ó ¡¼™\c@sddlmZddlmZddlmZmZmZddlmZmZm Z m Z m Z m Z ddlm Z ddlmZddlmZmZmZddlmZdd lmZdd lmZdd lmZd efd „ƒYZdS(iÿÿÿÿ(tprint_function(tRational(tretimt conjugate(tsqrttsintcostacostexptln(ttrigsimp(t integrate(tMatrixtAddtMul(tsympify(t SYMPY_INTS(tExpr(tIntegert QuaternioncBs‘eZdZdZeZdddded„Zed„ƒZ ed„ƒZ ed„ƒZ ed„ƒZ ed„ƒZ ed „ƒZed „ƒZed „ƒZd „Zd „Zd„Zd„Zd„Zd„Zd„Zd„Zd„Zd„Zd„Zed„ƒZd„Zd„Z d„Z!d„Z"d„Z#d„Z$d„Z%d„Z&d „Z'ed!„ƒZ(d"„Z)d$d#„Z+RS(%sÇProvides basic quaternion operations. Quaternion objects can be instantiated as Quaternion(a, b, c, d) as in (a + b*i + c*j + d*k). Examples ======== >>> from sympy.algebras.quaternion import Quaternion >>> q = Quaternion(1, 2, 3, 4) >>> q 1 + 2*i + 3*j + 4*k Quaternions over complex fields can be defined as : >>> from sympy.algebras.quaternion import Quaternion >>> from sympy import symbols, I >>> x = symbols('x') >>> q1 = Quaternion(x, x**3, x, x**2, real_field = False) >>> q2 = Quaternion(3 + 4*I, 2 + 5*I, 0, 7 + 8*I, real_field = False) >>> q1 x + x**3*i + x*j + x**2*k >>> q2 (3 + 4*I) + (2 + 5*I)*i + 0*j + (7 + 8*I)*k g&@icCs±t|ƒ}t|ƒ}t|ƒ}t|ƒ}td„||||gDƒƒratdƒ‚nLtj|||||ƒ}||_||_||_||_||_ |SdS(Ncss|]}|jtkVqdS(N(tis_commutativetFalse(t.0ti((s8lib/python2.7/site-packages/sympy/algebras/quaternion.pys 5ss arguments have to be commutative( Rtanyt ValueErrorRt__new__t_at_bt_ct_dt _real_field(tclstatbtctdt real_fieldtobj((s8lib/python2.7/site-packages/sympy/algebras/quaternion.pyR/s    "     cCs|jS(N(R(tself((s8lib/python2.7/site-packages/sympy/algebras/quaternion.pyR"@scCs|jS(N(R(R(((s8lib/python2.7/site-packages/sympy/algebras/quaternion.pyR#DscCs|jS(N(R(R(((s8lib/python2.7/site-packages/sympy/algebras/quaternion.pyR$HscCs|jS(N(R(R(((s8lib/python2.7/site-packages/sympy/algebras/quaternion.pyR%LscCs|jS(N(R (R(((s8lib/python2.7/site-packages/sympy/algebras/quaternion.pyR&Osc Cs¸|\}}}t|d|d|dƒ}||||||}}}t|tddƒƒ}t|tddƒƒ}||} ||} ||} ||| | | ƒjƒS(sÛReturns a rotation quaternion given the axis and the angle of rotation. Parameters ========== vector : tuple of three numbers The vector representation of the given axis. angle : number The angle by which axis is rotated (in radians). Returns ======= Quaternion The normalized rotation quaternion calculated from the given axis and the angle of rotation. Examples ======== >>> from sympy.algebras.quaternion import Quaternion >>> from sympy import pi, sqrt >>> q = Quaternion.from_axis_angle((sqrt(3)/3, sqrt(3)/3, sqrt(3)/3), 2*pi/3) >>> q 1/2 + 1/2*i + 1/2*j + 1/2*k ii(RRRRt normalize( R!tvectortangletxtytztnormtsR"R#R$R%((s8lib/python2.7/site-packages/sympy/algebras/quaternion.pytfrom_axis_angleSs     cCs>|jƒtddƒ}t||d|d|dƒd}t||d|d |d ƒd}t||d |d |d ƒd}t||d|d|dƒd}y^tj||d|dƒ}tj||d|dƒ}tj||d|dƒ}Wntk r*nXt||||ƒS(sReturns the equivalent quaternion of a matrix. The quaternion will be normalized only if