Rie˹ddlmZddlmZddlmZddlmZmZm Z m Z ddl m Z m ZddlmZddlmZmZmZddlmZmZdd lmZdd lmZdd lmZdd lmZm Z dd l!m"Z"ddl#m$Z$m%Z%ddl&m'Z'dZ(dZ)Gdde"Z*dS))Rational)S)is_eq) conjugateimresign)explog)sqrt)acosasinatan2)cossin)trigsimp integrate)MutableDenseMatrix)sympify_sympify)Expr) fuzzy_notfuzzy_or) prec_to_dpsc|u|jrp|jdurtdtd|D}|r?t |dzt d|DdurtddSdSdSdS)z$validate if input norm is consistentNFzInput norm must be positive.c36K|]}|jo|jduVdS)TN) is_numberis_real.0is 9lib/python3.11/site-packages/sympy/algebras/quaternion.py z_check_norm..s0LLa 9 T(9LLLLLLc3 K|] }|dzV dS)r&Nr s r#r$z_check_norm..s&+C+CQAqD+C+C+C+C+C+Cr%zIncompatible value for norm.)r is_positive ValueErrorallrsum)elementsnorm numericals r# _check_normr0s DN  u $ $;<< <LL8LLLLL  =tQw+C+C(+C+C+C(C(CDDMM;<< <   = =MMr%cFt|tkrtdt|dkr"td||}|}|s|std|\}}}||ks||krtdt|tdz }|r5tdd ||S) zGvalidate seq and return True if seq is lowercase and False if uppercasezExpected seq to be a string.zExpected 3 axes, got `{}`.zkseq must either be fully uppercase (for extrinsic rotations), or fully lowercase, for intrinsic rotations).z"Consecutive axes must be differentxyzXYZzNExpected axes from `seq` to be from ['x', 'y', 'z'] or ['X', 'Y', 'Z'], got {}) typestrr*lenformatisupperislowerlowersetjoin)seq intrinsic extrinsicr"jkbads r# _is_extrinsicrDs CyyC7888 3xx1}}5<>> c((S]] "C 8""(&"6"688 8 r%cLeZdZdZdZdZd>dZdZed Z ed Z ed Z ed Z ed Z edZedZd?dZedZedZd@dZedZedZdZdZdZdZdZdZdZdZdZdZ d Z!d!Z"d"Z#e$d#Z%d$Z&d%Z'd&Z(d'Z)d(Z*d)Z+d*Z,d+Z-d,Z.d-Z/d.Z0e$d/Z1d0Z2dAd1Z3d2Z4d3Z5d4Z6d5Z7d6Z8d7Z9d8Z:ed9Z;d:Zd=Z?dS)B QuaternionaProvides basic quaternion operations. Quaternion objects can be instantiated as Quaternion(a, b, c, d) as in (a + b*i + c*j + d*k). Parameters ========== norm : None or number Pre-defined quaternion norm. If a value is given, Quaternion.norm returns this pre-defined value instead of calculating the norm Examples ======== >>> from sympy 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 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 Defining symbolic unit quaternions: >>> from sympy import Quaternion >>> from sympy.abc import w, x, y, z >>> q = Quaternion(w, x, y, z, norm=1) >>> q w + x*i + y*j + z*k >>> q.norm() 1 References ========== .. [1] https://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions/ .. [2] https://en.wikipedia.org/wiki/Quaternion g&@FrTNc:tt||||f\}}}}td||||fDrtdt j|||||}||_||_||_||_ ||_ | ||S)Nc3(K|] }|jduVdS)FN)is_commutativer s r#r$z%Quaternion.__new__..ps*??Qq5(??????r%z arguments have to be commutative) mapranyr*r__new___a_b_c_d _real_fieldset_norm)clsabcd real_fieldr.objs r#rLzQuaternion.__new__ms1aA,// 1a ??1aA,??? ? ? ?@@ @,sAq!Q//CCFCFCFCF(CO LL   Jr%c\t|}t|j|||_dS)aSets norm of an already instantiated quaternion. Parameters ========== norm : None or number Pre-defined quaternion norm. If a value is given, Quaternion.norm returns this pre-defined value instead of calculating the norm Examples ======== >>> from sympy import Quaternion >>> from sympy.abc import a, b, c, d >>> q = Quaternion(a, b, c, d) >>> q.norm() sqrt(a**2 + b**2 + c**2 + d**2) Setting the norm: >>> q.set_norm(1) >>> q.norm() 1 Removing set norm: >>> q.set_norm(None) >>> q.norm() sqrt(a**2 + b**2 + c**2 + d**2) N)rr0args_norm)selfr.s r#rRzQuaternion.set_norm|s-@t}}DIt$$$ r%c|jSN)rMr]s r#rTz Quaternion.a wr%c|jSr_)rNr`s r#rUz Quaternion.brar%c|jSr_)rOr`s r#rVz Quaternion.crar%c|jSr_)rPr`s r#rWz Quaternion.drar%c|jSr_)rQr`s r#rXzQuaternion.real_fields r%c t|j|j |j |j g|j|j|j |jg|j|j|j|j g|j|j |j|jggS)aReturns 4 x 4 Matrix equivalent to a Hamilton product from the left. This can be useful when treating quaternion elements as column vectors. Given a quaternion $q = a + bi + cj + dk$ where a, b, c and d are real numbers, the product matrix from the left is: .. math:: M = \begin{bmatrix} a &-b &-c &-d \\ b & a &-d & c \\ c & d & a &-b \\ d &-c & b & a \end{bmatrix} Examples ======== >>> from sympy import Quaternion >>> from sympy.abc import a, b, c, d >>> q1 = Quaternion(1, 0, 0, 1) >>> q2 = Quaternion(a, b, c, d) >>> q1.product_matrix_left Matrix([ [1, 0, 0, -1], [0, 1, -1, 0], [0, 1, 1, 0], [1, 0, 0, 1]]) >>> q1.product_matrix_left * q2.to_Matrix() Matrix([ [a - d], [b - c], [b + c], [a + d]]) This is equivalent to: >>> (q1 * q2).to_Matrix() Matrix([ [a - d], [b - c], [b + c], [a + d]]) MatrixrTrUrVrWr`s r#product_matrix_leftzQuaternion.product_matrix_leftsxX$&46'DF73$&$&1$&1$&$&$&1 344 4r%c t|j|j |j |j g|j|j|j|j g|j|j |j|jg|j|j|j |jggS)aMReturns 4 x 4 Matrix equivalent to a Hamilton product from the right. This can be useful when treating quaternion elements as column vectors. Given a quaternion $q = a + bi + cj + dk$ where a, b, c and d are real numbers, the product matrix from the left is: .. math:: M = \begin{bmatrix} a &-b &-c &-d \\ b & a & d &-c \\ c &-d & a & b \\ d & c &-b & a \end{bmatrix} Examples ======== >>> from sympy import Quaternion >>> from sympy.abc import a, b, c, d >>> q1 = Quaternion(a, b, c, d) >>> q2 = Quaternion(1, 0, 0, 1) >>> q2.product_matrix_right Matrix([ [1, 0, 0, -1], [0, 1, 1, 0], [0, -1, 1, 0], [1, 0, 0, 1]]) Note the switched arguments: the matrix represents the quaternion on the right, but is still considered as a matrix multiplication from the left. >>> q2.product_matrix_right * q1.to_Matrix() Matrix([ [ a - d], [ b + c], [-b + c], [ a + d]]) This is equivalent to: >>> (q1 * q2).to_Matrix() Matrix([ [ a - d], [ b + c], [-b + c], [ a + d]]) rgr`s r#product_matrix_rightzQuaternion.product_matrix_rightsxb$&46'DF73$&1$&$&$&1$&$&1 344 4r%cf|rt|jddSt|jS)aReturns elements of quaternion as a column vector. By default, a Matrix of length 4 is returned, with the real part as the first element. If vector_only is True, returns only imaginary part as a Matrix of length 3. Parameters ========== vector_only : bool If True, only imaginary part is returned. Default value: False Returns ======= Matrix A column vector constructed by the elements of the quaternion. Examples ======== >>> from sympy import Quaternion >>> from sympy.abc import a, b, c, d >>> q = Quaternion(a, b, c, d) >>> q a + b*i + c*j + d*k >>> q.to_Matrix() Matrix([ [a], [b], [c], [d]]) >>> q.to_Matrix(vector_only=True) Matrix([ [b], [c], [d]]) N)rhr[)r] vector_onlys r# to_MatrixzQuaternion.to_Matrixs5X  %$)ABB-(( ($)$$ $r%ct|}|dkr(|dkr"td||dkr tdg|RSt|S)aReturns quaternion from elements of a column vector`. If vector_only is True, returns only imaginary part as a Matrix of length 3. Parameters ========== elements : Matrix, list or tuple of length 3 or 4. If length is 3, assume real part is zero. Default value: False Returns ======= Quaternion A quaternion created from the input elements. Examples ======== >>> from sympy import Quaternion >>> from sympy.abc import a, b, c, d >>> q = Quaternion.from_Matrix([a, b, c, d]) >>> q a + b*i + c*j + d*k >>> q = Quaternion.from_Matrix([b, c, d]) >>> q 0 + b*i + c*j + d*k r2z7Input elements must have length 3 or 4, got {} elementsr)r7r*r8rF)rSr-lengths r# from_MatrixzQuaternion.from_MatrixNsoBX Q;;6Q;;((.v88 8 