Ed}ddlmZddlmZddlmZmZmZmZddl m Z m Z ddl mZddlmZmZmZmZddlmZddlmZdd lmZdd lmZmZdd lmZdd l m!Z!m"Z"dd l#m$Z$GddeZ%dS))Rational)S) conjugateimresign)explog)sqrt)acoscossinatan2)trigsimp integrate)MutableDenseMatrix)sympify_sympify)Expr) fuzzy_notfuzzy_or) prec_to_dpsceZdZdZdZdZd6dZedZedZ ed Z ed Z ed Z e d Ze d ZdZdZdZdZdZdZdZdZdZdZdZdZdZedZdZdZ dZ!dZ"d Z#d!Z$d"Z%d#Z&d$Z'd%Z(ed&Z)d'Z*d7d)Z+d*Z,d+Z-d,Z.d-Z/d.Z0d/Z1d0Z2e d1Z3d2Z4d3Z5d4Z6d5Z7d(S)8 QuaternionaJProvides 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 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 References ========== .. [1] http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions/ .. [2] https://en.wikipedia.org/wiki/Quaternion g&@FrTctt||||f\}}}}td||||fDrtdt j|||||}||_||_||_||_ ||_ |S)Nc3(K|] }|jduVdS)FN)is_commutative).0is 9lib/python3.11/site-packages/sympy/algebras/quaternion.py z%Quaternion.__new__..8s*??Qq5(??????z arguments have to be commutative) maprany ValueErrorr__new___a_b_c_d _real_field)clsabcd real_fieldobjs r!r'zQuaternion.__new__5s1aA,// 1a ??1aA,??? ? ? ?@@ @,sAq!Q//CCFCFCFCF(COJr#c|jSN)r(selfs r!r.z Quaternion.aC wr#c|jSr5)r)r6s r!r/z Quaternion.bGr8r#c|jSr5)r*r6s r!r0z Quaternion.cKr8r#c|jSr5)r+r6s r!r1z Quaternion.dOr8r#c|jSr5)r,r6s r!r2zQuaternion.real_fieldSs 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 rrHalfr ) r-vectoranglexyznormsr.r/r0r1s r!from_axis_anglezQuaternion.from_axis_angleWs8 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 )rr)rIrI)r>r>r>)r>rI)rIr>)rr>)r>r)rIr)rrI)detrr rr)r-MabsQr.r/r0r1s r!from_rotation_matrixzQuaternion.from_rotation_matrixs?>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,||Sr5addr7others r!__add__zQuaternion.__add__xxr#c,||Sr5rPrRs r!__radd__zQuaternion.__radd__rUr#c2||dzSNrPrRs r!__sub__zQuaternion.__sub__sxxb!!!r#cH||t|Sr5 _generic_mulrrRs r!__mul__zQuaternion.__mul__s  x777r#cH|t||Sr5r]rRs r!__rmul__zQuaternion.__rmul__s  %$777r#c,||Sr5)pow)r7ps r!__pow__zQuaternion.__pow__sxx{{r#cVt|j |j |j |j Sr5)rr(r)r*r1r6s r!__neg__zQuaternion.__neg__s&47(TWHtwh@@@r#c,|t|dzzSrYrrRs r! __truediv__zQuaternion.__truediv__sgennb(((r#c,t||dzzSrYrirRs r! __rtruediv__zQuaternion.__rtruediv__su~~b((r#c|j|Sr5rr7argss r!_eval_IntegralzQuaternion._eval_Integralst~t$$r#cjdd|jfd|jDS)NevaluateTc*g|]}|jiS)diff)rr.kwargssymbolss r! z#Quaternion.diff..s*JJJ!61675f55JJJr#) setdefaultfuncro)r7rwrvs ``r!ruzQuaternion.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 r]rRs r!mulzQuaternion.muls!N  x777r#ct|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 zStdt|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 zzS)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.) r|rr2r}rrrr.r/r0r1r&)r~rs r!r^zQuaternion._generic_mul/s=V"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 !deee24%*rtBDy0249)r rr.r/r0r1rs r!rEzQuaternion.normysE HQS!Vac1f_qsAv5Q>??@@@r#c:|}|d|z zS)z.Returns the normalized form of the quaternion.rI)rErs r! normalizezQuaternion.normalizes AaffhhJr#c|}|stdt|d|dzz zS)z&Returns the inverse of the quaternion.z6Cannot compute inverse for a quaternion with zero normrIr>)rEr&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 rZrIrr>)rr is_IntegerNotImplemented)r7rdrress r!rczQuaternion.pows2 AJJ  7 99;; | "! ! q5 #99;;qA!e 1uz #g1AAA !e  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 r/r0r1r r.r rr)r7r vector_normr.r/r0r1s 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 r/r0r1rElnr r.r)r7rrq_normr.r/r0r1s r!