the matrix is special orthogonal (orthogonal and det(M) = 1). Parameters ========== M : Matrix Input matrix to be converted to equivalent quaternion. M must be special orthogonal (orthogonal and det(M) = 1) for the quaternion to be normalized. Returns ======= Quaternion The quaternion equivalent to given matrix. Examples ======== >>> from sympy.algebras.quaternion import Quaternion >>> from sympy import Matrix, symbols, cos, sin, trigsimp >>> x = symbols('x') >>> M = Matrix([[cos(x), -sin(x), 0], [sin(x), cos(x), 0], [0, 0, 1]]) >>> q = trigsimp(Quaternion.from_rotation_matrix(M)) >>> q sqrt(2)*sqrt(cos(x) + 1)/2 + 0*i + 0*j + sqrt(2 - 2*cos(x))/2*k iiii(ii(ii(ii(ii(ii(ii(ii(ii(ii(ii(ii(ii(ii(ii(ii(ii(ii(ii(tdetRRRt_Quaternion__copysignt Exception(R!tMtabsQR"R#R$R%((s8lib/python2.7/site-packages/sympy/algebras/quaternion.pytfrom_rotation_matrixys((((" cCs)|dkrdS||dkr$|S| S(Ni((R,R-((s8lib/python2.7/site-packages/sympy/algebras/quaternion.pyt __copysign¨s cCs |j|ƒS(N(tadd(R(tother((s8lib/python2.7/site-packages/sympy/algebras/quaternion.pyt__add__²scCs |j|ƒS(N(R9(R(R:((s8lib/python2.7/site-packages/sympy/algebras/quaternion.pyt__radd__µscCs|j|dƒS(Niÿÿÿÿ(R9(R(R:((s8lib/python2.7/site-packages/sympy/algebras/quaternion.pyt__sub__¸scCs|j||ƒS(N(t _generic_mul(R(R:((s8lib/python2.7/site-packages/sympy/algebras/quaternion.pyt__mul__»scCs|j||ƒS(N(R>(R(R:((s8lib/python2.7/site-packages/sympy/algebras/quaternion.pyt__rmul__¾scCs |j|ƒS(N(tpow(R(tp((s8lib/python2.7/site-packages/sympy/algebras/quaternion.pyt__pow__ÁscCs#t|j |j |j |j ƒS(N(RRRRR%(R(((s8lib/python2.7/site-packages/sympy/algebras/quaternion.pyt__neg__ÄscGs |j|ŒS(N(R (R(targs((s8lib/python2.7/site-packages/sympy/algebras/quaternion.pyt_eval_IntegralÇscOs?|jdtƒ|jg|jD]}|j||Ž^q ŒS(Ntevaluate(t setdefaulttTruetfuncREtdiff(R(tsymbolstkwargsR"((s8lib/python2.7/site-packages/sympy/algebras/quaternion.pyRKÊscCs×|}t|ƒ}t|tƒsœ|jrv|jrftt|ƒ|jt|ƒ|j|j |j ƒSt ||ƒSqœt|j||j|j |j ƒSnt|j|j|j|j|j |j |j |j ƒS(sÖAdds quaternions. Parameters ========== other : Quaternion The quaternion to add to current (self) quaternion. Returns ======= Quaternion The resultant quaternion after adding self to other Examples ======== >>> from sympy.algebras.quaternion import Quaternion >>> from sympy import symbols >>> q1 = Quaternion(1, 2, 3, 4) >>> q2 = Quaternion(5, 6, 7, 8) >>> q1.add(q2) 6 + 8*i + 10*j + 12*k >>> q1 + 5 6 + 2*i + 3*j + 4*k >>> x = symbols('x', real = True) >>> q1.add(x) (x + 1) + 2*i + 3*j + 4*k Quaternions over complex fields : >>> from sympy.algebras.quaternion import Quaternion >>> from sympy import I >>> q3 = Quaternion(3 + 4*I, 2 + 5*I, 0, 7 + 8*I, real_field = False) >>> q3.add(2 + 3*I) (5 + 7*I) + (2 + 5*I)*i + 0*j + (7 + 8*I)*k ( Rt isinstanceRR&t is_complexRR"RR#R$R%R(R(R:tq1tq2((s8lib/python2.7/site-packages/sympy/algebras/quaternion.pyR9Îs&   