Q;;a+(+++ +x( (r%c t|dkrtdt|}|\ fddD} fddD} fddD}|||d}|||d}|||d } |rt | |z|zSt ||z| zS) aReturns quaternion equivalent to rotation represented by the Euler angles, in the sequence defined by ``seq``. Parameters ========== angles : list, tuple or Matrix of 3 numbers The Euler angles (in radians). seq : string of length 3 Represents the sequence of rotations. For intrinsic rotations, seq must be all lowercase and its elements must be from the set ``{'x', 'y', 'z'}`` For extrinsic rotations, seq must be all uppercase and its elements must be from the set ``{'X', 'Y', 'Z'}`` Returns ======= Quaternion The normalized rotation quaternion calculated from the Euler angles in the given sequence. Examples ======== >>> from sympy import Quaternion >>> from sympy import pi >>> q = Quaternion.from_euler([pi/2, 0, 0], 'xyz') >>> q sqrt(2)/2 + sqrt(2)/2*i + 0*j + 0*k >>> q = Quaternion.from_euler([0, pi/2, pi] , 'zyz') >>> q 0 + (-sqrt(2)/2)*i + 0*j + sqrt(2)/2*k >>> q = Quaternion.from_euler([0, pi/2, pi] , 'ZYZ') >>> q 0 + sqrt(2)/2*i + 0*j + sqrt(2)/2*k r2z3 angles must be given.c$g|] }|krdnd Srmrr()r!nr"s r# z)Quaternion.from_euler..% 0 0 0Q166aaq 0 0 0r%xyzc$g|] }|krdnd Srvr()r!rwrAs r#rxz)Quaternion.from_euler..ryr%c$g|] }|krdnd Srvr()r!rwrBs r#rxz)Quaternion.from_euler..ryr%rrmr&)r7r*rDr;from_axis_angler) rSanglesr>r@eiejekqiqjqkr"rArBs @@@r# from_eulerzQuaternion.from_eulerysV v;;!  677 7!#&& ))++1a1 0 0 0% 0 0 0 0 0 0 0% 0 0 0 0 0 0 0% 0 0 0 VAY / /  VAY / /  VAY / /  *BGbL)) )BGbL)) )r%c V|rtdgd}t|}|\}}}d|dz}d|dz}d|dz}|s||}}||k} | rd|z |z }||z ||z z||z zdz} |j|j|j|jg} | d} | |} | |}| || z}| s| |z | |z|| z|| z f\} } }}|r| rB| dz}t| | z| | zz||zz ||zz |z |d<nd| dzz}t||z||zz| | zz | | zz |z |d<nadtt||z||zzt| | z| | zzz|d<| s|dxxtjdz zcc<d}t!|tjrt!|tjrd}t!| tjrt!| tjrd}|dkr|rIt| | t||z|d<t| | t||z |d<nt| |z| |zz| |z| |zz |d<t| |z| |zz | |z| |zz|d<nctj|d| z<|dkrdt| | z|d|z<n0dt||z|d|z<|d|zxx|rdndzcc<| s|dxx| zcc<|rt%|d d dSt%|S) a}Returns Euler angles representing same rotation as the quaternion, in the sequence given by ``seq``. This implements the method described in [1]_. For degenerate cases (gymbal lock cases), the third angle is set to zero. Parameters ========== seq : string of length 3 Represents the sequence of rotations. For intrinsic rotations, seq must be all lowercase and its elements must be from the set ``{'x', 'y', 'z'}`` For extrinsic rotations, seq must be all uppercase and its elements must be from the set ``{'X', 'Y', 'Z'}`` angle_addition : bool When True, first and third angles are given as an addition and subtraction of two simpler ``atan2`` expressions. When False, the first and third angles are each given by a single more complicated ``atan2`` expression. This equivalent expression is given by: .. math:: \operatorname{atan_2} (b,a) \pm \operatorname{atan_2} (d,c) = \operatorname{atan_2} (bc\pm ad, ac\mp bd) Default value: True avoid_square_root : bool When True, the second angle is calculated with an expression based on ``acos``, which is slightly more complicated but avoids a square root. When False, second angle is calculated with ``atan2``, which is simpler and can be better for numerical reasons (some numerical implementations of ``acos`` have problems near zero). Default value: False Returns ======= Tuple The Euler angles calculated from the quaternion Examples ======== >>> from sympy import Quaternion >>> from sympy.abc import a, b, c, d >>> euler = Quaternion(a, b, c, d).to_euler('zyz') >>> euler (-atan2(-b, c) + atan2(d, a), 2*atan2(sqrt(b**2 + c**2), sqrt(a**2 + d**2)), atan2(-b, c) + atan2(d, a)) References ========== .. [1] https://doi.org/10.1371/journal.pone.0276302 z(Cannot convert a quaternion with norm 0.)rrrrzrmr&rN)is_zero_quaternionr*rDr;indexrTrUrVrWr.r rrr rPirZerotuple)r]r>angle_additionavoid_square_rootr~r@r"rArB symmetricr r-rTrUrVrWn2cases r#to_eulerzQuaternion.to_eulers*@  " " $ $ IGHH H!