒ}t|dƒ} t|dƒ} t|ƒ}t|ƒ}t|ƒ}|jrat ||ƒt | |ƒt | |ƒ} n)t | |ƒt | |ƒt ||ƒ} | j } t t |ƒj ||||t|ƒƒ} || _|| _|| _|| _| | _| S(Nt123t231t312t132t213t321t121t131t212t232t313t323tis%rot_order should be a str of length 3tXt1tYt2tZt3sInvalid rot_type parameteriii( R+R,R-R.R/R0R1R2R3R4R5R6R7(tstrtuppertlenRtreplacetjointintRt _in_ordert_rotR$RR*RR t_angle1t_angle2t_angle3t _rot_orderR ( Rtangle1tangle2tangle3t rot_ordertapproved_orderstoriginal_rot_ordertita1ta2ta3R(R((s5lib/python2.7/site-packages/sympy/vector/orienters.pyRksB%%%           cCs|jS(N(RF(R ((s5lib/python2.7/site-packages/sympy/vector/orienters.pyRJ“scCs|jS(N(RG(R ((s5lib/python2.7/site-packages/sympy/vector/orienters.pyRK—scCs|jS(N(RH(R ((s5lib/python2.7/site-packages/sympy/vector/orienters.pyRL›scCs|jS(N(RI(R ((s5lib/python2.7/site-packages/sympy/vector/orienters.pyRMŸs( RRRRR)RJRKRLRM(((s5lib/python2.7/site-packages/sympy/vector/orienters.pyR*fs  (t BodyOrientercBs&eZdZeZd„Zd„ZRS(s* Class to denote a body-orienter. cCstj|||||ƒ}|S(N(R*R(RRJRKRLRMR((s5lib/python2.7/site-packages/sympy/vector/orienters.pyR«s cCsdS(s’ 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 ========== angle1, angle2, angle3 : Expr Three successive angles to rotate the coordinate system by rotation_order : string String defining the order of axes for rotation Examples ======== >>> from sympy.vector import CoordSys3D, BodyOrienter >>> 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, >>> body_orienter = BodyOrienter(q1, q2, q3, '123') >>> D = N.orient_new('D', (body_orienter, )) is same as >>> from sympy.vector import AxisOrienter >>> axis_orienter1 = AxisOrienter(q1, N.i) >>> D = N.orient_new('D', (axis_orienter1, )) >>> axis_orienter2 = AxisOrienter(q2, D.j) >>> D = D.orient_new('D', (axis_orienter2, )) >>> axis_orienter3 = AxisOrienter(q3, D.k) >>> D = D.orient_new('D', (axis_orienter3, )) 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. >>> body_orienter1 = BodyOrienter(q1, q2, q3, '123') >>> body_orienter2 = BodyOrienter(q1, q2, 0, 'ZXZ') >>> body_orienter3 = BodyOrienter(0, 0, 0, 'XYX') N((R RJRKRLRM((s5lib/python2.7/site-packages/sympy/vector/orienters.pyR °s7(RRRtTrueRDRR (((s5lib/python2.7/site-packages/sympy/vector/orienters.pyRT¤s t SpaceOrientercBs&eZdZeZd„Zd„ZRS(s+ Class to denote a space-orienter. cCstj|||||ƒ}|S(N(R*R(RRJRKRLRMR((s5lib/python2.7/site-packages/sympy/vector/orienters.pyRñs cCsdS(s¢ Space rotation is similar to Body rotation, but the rotations are applied in the opposite order. Parameters ========== angle1, angle2, angle3 : Expr Three successive angles to rotate the coordinate system by rotation_order : string String defining the order of axes for rotation See Also ======== BodyOrienter : Orienter to orient systems wrt Euler angles. Examples ======== >>> from sympy.vector import CoordSys3D, SpaceOrienter >>> 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, >>> space_orienter = SpaceOrienter(q1, q2, q3, '312') >>> D = N.orient_new('D', (space_orienter, )) is same as >>> from sympy.vector import AxisOrienter >>> axis_orienter1 = AxisOrienter(q1, N.i) >>> B = N.orient_new('B', (axis_orienter1, )) >>> axis_orienter2 = AxisOrienter(q2, N.j) >>> C = B.orient_new('C', (axis_orienter2, )) >>> axis_orienter3 = AxisOrienter(q3, N.k) >>> D = C.orient_new('C', (axis_orienter3, )) N((R RJRKRLRM((s5lib/python2.7/site-packages/sympy/vector/orienters.pyR ös0(RRRtFalseRDRR (((s5lib/python2.7/site-packages/sympy/vector/orienters.pyRVês tQuaternionOrientercBs\eZdZd„Zd„Zed„ƒZed„ƒZed„ƒZed„ƒZ RS(s0 Class to denote a quaternion-orienter. cCsrt|ƒ}t|ƒ}t|ƒ}t|ƒ}t|d|d|d|dd||||d||||gd|||||d|d|d|dd||||gd||||d|||||d|d|d|dggƒ}|j}tt|ƒj|||||ƒ}||_||_||_||_ ||_ |S(Ni( RR%R$RRXRt_q0t_q1t_q2t_q3R (Rtq0tq1tq2tq3R(R((s5lib/python2.7/site-packages/sympy/vector/orienters.pyR.s,    + $     cCsdS(s¨ 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 ========== q0, q1, q2, q3 : Expr The quaternions to rotate the coordinate system by Examples ======== >>> from sympy.vector import CoordSys3D >>> from sympy import symbols >>> q0, q1, q2, q3 = symbols('q0 q1 q2 q3') >>> N = CoordSys3D('N') >>> from sympy.vector import QuaternionOrienter >>> q_orienter = QuaternionOrienter(q0, q1, q2, q3) >>> B = N.orient_new('B', (q_orienter, )) N((R RJRKRLRM((s5lib/python2.7/site-packages/sympy/vector/orienters.pyR Js%cCs|jS(N(RY(R ((s5lib/python2.7/site-packages/sympy/vector/orienters.pyR]qscCs|jS(N(RZ(R ((s5lib/python2.7/site-packages/sympy/vector/orienters.pyR^uscCs|jS(N(R[(R ((s5lib/python2.7/site-packages/sympy/vector/orienters.pyR_yscCs|jS(N(R\(R ((s5lib/python2.7/site-packages/sympy/vector/orienters.pyR`}s( RRRRR R)R]R^R_R`(((s5lib/python2.7/site-packages/sympy/vector/orienters.pyRX)s  'cCsa|dkrtt|ƒjƒS|dkr>tt|ƒjƒS|dkr]tt|ƒjƒSdS(s)DCM for simple axis 1, 2 or 3 rotations. iiiN(R%RR$RR(RR((s5lib/python2.7/site-packages/sympy/vector/orienters.pyRE‚s    (tsympy.core.basicRRRRRRRRRRR%R tsympy.core.cacheR t sympy.vectorR RR*RTRVRXRE(((s5lib/python2.7/site-packages/sympy/vector/orienters.pyts@  Q>F?Y