a RG5d}@sddlmZddlmZddlmZmZmZmZddl m Z m Z ddl mZddlmZmZmZmZddlmZddlmZdd lmZdd lmZmZdd lmZdd l m!Z!m"Z"dd l#m$Z$GdddeZ%dS))Rational)S) conjugateimresign)explog)sqrt)acoscossinatan2)trigsimp integrate)MutableDenseMatrix)sympify_sympify)Expr) fuzzy_notfuzzy_or) prec_to_dpsc@seZdZdZdZdZdeddZedd Zed d Z ed d Z eddZ eddZ e ddZe ddZddZddZddZddZddZd d!Zd"d#Zd$d%Zd&d'Zd(d)Zd*d+Zd,d-Zd.d/Zed0d1Zd2d3Zd4d5Z d6d7Z!d8d9Z"d:d;Z#dd?Z%d@dAZ&dBdCZ'dDdEZ(edFdGZ)dHdIZ*dfdKdLZ+dMdNZ,dOdPZ-dQdRZ.dSdTZ/dUdVZ0dWdXZ1dYdZZ2e d[d\Z3d]d^Z4d_d`Z5dadbZ6dcddZ7dJS)g 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&@FrTcCsvtt||||f\}}}}tdd||||fDr>tdn4t|||||}||_||_||_||_ ||_ |SdS)Ncss|]}|jduVqdS)FN)is_commutative).0irU/var/www/html/django/DPS/env/lib/python3.9/site-packages/sympy/algebras/quaternion.py 8z%Quaternion.__new__..z arguments have to be commutative) maprany ValueErrorr__new___a_b_c_d _real_field)clsabcd real_fieldobjrrrr$5s zQuaternion.__new__cCs|jSN)r%selfrrrr+Csz Quaternion.acCs|jSr1)r&r2rrrr,Gsz Quaternion.bcCs|jSr1)r'r2rrrr-Ksz Quaternion.ccCs|jSr1)r(r2rrrr.Osz Quaternion.dcCs|jSr1)r)r2rrrr/SszQuaternion.real_fieldc Cs|\}}}t|d|d|d}||||||}}}t|tj}t|tj}||} ||} ||} ||| | | 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 rHalfr ) r*vectoranglexyznormsr+r,r-r.rrrfrom_axis_angleWs zQuaternion.from_axis_anglecCs|tdd}t||d|d|dd}t||d|d|dd}t||d|d|dd}t||d|d|dd}|t|d|d}|t|d |d }|t|d |d }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)r>r>)r4r4r4)r4r>)r>r4)rr4)r4r)r>r)rr>)detrr rr)r*MZabsQr+r,r-r.rrrfrom_rotation_matrixs$$$$zQuaternion.from_rotation_matrixcCs ||Sr1addr3otherrrr__add__szQuaternion.__add__cCs ||Sr1rCrErrr__radd__szQuaternion.__radd__cCs||dSNrCrErrr__sub__szQuaternion.__sub__cCs||t|Sr1 _generic_mulrrErrr__mul__szQuaternion.__mul__cCs|t||Sr1rLrErrr__rmul__szQuaternion.__rmul__cCs ||Sr1)pow)r3prrr__pow__szQuaternion.__pow__cCst|j |j |j |j Sr1)rr%r&r'r.r2rrr__neg__szQuaternion.__neg__cCs|t|dSrIrrErrr __truediv__szQuaternion.__truediv__cCst||dSrIrTrErrr __rtruediv__szQuaternion.__rtruediv__cGs |j|Sr1rr3argsrrr_eval_IntegralszQuaternion._eval_Integralcs(dd|jfdd|jDS)NevaluateTcsg|]}|jiqSr)diff)rr+kwargssymbolsrr r z#Quaternion.diff..) setdefaultfuncrX)r3r^r]rr\rr[s zQuaternion.diffcCs|}t|}t|tsp|jrH|jrHtt||jt||j|j |j S|j rht|j||j|j |j St dt|j|j|j|j|j |j |j |j S)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>> 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 rLrErrrmuls'zQuaternion.mulcCst|tst|ts||St|ts|jrL|jrLtt|t|dd|S|jrxt||j||j||j ||j St dt|ts|jr|jr|tt|t|ddS|jrt||j||j||j ||j St dt|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 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.) rbrr/rcrrrr+r,r-r.r#)rdrerrrrM/s&+  &  &2.0.zQuaternion._generic_mulcCs |}t|j|j |j |j S)z(Returns the conjugate of the quaternion.)rr+r,r-r.r3qrrr_eval_conjugatetszQuaternion._eval_conjugatecCs4|}tt|jd|jd|jd|jdS)z#Returns the norm of the quaternion.r4)r rr+r,r-r.rgrrrr;yszQuaternion.normcCs|}|d|S)z.Returns the normalized form of the quaternion.r>)r;rgrrr normalizeszQuaternion.normalizecCs,|}|stdt|d|dS)z&Returns the inverse of the quaternion.z6Cannot compute inverse for a quaternion with zero normr>r4)r;r#rrgrrrinverseszQuaternion.inversecCstt|}|}|dkr|Sd}|js*tS|dkrB|| }}|dkrp|ddkr^||}|d}||}qB|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 rJr>rr4)rrk is_IntegerNotImplemented)r3rQrhresrrrrPs  zQuaternion.powcCs|}t|jd|jd|jd}t|jt|}t|jt||j|}t|jt||j|}t|jt||j|}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 r4) r r,r-r.rr+r r r)r3rh vector_normr+r,r-r.rrrrs"zQuaternion.expcCs|}t|jd|jd|jd}|}t|}|jt|j||}|jt|j||}|jt|j||}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 r4) r r,r-r.r;lnr