#&& ))++1a KKNNQ  KKNNQ  KKNNQ  aqAF  A AA!a% AE*a/FDFDFDF3 QK QK QK QK$  4QAq1ua!e3JAq!Q  & GYY[[!^ !a%!a%-!a%"7!a%"?2!EFFq a' !a%!a%-!a%"7!a%"?2!EFFq E$q1uq1u}"5"5tAEAEM7J7JKKKF1I &q QTAX%  AF   a 0 0 D AF   a 0 0 D 199 8!!QKK%1++5q !!QKK%1++5q !!A#!)QqS1Q3Y77q !!A#!)QqS1Q3Y77q +,&F1I & 'qyy()E!QKKq9}%%()E!QKKq9}%q9}%%% *@""qA%%%  1III III  !"&& &== r%c|\}}}t|dz|dzz|dzz}||z ||z ||z }}}t|tjz}t |tjz}||z} ||z} ||z} ||| | | S)aReturns a rotation quaternion given the axis and the angle of rotation. Parameters ========== vector : tuple of three numbers The vector representation of the given axis. angle : number The angle by which axis is rotated (in radians). Returns ======= Quaternion The normalized rotation quaternion calculated from the given axis and the angle of rotation. Examples ======== >>> from sympy import Quaternion >>> from sympy import pi, sqrt >>> q = Quaternion.from_axis_angle((sqrt(3)/3, sqrt(3)/3, sqrt(3)/3), 2*pi/3) >>> q 1/2 + 1/2*i + 1/2*j + 1/2*k r&)r rrHalfr) rSvectoranglexyzr.srTrUrVrWs r#r}zQuaternion.from_axis_angleGs8 AqAqD1a4K!Q$&''Xq4xTqA       E E E s1aAr%c|tddz}t||dz|dz|dzdz }t||dz|dz |dz dz }t||dz |dz|dz dz }t||dz |dz |dzdz }|t|d|dz z}|t|d |d z z}|t|d |d z z}t ||||S) aReturns 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 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))*sign(sin(x))/2*k rmr2)rr)rmrm)r&r&r&)r&rm)rmr&)rr&)r&rrv)rrm)detrr r rF)rSMabsQrTrUrVrWs r#from_rotation_matrixzQuaternion.from_rotation_matrixrs?>uuwwA& $!D')AdG3 4 4q 8 $!D')AdG3 4 4q 8 $!D')AdG3 4 4q 8 $!D')AdG3 4 4q 8 QtWqw&'' ' QtWqw&'' ' QtWqw&'' '!Q1%%%r%c,||Sr_addr]others r#__add__zQuaternion.__add__xxr%c,||Sr_rrs r#__radd__zQuaternion.__radd__rr%c2||dzSNrrrs r#__sub__zQuaternion.__sub__sxxb!!!r%cH||t|Sr_ _generic_mulrrs r#__mul__zQuaternion.__mul__s  x777r%cH|t||Sr_rrs r#__rmul__zQuaternion.__rmul__s  %$777r%c,||Sr_)pow)r]ps r#__pow__zQuaternion.__pow__sxx{{r%cVt|j |j |j |j Sr_)rFrMrNrOrWr`s r#__neg__zQuaternion.__neg__s&47(TWHtwh@@@r%c,|t|dzzSrrrs r# __truediv__zQuaternion.__truediv__sgennb(((r%c,t||dzzSrrrs r# __rtruediv__zQuaternion.__rtruediv__su~~b((r%c|j|Sr_rr]r[s r#_eval_IntegralzQuaternion._eval_Integralst~t$$r%cjdd|jfd|jDS)NevaluateTc*g|]}|jiSr()diff)r!rTkwargssymbolss r#rxz#Quaternion.diff..s*JJJ!61675f55JJJr%) setdefaultfuncr[)r]rrs ``r#rzQuaternion.diffsC*d+++tyJJJJJ JJJKKr%c|}t|}t|ts|jrM|jrFtt ||jzt||jz|j |j S|j r)t|j|z|j|j |j Stdt|j|jz|j|jz|j |j z|j |j zS)aAdds 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 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 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 z>> !