# References : # http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions/ # https://en.wikipedia.org/wiki/Quaternion from sympy import S, Rational from sympy import re, im, conjugate, sign from sympy import sqrt, sin, cos, acos, exp, ln from sympy import trigsimp from sympy import integrate from sympy import Matrix from sympy import sympify from sympy.core.evalf import prec_to_dps from sympy.core.expr import Expr class Quaternion(Expr): """Provides 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 """ _op_priority = 11.0 is_commutative = False def __new__(cls, a=0, b=0, c=0, d=0, real_field=True): a = sympify(a) b = sympify(b) c = sympify(c) d = sympify(d) if any(i.is_commutative is False for i in [a, b, c, d]): raise ValueError("arguments have to be commutative") else: obj = Expr.__new__(cls, a, b, c, d) obj._a = a obj._b = b obj._c = c obj._d = d obj._real_field = real_field return obj @property def a(self): return self._a @property def b(self): return self._b @property def c(self): return self._c @property def d(self): return self._d @property def real_field(self): return self._real_field @classmethod def from_axis_angle(cls, vector, angle): """Returns 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 """ (x, y, z) = vector norm = sqrt(x**2 + y**2 + z**2) (x, y, z) = (x / norm, y / norm, z / norm) s = sin(angle * S.Half) a = cos(angle * S.Half) b = x * s c = y * s d = z * s # note that this quaternion is already normalized by construction: # c^2 + (s*x)^2 + (s*y)^2 + (s*z)^2 = c^2 + s^2*(x^2 + y^2 + z^2) = c^2 + s^2 * 1 = c^2 + s^2 = 1 # so, what we return is a normalized quaternion return cls(a, b, c, d) @classmethod def from_rotation_matrix(cls, M): """Returns 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))*sign(sin(x))/2*k """ absQ = M.det()**Rational(1, 3) a = sqrt(absQ + M[0, 0] + M[1, 1] + M[2, 2]) / 2 b = sqrt(absQ + M[0, 0] - M[1, 1] - M[2, 2]) / 2 c = sqrt(absQ - M[0, 0] + M[1, 1] - M[2, 2]) / 2 d = sqrt(absQ - M[0, 0] - M[1, 1] + M[2, 2]) / 2 b = b * sign(M[2, 1] - M[1, 2]) c = c * sign(M[0, 2] - M[2, 0]) d = d * sign(M[1, 0] - M[0, 1]) return Quaternion(a, b, c, d) def __add__(self, other): return self.add(other) def __radd__(self, other): return self.add(other) def __sub__(self, other): return self.add(other*-1) def __mul__(self, other): return self._generic_mul(self, other) def __rmul__(self, other): return self._generic_mul(other, self) def __pow__(self, p): return self.pow(p) def __neg__(self): return Quaternion(-self._a, -self._b, -self._c, -self.d) def __truediv__(self, other): return self * sympify(other)**-1 def __rtruediv__(self, other): return sympify(other) * self**-1 def _eval_Integral(self, *args): return self.integrate(*args) def diff(self, *symbols, **kwargs): kwargs.setdefault('evaluate', True) return self.func(*[a.diff(*symbols, **kwargs) for a in self.args]) def add(self, other): """Adds 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 """ q1 = self q2 = sympify(other) # If q2 is a number or a sympy expression instead of a quaternion if not isinstance(q2, Quaternion): if q1.real_field and q2.is_complex: return Quaternion(re(q2) + q1.a, im(q2) + q1.b, q1.c, q1.d) elif q2.is_commutative: return Quaternion(q1.a + q2, q1.b, q1.c, q1.d) else: raise ValueError("Only commutative expressions can be added with a Quaternion.") return Quaternion(q1.a + q2.a, q1.b + q2.b, q1.c + q2.c, q1.d + q2.d) def mul(self, other): """Multiplies 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 """ return self._generic_mul(self, other) @staticmethod def _generic_mul(q1, q2): """Generic 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 Symbol >>> 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 = 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.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 """ q1 = sympify(q1) q2 = sympify(q2) # None is a Quaternion: if not isinstance(q1, Quaternion) and not isinstance(q2, Quaternion): return q1 * q2 # If q1 is a number or a sympy expression instead of a quaternion if not isinstance(q1, Quaternion): if q2.real_field and q1.is_complex: return Quaternion(re(q1), im(q1), 0, 0) * q2 elif q1.is_commutative: return Quaternion(q1 * q2.a, q1 * q2.b, q1 * q2.c, q1 * q2.d) else: raise ValueError("Only commutative expressions can be multiplied with a Quaternion.") # If q2 is a number or a sympy expression instead of a quaternion if not isinstance(q2, Quaternion): if q1.real_field and q2.is_complex: return q1 * Quaternion(re(q2), im(q2), 0, 0) elif q2.is_commutative: return Quaternion(q2 * q1.a, q2 * q1.b, q2 * q1.c, q2 * q1.d) else: raise ValueError("Only commutative expressions can be