3&0cCs|j||ƒS(sMultiplies quaternions. Parameters ========== other : Quaternion or symbol The quaternion to multiply to current (self) quaternion. Returns ======= Quaternion The resultant quaternion after multiplying self with other Examples ======== >>> from sympy.algebras.quaternion import Quaternion >>> from sympy import symbols >>> q1 = Quaternion(1, 2, 3, 4) >>> q2 = Quaternion(5, 6, 7, 8) >>> q1.mul(q2) (-60) + 12*i + 30*j + 24*k >>> q1.mul(2) 2 + 4*i + 6*j + 8*k >>> x = symbols('x', real = True) >>> q1.mul(x) x + 2*x*i + 3*x*j + 4*x*k Quaternions over complex fields : >>> from sympy.algebras.quaternion import Quaternion >>> from sympy import I >>> q3 = Quaternion(3 + 4*I, 2 + 5*I, 0, 7 + 8*I, real_field = False) >>> q3.mul(2 + 3*I) (2 + 3*I)*(3 + 4*I) + (2 + 3*I)*(2 + 5*I)*i + 0*j + (2 + 3*I)*(7 + 8*I)*k (R>(R(R:((s8lib/python2.7/site-packages/sympy/algebras/quaternion.pytmuls&cCs1t|ƒ}t|ƒ}t|tƒ r@t|tƒ r@||St|tƒsÆ|jr”|jr„|tt|ƒt|ƒddƒSt||ƒSqÆt||j||j ||j ||j ƒSnt|tƒsL|jr|jr |tt|ƒt|ƒddƒSt||ƒSqLt||j||j ||j ||j ƒSnt|j |j |j |j |j |j |j|j|j |j|j |j |j |j |j|j |j |j |j |j|j |j |j|j |j |j |j |j |j |j|j|j ƒS(s¹Generic multiplication. Parameters ========== q1 : Quaternion or symbol q2 : Quaternion or symbol It's important to note that if neither q1 nor q2 is a Quaternion, this function simply returns q1 * q2. Returns ======= Quaternion The resultant quaternion after multiplying q1 and q2 Examples ======== >>> from sympy.algebras.quaternion import Quaternion >>> from sympy import symbols >>> q1 = Quaternion(1, 2, 3, 4) >>> q2 = Quaternion(5, 6, 7, 8) >>> Quaternion._generic_mul(q1, q2) (-60) + 12*i + 30*j + 24*k >>> Quaternion._generic_mul(q1, 2) 2 + 4*i + 6*j + 8*k >>> x = symbols('x', real = True) >>> Quaternion._generic_mul(q1, x) x + 2*x*i + 3*x*j + 4*x*k Quaternions over complex fields : >>> from sympy.algebras.quaternion import Quaternion >>> from sympy import I >>> q3 = Quaternion(3 + 4*I, 2 + 5*I, 0, 7 + 8*I, real_field = False) >>> Quaternion._generic_mul(q3, 2 + 3*I) (2 + 3*I)*(3 + 4*I) + (2 + 3*I)*(2 + 5*I)*i + 0*j + (2 + 3*I)*(7 + 8*I)*k i( RRNRR&RORRRR"R#R$R%(RPRQ((s8lib/python2.7/site-packages/sympy/algebras/quaternion.pyR>-s(*     #2  #2;78cCs(|}t|j|j |j |j ƒS(s(Returns the conjugate of the quaternion.(RR"R#R$R%(R(tq((s8lib/python2.7/site-packages/sympy/algebras/quaternion.pyt_eval_conjugatexscCs>|}tt|jd|jd|jd|jdƒƒS(s#Returns the norm of the quaternion.i(RR R"R#R$R%(R(RS((s8lib/python2.7/site-packages/sympy/algebras/quaternion.pyR/}scCs|}|d|jƒS(s.Returns the normalized form of the quaternion.i(R/(R(RS((s8lib/python2.7/site-packages/sympy/algebras/quaternion.pyR)„scCs=|}|jƒs!tdƒ‚nt|ƒd|jƒdS(s&Returns the inverse of the quaternion.s6Cannot compute inverse for a quaternion with zero normii(R/RR(R(RS((s8lib/python2.7/site-packages/sympy/algebras/quaternion.pytinverse‰s cCs |}|dkr|jƒSd}|dkrE|jƒ| }}nt|ttfƒs^tSx;|dkr›|d@r„||}n|d?