_```"$+rtbd{BD24KDB!"" "r%cH||t|S)aMultiplies 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 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 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 rrs r#mulzQuaternion.muls!N  x777r%cPt|tst|ts||zSt|ts|jr6|jr/tt |t |dd|zS|jr2t||jz||jz||j z||j zStdt|ts|jr6|jr/|tt |t |ddzS|jr2t||jz||jz||j z||j zStd|j |j d}n)| | z}t|j |jz|j |j zz |j |j zz |j|jzz|j|jz|j |j zz|j |j zz |j|jzz|j |j z|j |jzz|j |jzz|j|j zz|j|j z|j |jzz |j |jzz|j|j zz|S)anGeneric multiplication. Parameters ========== q1 : Quaternion or symbol q2 : Quaternion or symbol It is 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 import Quaternion >>> from sympy import Symbol, S >>> 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, S(2)) 2 + 4*i + 6*j + 8*k >>> x = Symbol('x', real = True) >>> Quaternion._generic_mul(q1, x) x + 2*x*i + 3*x*j + 4*x*k Quaternions over complex fields : >>> 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 rzAOnly commutative expressions can be multiplied with a Quaternion.Nr.)rrFrXrrrrIrTrUrVrWr*r\r.)rrr.s r#rzQuaternion._generic_mulstV"j)) *R2L2L 7N"j)) f} f f!"R&&"R&&!Q77"<<" f!"rt)R"$YRT 29MMM !deee"j)) f} f fJr"vvr"vvq!<<<<" f!"rt)R"$YRT 29MMM !deee 8  0DD7799rwwyy(D24%*rtBDy0249>>>r%c|jL|}tt|jdz|jdzz|jdzz|jdzz|_|jS)z#Returns the norm of the quaternion.Nr&)r\r rrTrUrVrWrs r#r.zQuaternion.normpsV : AhqsAvQa'?!#q&'HIIJJDJzr%c:|}|d|z zS)z.Returns the normalized form of the quaternion.rmrrs r# normalizezQuaternion.normalizezs AaffhhJr%c|}|stdt|d|dzz zS)z&Returns the inverse of the quaternion.z6Cannot compute inverse for a quaternion with zero normrmr&)r.r*rrs r#inversezQuaternion.inversesH vvxx WUVV V||q1}--r%ct|}|}|dkr|Sd}|jstS|dkr|| }}|dkr|dzdkr||z}|dz}||z}|dk|S)aFinds 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 import Quaternion >>> q = Quaternion(1, 2, 3, 4) >>> q.pow(4) 668 + (-224)*i + (-336)*j + (-448)*k rrmrr&)rr is_IntegerNotImplemented)r]rrress r#rzQuaternion.pows2 AJJ  7799;; | "! ! q5599;;qA!ee1uzz#g1AAA !ee r%c|}t|jdz|jdzz|jdzz}t |jt |z}t |jt|z|jz|z }t |jt|z|jz|z }t |jt|z|jz|z }t||||S)aReturns the exponential of q (e^q). Returns ======= Quaternion Exponential of q (e^q). Examples ======== >>> from sympy 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 r&) r rUrVrWr rTrrrF)r]r vector_normrTrUrVrWs r#r zQuaternion.exps, 136ACF?QS!V344 HHs;'' ' HHs;'' '!# - ; HHs;'' '!# - ; HHs;'' '!# - ;!Q1%%%r%c|}t|jdz|jdzz|jdzz}|}t |}|jt |j|z z|z }|jt |j|z z|z }|jt |j|z z|z }t||||S)aeReturns the natural logarithm of the quaternion (_ln(q)). Examples ======== >>> from sympy 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 r&) r rUrVrWr.lnr rTrF)r]rrq_normrTrUrVrWs r#_lnzQuaternion._lns 136ACF?QS!V344  vJJ C$qsV|$$ ${ 2 C$qsV|$$ ${ 2 C$qsV|$$ ${ 2!Q1%%%r%cfd|jD}|j} |j}n#t$rYnwxYwt ||t |d|iS)Nc$g|] }|j Sr()subs)r!r"r[s r#rxz)Quaternion._eval_subs..s!555aFAFDM555r%r.)r[r\rAttributeErrorr0rF)r]r[r-r.s ` r# _eval_subszQuaternion._eval_subss}555549555z 49d#DD    D Hd###8/$///s ( 55cVt|tfd|jDS)a Returns the floating point approximations (decimal numbers) of the quaternion. Returns ======= Quaternion Floating point approximations of quaternion(self) Examples ======== >>> from sympy import Quaternion >>> from sympy import sqrt >>> q = Quaternion(1/sqrt(1), 1/sqrt(2), 1/sqrt(3), 1/sqrt(4)) >>> q.evalf() 1.00000000000000 + 0.707106781186547*i + 0.577350269189626*j + 0.500000000000000*k c<g|]}|S))rw)evalf)r!argnprecs r#rxz*Quaternion._eval_evalf..s'DDD3CIII..DDDr%)rrFr[)r]precrs @r# _eval_evalfzQuaternion._eval_evalfs4,D!!DDDD$)DDDEEr%c|}|\}}t|||z}|||zzS)aYComputes 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 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 ) to_axis_anglerFr}r.)r]rrvrrs r# pow_cos_sinzQuaternion.pow_cos_sinsL< __&& E  ' '1u9 5 5QVVXXq[!!r%c tt|jg|Rt|jg|Rt|jg|Rt|jg|RS)aComputes integration of quaternion. Returns ======= Quaternion Integration of the quaternion(self) with the given variable. Examples ======== Indefinite Integral of quaternion : >>> from sympy import