multiplied with a Quaternion.") return Quaternion(-q1.b*q2.b - q1.c*q2.c - q1.d*q2.d + q1.a*q2.a, q1.b*q2.a + q1.c*q2.d - q1.d*q2.c + q1.a*q2.b, -q1.b*q2.d + q1.c*q2.a + q1.d*q2.b + q1.a*q2.c, q1.b*q2.c - q1.c*q2.b + q1.d*q2.a + q1.a * q2.d) def _eval_conjugate(self): """Returns the conjugate of the quaternion.""" q = self return Quaternion(q.a, -q.b, -q.c, -q.d) def norm(self): """Returns the norm of the quaternion.""" q = self # trigsimp is used to simplify sin(x)^2 + cos(x)^2 (these terms # arise when from_axis_angle is used). return sqrt(trigsimp(q.a**2 + q.b**2 + q.c**2 + q.d**2)) def normalize(self): """Returns the normalized form of the quaternion.""" q = self return q * (1/q.norm()) def inverse(self): """Returns the inverse of the quaternion.""" q = self if not q.norm(): raise ValueError("Cannot compute inverse for a quaternion with zero norm") return conjugate(q) * (1/q.norm()**2) def pow(self, p): """Finds 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 """ p = sympify(p) q = self if p == -1: return q.inverse() res = 1 if not p.is_Integer: return NotImplemented if p < 0: q, p = q.inverse(), -p while p > 0: if p % 2 == 1: res = q * res p = p//2 q = q * q return res def exp(self): """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 """ # exp(q) = e^a(cos||v|| + v/||v||*sin||v||) q = self vector_norm = sqrt(q.b**2 + q.c**2 + q.d**2) a = exp(q.a) * cos(vector_norm) b = exp(q.a) * sin(vector_norm) * q.b / vector_norm c = exp(q.a) * sin(vector_norm) * q.c / vector_norm d = exp(q.a) * sin(vector_norm) * q.d / vector_norm return Quaternion(a, b, c, d) def _ln(self): """Returns 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 """ # _ln(q) = _ln||q|| + v/||v||*arccos(a/||q||) q = self vector_norm = sqrt(q.b**2 + q.c**2 + q.d**2) q_norm = q.norm() a = ln(q_norm) b = q.b * acos(q.a / q_norm) / vector_norm c = q.c * acos(q.a / q_norm) / vector_norm d = q.d * acos(q.a / q_norm) / vector_norm return Quaternion(a, b, c, d) def _eval_evalf(self, prec): """Returns the floating point approximations (decimal numbers) of the quaternion. Returns ======= Quaternion Floating point approximations of quaternion(self) Examples ======== >>> from sympy.algebras.quaternion 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 """ return Quaternion(*[arg.evalf(n=prec_to_dps(prec)) for arg in self.args]) def pow_cos_sin(self, p): """Computes 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 """ # q = ||q||*(cos(a) + u*sin(a)) # q^p = ||q||^p * (cos(p*a) + u*sin(p*a)) q = self (v, angle) = q.to_axis_angle() q2 = Quaternion.from_axis_angle(v, p * angle) return q2 * (q.norm()**p) def integrate(self, *args): """Computes integration of quaternion. Returns ======= Quaternion Integration of the quaternion(self) with the given variable. Examples ======== Indefinite Integral of quaternion : >>> from sympy.algebras.quaternion 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.algebras.quaternion 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 """ # TODO: is this expression correct? return Quaternion(integrate(self.a, *args), integrate(self.b, *args), integrate(self.c, *args), integrate(self.d, *args)) @staticmethod def rotate_point(pin, r): """Returns 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.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) """ if isinstance(r, tuple): # if r is of the form (vector, angle) q = Quaternion.from_axis_angle(r[0], r[1]) else: # if r is a quaternion q = r.normalize() pout = q * Quaternion(0, pin[0], pin[1], pin[2]) * conjugate(q) return (pout.b, pout.c, pout.d) def to_axis_angle(self): """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 """ q = self if q.a.is_negative: q = q * -1 q = q.normalize() angle = trigsimp(2 * acos(q.a)) # Since quaternion is normalised, q.a is less than 1. s = sqrt(1 - q.a*q.a) x = trigsimp(q.b / s) y = trigsimp(q.c / s) z = trigsimp(q.d / s) v = (x, y, z) t = (v, angle) return t def to_rotation_matrix(self, v=None): """Returns 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]]) """ q = self s = q.norm()**-2 m00 = 1 - 2*s*(q.c**2 + q.d**2) m01 = 2*s*(q.b*q.c - q.d*q.a) m02 = 2*s*(q.b*q.d + q.c*q.a) m10 = 2*s*(q.b*q.c + q.d*q.a) m11 = 1 - 2*s*(q.b**2 + q.d**2) m12 = 2*s*(q.c*q.d - q.b*q.a) m20 = 2*s*(q.b*q.d - q.c*q.a) m21 = 2*s*(q.c*q.d + q.b*q.a) m22 = 1 - 2*s*(q.b**2 + q.c**2) if not v: return Matrix([[m00, m01, m02], [m10, m11, m12], [m20, m21, m22]]) else: (x, y, z) = v m03 = x - x*m00 - y*m01 - z*m02 m13 = y - x*m10 - y*m11 - z*m12 m23 = z - x*m20 - y*m21 - z*m22 m30 = m31 = m32 = 0 m33 = 1 return Matrix([[m00, m01, m02, m03], [m10, m11, m12, m13], [m20, m21, m22, m23], [m30, m31, m32, m33]])