}||}qaW|S(sþFinds the pth power of the quaternion. Parameters ========== p : int Power to be applied on quaternion. Returns ======= Quaternion Returns the p-th power of the current quaternion. Returns the inverse if p = -1. Examples ======== >>> from sympy.algebras.quaternion import Quaternion >>> q = Quaternion(1, 2, 3, 4) >>> q.pow(4) 668 + (-224)*i + (-336)*j + (-448)*k iÿÿÿÿii(RURNRRtNotImplemented(R(RBRStres((s8lib/python2.7/site-packages/sympy/algebras/quaternion.pyRAs      cCsÇ|}t|jd|jd|jdƒ}t|jƒt|ƒ}t|jƒt|ƒ|j|}t|jƒt|ƒ|j|}t|jƒt|ƒ|j|}t||||ƒS(s·Returns the exponential of q (e^q). Returns ======= Quaternion Exponential of q (e^q). Examples ======== >>> from sympy.algebras.quaternion import Quaternion >>> q = Quaternion(1, 2, 3, 4) >>> q.exp() E*cos(sqrt(29)) + 2*sqrt(29)*E*sin(sqrt(29))/29*i + 3*sqrt(29)*E*sin(sqrt(29))/29*j + 4*sqrt(29)*E*sin(sqrt(29))/29*k i( RR#R$R%R R"RRR(R(RSt vector_normR"R#R$R%((s8lib/python2.7/site-packages/sympy/algebras/quaternion.pyR ¼s)$$$cCs´|}t|jd|jd|jdƒ}|jƒ}t|ƒ}|jt|j|ƒ|}|jt|j|ƒ|}|jt|j|ƒ|}t||||ƒS(sxReturns the natural logarithm of the quaternion (_ln(q)). Examples ======== >>> from sympy.algebras.quaternion import Quaternion >>> q = Quaternion(1, 2, 3, 4) >>> q._ln() log(sqrt(30)) + 2*sqrt(29)*acos(sqrt(30)/30)/29*i + 3*sqrt(29)*acos(sqrt(30)/30)/29*j + 4*sqrt(29)*acos(sqrt(30)/30)/29*k i( RR#R$R%R/R RR"R(R(RSRXtq_normR"R#R$R%((s8lib/python2.7/site-packages/sympy/algebras/quaternion.pyt_lnÚs)  cCs@|}|jƒ\}}tj|||ƒ}||jƒ|S(slComputes the pth power in the cos-sin form. Parameters ========== p : int Power to be applied on quaternion. Returns ======= Quaternion The p-th power in the cos-sin form. Examples ======== >>> from sympy.algebras.quaternion import Quaternion >>> q = Quaternion(1, 2, 3, 4) >>> q.pow_cos_sin(4) 900*cos(4*acos(sqrt(30)/30)) + 1800*sqrt(29)*sin(4*acos(sqrt(30)/30))/29*i + 2700*sqrt(29)*sin(4*acos(sqrt(30)/30))/29*j + 3600*sqrt(29)*sin(4*acos(sqrt(30)/30))/29*k (t to_axis_angleRR1R/(R(RBRStvR+RQ((s8lib/python2.7/site-packages/sympy/algebras/quaternion.pyt pow_cos_sinóscGsCtt|j|Œt|j|Œt|j|Œt|j|ŒƒS(N(RR R"R#R$R%(R(RE((s8lib/python2.7/site-packages/sympy/algebras/quaternion.pyR s!cCs}t|tƒr,tj|d|dƒ}n |jƒ}|td|d|d|dƒt|ƒ}|j|j|jfS(s¢Returns the coordinates of the point pin(a 3 tuple) after rotation. Parameters ========== pin : tuple A 3-element tuple of coordinates of a point. This point will be the axis of rotation. r Angle to be rotated. Returns ======= tuple The coordinates of the quaternion after rotation. Examples ======== >>> from sympy.algebras.quaternion import Quaternion >>> from sympy import symbols, trigsimp, cos, sin >>> x = symbols('x') >>> q = Quaternion(cos(x/2), 0, 0, sin(x/2)) >>> trigsimp(Quaternion.rotate_point((1, 