Quaternion >>> from sympy.abc import x >>> q = Quaternion(1, 2, 3, 4) >>> q.integrate(x) x + 2*x*i + 3*x*j + 4*x*k Definite integral of quaternion : >>> from sympy import Quaternion >>> from sympy.abc import x >>> q = Quaternion(1, 2, 3, 4) >>> q.integrate((x, 1, 5)) 4 + 8*i + 12*j + 16*k )rFrrTrUrVrWrs r#rzQuaternion.integrate3sj<)DF2T222Idf4Lt4L4L4L#DF2T222Idf4Lt4L4L4LNN Nr%c:t|tr(t|d|d}n|}|td|d|d|dzt |z}|j|j|jfS)aReturns the coordinates of the point pin(a 3 tuple) after rotation. Parameters ========== pin : tuple A 3-element tuple of coordinates of a point which needs to be rotated. r : Quaternion or tuple Axis and angle of rotation. It's important to note that when r is a tuple, it must be of the form (axis, angle) Returns ======= tuple The coordinates of the point after rotation. Examples ======== >>> from sympy 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) rrmr&) rrrFr}rrrUrVrW)pinrrpouts r# rotate_pointzQuaternion.rotate_pointTsH a   **1Q4166AA A:aQQQ8889Q<<G''r%cv|}|jjr|dz}|}tdt |jz}t d|j|jzz }t|j|z }t|j|z }t|j|z }|||f}||f}|S)aReturns the axis and angle of rotation of a quaternion. Returns ======= tuple Tuple of (axis, angle) Examples ======== >>> from sympy 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 rr&rm) rT is_negativerrr r rUrVrW) r]rrrrrrrts r#rzQuaternion.to_axis_angles*  3? BA KKMMT!#YY'' QSW   QS1W   QS1W   QS1W   1I Jr%c |}|dz}|r||jdz|jdzz|jdzz |jdzz z}||jdz|jdzz |jdzz|jdzz z}||jdz|jdzz |jdzz |jdzzz}nZdd|z|jdz|jdzzzz }dd|z|jdz|jdzzzz }dd|z|jdz|jdzzzz }d|z|j|jz|j|jzz z}d|z|j|jz|j|jzzz} d|z|j|jz|j|jzzz} d|z|j|jz|j|jzz z} d|z|j|jz|j|jzz z} d|z|j|jz|j|jzzz} |st ||| g| || g| | |ggS|\}}}|||zz ||zz || zz }||| zz ||zz || zz }||| zz || zz ||zz }dx}x}}d}t ||| |g| || |g| | ||g||||ggS)aReturns 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 homogeneous : bool When True, gives an expression that may be more efficient for symbolic calculations but less so for direct evaluation. Both formulas are mathematically equivalent. Default value: True 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 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. r&rmr)r.rTrUrVrWrh)r]r homogeneousrrm00m11m22m01m02m10m12m20m21rrrm03m13m23m30m31m32m33s r#to_rotation_matrixzQuaternion.to_rotation_matrixsN  FFHHbL  ,QS!Vac1f_qsAv-Q67CQS!Vac1f_qsAv-Q67CQS!Vac1f_qsAv-Q67CCac136ACF?++Cac136ACF?++Cac136ACF?++Cc13qs7QSW$%c13qs7QSW$%c13qs7QSW$%c13qs7QSW$%c13qs7QSW$%c13qs7QSW$% GCc?S#sOc3_MNN NIQ1ae)ae#ae+Cae)ae#ae+Cae)ae#ae+C C #CCc3/#sC1ES#.c30DFGG Gr%c|jS)amReturns scalar part($\mathbf{S}(q)$) of the quaternion q. Explanation =========== Given a quaternion $q = a + bi + cj + dk$, returns $\mathbf{S}(q) = a$. Examples ======== >>> from sympy.algebras.quaternion import Quaternion >>> q = Quaternion(4, 8, 13, 12) >>> q.scalar_part() 4 )rTr`s r# scalar_partzQuaternion.scalar_parts $v r%cDtd|j|j|jS)a Returns vector part($\mathbf{V}(q)$) of the quaternion q. Explanation =========== Given a quaternion $q = a + bi + cj + dk$, returns $\mathbf{V}(q) = bi + cj + dk$. Examples ======== >>> from sympy.algebras.quaternion import Quaternion >>> q = Quaternion(1, 1, 1, 1) >>> q.vector_part() 0 + 1*i + 1*j + 1*k >>> q = Quaternion(4, 8, 13, 12) >>> q.vector_part() 0 + 8*i + 13*j + 12*k r)rFrUrVrWr`s r# vector_partzQuaternion.vector_part s.!TVTVTV444r%c|}td|j|j|jS)a Returns the axis($\mathbf{Ax}(q)$) of the quaternion. Explanation =========== Given a quaternion $q = a + bi + cj + dk$, returns $\mathbf{Ax}(q)$ i.e., the versor of the vector part of that quaternion equal to $\mathbf{U}[\mathbf{V}(q)]$. The axis is always an imaginary unit with square equal to $-1 + 0i + 0j + 0k$. Examples ======== >>> from sympy.algebras.quaternion import Quaternion >>> q = Quaternion(1, 1, 1, 1) >>> q.axis() 0 + sqrt(3)/3*i + sqrt(3)/3*j + sqrt(3)/3*k See Also ======== vector_part r)rrrFrUrVrW)r]axiss r#rzQuaternion.axis"s;2!!