_lnzQuaternion._lns 136ACF?QS!V344  vJJ C$qsV|$$ ${ 2 C$qsV|$$ ${ 2 C$qsV|$$ ${ 2!Q1%%%r#cVt|tfd|jDS)a Returns the floating point approximations (decimal numbers) of the quaternion. Returns ======= Quaternion Floating point approximations of quaternion(self) Examples ======== >>> from sympy import Quaternion >>> from sympy import sqrt >>> q = Quaternion(1/sqrt(1), 1/sqrt(2), 1/sqrt(3), 1/sqrt(4)) >>> q.evalf() 1.00000000000000 + 0.707106781186547*i + 0.577350269189626*j + 0.500000000000000*k c<g|]}|S))n)evalf)rargnprecs r!rxz*Quaternion._eval_evalf.. s'DDD3CIII..DDDr#)rrro)r7precrs @r! _eval_evalfzQuaternion._eval_evalfs4,D!!DDDD$)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_anglerrGrE)r7rdrvrArs r! pow_cos_sinzQuaternion.pow_cos_sin sL< __&& 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 )rrr.r/r0r1rns r!rzQuaternion.integrate0sj<)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) rrIr>) r|tuplerrGrrr/r0r1)pinrrpouts r! rotate_pointzQuaternion.rotate_pointQsH 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 rZr>rI) r. is_negativerrr r r/r0r1) r7rrArFrBrCrDrts r!rzQuaternion.to_axis_angle~s*  3? BA KKMMT!#YY'' QSW   QS1W   QS1W   QS1W   1I Jr#Nc |}|dz}dd|z|jdz|jdzzzz }d|z|j|jz|j|jzz z}d|z|j|jz|j|jzzz}d|z|j|jz|j|jzzz}dd|z|jdz|jdzzzz }d|z|j|jz|j|jzz z} d|z|j|jz|j|jzz z} d|z|j|jz|j|jzzz} dd|z|jdz|jdzzzz } |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 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. 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((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]]) rIr>r)rEr0r1r/r.Matrix)r7rrrFm00m01m02m10m11m12m20m21m22rBrCrDm03m13m23m30m31m32m33s r!to_rotation_matrixzQuaternion.to_rotation_matrixsLb  FFHHbL!A#qsAvQ''c13qs7QSW$%c13qs7QSW$%c13qs7QSW$%!A#qsAvQ''c13qs7QSW$%c13qs7QSW$%c13qs7QSW$%!A#qsAvQ'' 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 )r.r6s 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)rr/r0r1r6s r! vector_partzQuaternion.vector_parts.!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)rrrr/r0r1)r7axiss 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 )r.is_zeror6s 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 )rErr6s r!is_zero_quaternionzQuaternion.is_zero_quaternionYs<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)) )rrrErr6s r!rAzQuaternion.angleys8.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&rrrRs 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&rr/r0r1rKr)r-r~rq3rLs r!vector_coplanarzQuaternion.vector_coplanarsf 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&rrRs 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 rrrRs 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 )rErr6s r! index_vectorzQuaternion.index_vectorOs>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 )rrEr6s r!mensorzQuaternion.mensorps*$))++r#)rrrrTr5)8__name__ __module__ __qualname____doc__ _op_priorityrr'propertyr.r/r0r1r2 classmethodrGrNrTrWr[r_rarergrjrlrprurQr staticmethodr^rrErrrcr rrrrrrrrrrrrrArrrrrrrtr#r!rrs>LN    XXXX  X (([(T)&)&[)&V"""888888AAA))))))%%%LLL4"4"4"l'8'8'8RBKBK\BKH111 AAA   ...,,,\&&&>&&&4FFF2!"!"!"FNNNB*(*(\*(X&&&PLGLGLGLG\(5552555:6###@DDD4-@-@-@^66[6p(>(>(>T(>(>(>T)))Br#rN)&sympy.core.numbersrsympy.core.singletonr$sympy.functions.elementary.complexesrrrr&sympy.functions.elementary.exponentialr r r(sympy.functions.elementary.miscellaneousr (sympy.functions.elementary.trigonometricr r rrsympy.simplify.trigsimprsympy.integrals.integralsrsympy.matrices.denserrsympy.core.sympifyrrsympy.core.exprrsympy.core.logicrrmpmath.libmp.libmpfrrrtr#r!rsc''''''""""""JJJJJJJJJJJJCCCCCCCC999999LLLLLLLLLLLL,,,,,,//////======00000000 00000000++++++ttttttttttr#