1, 1), q)) (sqrt(2)*cos(x + pi/4), sqrt(2)*sin(x + pi/4), 1) >>> (axis, angle) = q.to_axis_angle() >>> trigsimp(Quaternion.rotate_point((1, 1, 1), (axis, angle))) (sqrt(2)*cos(x + pi/4), sqrt(2)*sin(x + pi/4), 1) iii( RNttupleRR1R)RR#R$R%(tpintrRStpout((s8lib/python2.7/site-packages/sympy/algebras/quaternion.pyt rotate_points  /c CsÑ|}y |jdkr%|d}nWntk r9nX|jƒ}tdt|jƒƒ}td|j|jƒ}t|j|ƒ}t|j|ƒ}t|j|ƒ}|||f}||f}|S(s¤Returns the axis and angle of rotation of a quaternion Returns ======= tuple Tuple of (axis, angle) Examples ======== >>> from sympy.algebras.quaternion import Quaternion >>> q = Quaternion(1, 1, 1, 1) >>> (axis, angle) = q.to_axis_angle() >>> axis (sqrt(3)/3, sqrt(3)/3, sqrt(3)/3) >>> angle 2*pi/3 iiÿÿÿÿii( R"t BaseExceptionR)R RRR#R$R%( R(RSR+R0R,R-R.R\tt((s8lib/python2.7/site-packages/sympy/algebras/quaternion.pyR[Cs   cCsa|}|jƒd}dd||jd|jd}d||j|j|j|j}d||j|j|j|j}d||j|j|j|j}dd||jd|jd}d||j|j|j|j} d||j|j|j|j} d||j|j|j|j} dd||jd|jd} |sšt|||g||| g| | | ggƒS|\} }}| | |||||}|| ||||| }|| | || || }d}}}d}t||||g||| |g| | | |g||||ggƒSdS(síReturns the equivalent rotation transformation matrix of the quaternion which represents rotation about the origin if v is not passed. Parameters ========== v : tuple or None Default value: None Returns ======= tuple Returns the equivalent rotation transformation matrix of the quaternion which represents rotation about the origin if v is not passed. Examples ======== >>> from sympy.algebras.quaternion import Quaternion >>> from sympy import symbols, trigsimp, cos, sin >>> x = symbols('x') >>> q = Quaternion(cos(x/2), 0, 0, sin(x/2)) >>> trigsimp(q.to_rotation_matrix()) Matrix([ [cos(x), -sin(x), 0], [sin(x), cos(x), 0], [ 0, 0, 1]]) Generates a 4x4 transformation matrix (used for rotation about a point other than the origin) if the point(v) is passed as an argument. Examples ======== >>> from sympy.algebras.quaternion import Quaternion >>> from sympy import symbols, trigsimp, cos, sin >>> x = symbols('x') >>> q = Quaternion(cos(x/2), 0, 0, sin(x/2)) >>> trigsimp(q.to_rotation_matrix((1, 1, 1))) Matrix([ [cos(x), -sin(x), 0, sin(x) - cos(x) + 1], [sin(x), cos(x), 0, -sin(x) - cos(x) + 1], [ 0, 0, 1, 0], [ 0, 0, 0, 1]]) iþÿÿÿiiiN(R/R$R%R#R"R (R(R\RSR0tm00tm01tm02tm10tm11tm12tm20tm21tm22R,R-R.tm03tm13tm23tm30tm31tm32tm33((s8lib/python2.7/site-packages/sympy/algebras/quaternion.pytto_rotation_matrixps*0$&&&$&&&$.!N(,t__name__t __module__t__doc__t _op_priorityRRRIRtpropertyR"R#R$R%R&t classmethodR1R7t staticmethodR3R;R<R=R?R@RCRDRFRKR9RRR>RTR/R)RURAR RZR]R RbR[tNoneRu(((s8lib/python2.7/site-packages/sympy/algebras/quaternion.pyRsH&/          7 (K     ,   " ) -N(t __future__RtsympyRRRRRRRRR R R R R RRRtsympy.core.compatibilityRtsympy.core.exprRtsympy.core.numbersRR(((s8lib/python2.7/site-packages/sympy/algebras/quaternion.pyts.