++--!TVTVTV444r%c|jjS)a Returns true if the quaternion is pure, false if the quaternion is not pure or returns none if it is unknown. Explanation =========== A pure quaternion (also a vector quaternion) is a quaternion with scalar part equal to 0. Examples ======== >>> from sympy.algebras.quaternion import Quaternion >>> q = Quaternion(0, 8, 13, 12) >>> q.is_pure() True See Also ======== scalar_part )rTis_zeror`s r#is_purezQuaternion.is_pure?s2v~r%c4|jS)a Returns true if the quaternion is a zero quaternion or false if it is not a zero quaternion and None if the value is unknown. Explanation =========== A zero quaternion is a quaternion with both scalar part and vector part equal to 0. Examples ======== >>> from sympy.algebras.quaternion import Quaternion >>> q = Quaternion(1, 0, 0, 0) >>> q.is_zero_quaternion() False >>> q = Quaternion(0, 0, 0, 0) >>> q.is_zero_quaternion() True See Also ======== scalar_part vector_part )r.rr`s r#rzQuaternion.is_zero_quaternionZs<yy{{""r%ct||S)a Returns the angle of the quaternion measured in the real-axis plane. Explanation =========== Given a quaternion $q = a + bi + cj + dk$ where a, b, c and d are real numbers, returns the angle of the quaternion given by .. math:: angle := atan2(\sqrt{b^2 + c^2 + d^2}, {a}) Examples ======== >>> from sympy.algebras.quaternion import Quaternion >>> q = Quaternion(1, 4, 4, 4) >>> q.angle() atan(4*sqrt(3)) )rrr.rr`s r#rzQuaternion.anglezs8.T%%'',,..0@0@0B0BCCCr%cv|s|rtdt||z ||zgS)aS Returns True if the transformation arcs represented by the input quaternions happen in the same plane. Explanation =========== Two quaternions are said to be coplanar (in this arc sense) when their axes are parallel. The plane of a quaternion is the one normal to its axis. Parameters ========== other : a Quaternion Returns ======= True : if the planes of the two quaternions are the same, apart from its orientation/sign. False : if the planes of the two quaternions are not the same, apart from its orientation/sign. None : if plane of either of the quaternion is unknown. Examples ======== >>> from sympy.algebras.quaternion import Quaternion >>> q1 = Quaternion(1, 4, 4, 4) >>> q2 = Quaternion(3, 8, 8, 8) >>> Quaternion.arc_coplanar(q1, q2) True >>> q1 = Quaternion(2, 8, 13, 12) >>> Quaternion.arc_coplanar(q1, q2) False See Also ======== vector_coplanar is_pure z)Neither of the given quaternions can be 0)rr*rrrs r# arc_coplanarzQuaternion.arc_coplanarsT  # # % % J5+C+C+E+E JHII I$))++ 4HHJJTYY[[[`[e[e[g[gMgL{L{L}L}~r%ct|sBt|s!t|rtdt|j|j|jg|j|j|jg|j|j|jgg}|jS)a Returns True if the axis of the pure quaternions seen as 3D vectors q1, q2, and q3 are coplanar. Explanation =========== Three pure quaternions are vector coplanar if the quaternions seen as 3D vectors are coplanar. Parameters ========== q1 A pure Quaternion. q2 A pure Quaternion. q3 A pure Quaternion. Returns ======= True : if the axis of the pure quaternions seen as 3D vectors q1, q2, and q3 are coplanar. False : if the axis of the pure quaternions seen as 3D vectors q1, q2, and q3 are not coplanar. None : if the axis of the pure quaternions seen as 3D vectors q1, q2, and q3 are coplanar