r+r)r3rhroZq_normr+r,r-r.rrr_lns"zQuaternion._lncs t|tfdd|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 csg|]}|jdqS))n)evalf)rargnprecrrr_ r z*Quaternion._eval_evalf..)rrrX)r3precrrur _eval_evalfszQuaternion._eval_evalfcCs0|}|\}}t|||}|||S)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_anglerr=r;)r3rQrhvr7rerrr pow_cos_sin s zQuaternion.pow_cos_sincGsFtt|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,r-r.rWrrrr0s" zQuaternion.integratecCs^t|tr t|d|d}n|}|td|d|d|dt|}|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) rr>r4) rbtuplerr=rjrr,r-r.)ZpinrrhZpoutrrr rotate_pointQs $ &zQuaternion.rotate_pointc Cs|}|jjr|d}|}tdt|j}td|j|j}t|j|}t|j|}t|j|}|||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 rJr4r>) r+ is_negativerjrr r r,r-r.) r3rhr7r<r8r9r:rztrrrry~s zQuaternion.to_axis_angleNcCs|}|d}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} |sVt|||g||| g| | | ggS|\} }}| | |||||}|| ||||| }|| | || || }d}}}d}t||||g||| |g| | | |g||||ggSdS)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]]) r>r4rN)r;r-r.r,r+Matrix)r3rzrhr<Zm00Zm01Zm02Zm10Zm11Zm12Zm20Zm21Zm22r8r9r:Zm03Zm13Zm23Zm30Zm31Zm32Zm33rrrto_rotation_matrixs,1             zQuaternion.to_rotation_matrixcCs|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+r2rrr scalar_partszQuaternion.scalar_partcCstd|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,r-r.r2rrr vector_partszQuaternion.vector_partcCs |}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)rrjrr,r-r.)r3axisrrrr!s zQuaternion.axiscCs|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_zeror2rrris_pure>szQuaternion.is_purecCs |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;rr2rrris_zero_quaternionYszQuaternion.is_zero_quaternioncCst||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;rr2rrrr7yszQuaternion.anglecCsD|s|rtdt||||gS)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#rrrErrr arc_coplanars*zQuaternion.arc_coplanarcCsht|s$t|s$t|r,tdt|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,r-r.r@r)r*rdreZq3rArrrvector_coplanars3$6zQuaternion.vector_coplanarcCs4t|st|r td||||S)a Returns True if the two pure quaternions seen as 3D vectors are parallel. Explanation =========== Two pure quaternions are called parallel when their vector product is commutative which implies that the quaternions seen as 3D vectors have same direction. Parameters ========== other : a Quaternion Returns ======= True : if the two pure quaternions seen as 3D vectors are parallel. False : if the two pure quaternions seen as 3D vectors are not parallel. None : if the two pure quaternions seen as 3D vectors are parallel is unknown. Examples ======== >>> 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#rrErrrparallels%zQuaternion.parallelcCs4t|st|r td||||S)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 rrrErrr orthogonal%s%zQuaternion.orthogonalcCs||S)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;rr2rrr index_vectorOszQuaternion.index_vectorcCs t|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 )rpr;r2rrrmensorpszQuaternion.mensor)rrrrT)N)8__name__ __module__ __qualname____doc__ _op_priorityrr$propertyr+r,r-r.r/ classmethodr=rBrGrHrKrNrOrRrSrUrVrYr[rDrf staticmethodrMrir;rjrkrPrrqrxr{rr~ryrrrrrrr7rrrrrrrrrrrsx       * +6) D.#! ,( N / 8**!rN)&sympy.core.numbersrZsympy.core.singletonr$sympy.functions.elementary.complexesrrrr&sympy.functions.elementary.exponentialrr rp(sympy.functions.elementary.miscellaneousr (sympy.functions.elementary.trigonometricr r r rsympy.simplify.trigsimprsympy.integrals.integralsrsympy.matrices.denserrsympy.core.sympifyrrsympy.core.exprrZsympy.core.logicrrmpmath.libmp.libmpfrrrrrrs