is unknown. Examples ======== >>> from sympy.algebras.quaternion import Quaternion >>> q1 = Quaternion(0, 4, 4, 4) >>> q2 = Quaternion(0, 8, 8, 8) >>> q3 = Quaternion(0, 24, 24, 24) >>> Quaternion.vector_coplanar(q1, q2, q3) True >>> q1 = Quaternion(0, 8, 16, 8) >>> q2 = Quaternion(0, 8, 3, 12) >>> Quaternion.vector_coplanar(q1, q2, q3) False See Also ======== axis is_pure "The given quaternions must be pure) rrr*rhrUrVrWrr)rSrrq3rs r#vector_coplanarzQuaternion.vector_coplanarsl RZZ\\ " " Ci &=&= C2::<>> from sympy.algebras.quaternion import Quaternion >>> q = Quaternion(0, 4, 4, 4) >>> q1 = Quaternion(0, 8, 8, 8) >>> q.parallel(q1) True >>> q1 = Quaternion(0, 8, 13, 12) >>> q.parallel(q1) False z%The provided quaternions must be purerrr*rrs r#parallelzQuaternion.parallelsdJ T\\^^ $ $ F %--//(B(B FDEE EU U4Z';;===r%ct|s!t|rtd||z||zzS)a| Returns the orthogonality of two quaternions. Explanation =========== Two pure quaternions are called orthogonal when their product is anti-commutative. Parameters ========== other : a Quaternion Returns ======= True : if the two pure quaternions seen as 3D vectors are orthogonal. False : if the two pure quaternions seen as 3D vectors are not orthogonal. None : if the two pure quaternions seen as 3D vectors are orthogonal is unknown. Examples ======== >>> from sympy.algebras.quaternion import Quaternion >>> q = Quaternion(0, 4, 4, 4) >>> q1 = Quaternion(0, 8, 8, 8) >>> q.orthogonal(q1) False >>> q1 = Quaternion(0, 2, 2, 0) >>> q = Quaternion(0, 2, -2, 0) >>> q.orthogonal(q1) True r r$rs r# orthogonalzQuaternion.orthogonal)sdJ T\\^^ $ $ C %--//(B(B CABB BU U4Z';;===r%cT||zS)a Returns the index vector of the quaternion. Explanation =========== Index vector is given by $\mathbf{T}(q)$ multiplied by $\mathbf{Ax}(q)$ where $\mathbf{Ax}(q)$ is the axis of the quaternion q, and mod(q) is the $\mathbf{T}(q)$ (magnitude) of the quaternion. Returns ======= Quaternion: representing index vector of the provided quaternion. Examples ======== >>> from sympy.algebras.quaternion import Quaternion >>> q = Quaternion(2, 4, 2, 4) >>> q.index_vector() 0 + 4*sqrt(10)/3*i + 2*sqrt(10)/3*j + 4*sqrt(10)/3*k See Also ======== axis norm )r.rr`s r# index_vectorzQuaternion.index_vectorSs>yy{{TYY[[((r%cDt|S)aj Returns the natural logarithm of the norm(magnitude) of the quaternion. Examples ======== >>> from sympy.algebras.quaternion import Quaternion >>> q = Quaternion(2, 4, 2, 4) >>> q.mensor() log(2*sqrt(10)) >>> q.norm() 2*sqrt(10) See Also ======== norm )rr.r`s r#mensorzQuaternion.mensorts*$))++r%)rrrrTN)F)TF)NT)@__name__ __module__ __qualname____doc__ _op_priorityrIrLrRpropertyrTrUrVrWrXrirkro classmethodrsrrr}rrrrrrrrrrrrrr staticmethodrrr.rrrr rrrrrrrrrrrrrrrr"r%r'r)r+r(r%r#rFrF9sY..^LN    """HXXXX  X /4/4X/4b4444X44l/%/%/%/%b()()[()T=*=*[=*~L!L!L!L!\(([(T)&)&[)&V"""888888AAA))))))%%%LLL4"4"4"l'8'8'8RI%I%\I%V???    ...,,,\&&&>&&&4000FFF2!"!"!"FNNNB*(*(\*(X&&&PJGJGJGJGX(5552555:6###@DDD4-@-@-@^99[9v(>(>(>T(>(>(>T)))Br%rFN)+sympy.core.numbersrsympy.core.singletonrsympy.core.relationalr$sympy.functions.elementary.complexesrrrr &sympy.functions.elementary.exponentialr r r(sympy.functions.elementary.miscellaneousr (sympy.functions.elementary.trigonometricr rrrrsympy.simplify.trigsimprsympy.integrals.integralsrsympy.matrices.denserrhsympy.core.sympifyrrsympy.core.exprrsympy.core.logicrrmpmath.libmp.libmpfrr0rDrFr(r%r#rBs''''''""""""''''''JJJJJJJJJJJJCCCCCCCC999999HHHHHHHHHH????????,,,,,,//////======00000000 00000000++++++===6PPPPPPPPPPr%