biquaternion_py package
Submodules
biquaternion_py.biquaternion module
Calculate with BiQuaternions.
This module implements BiQuaternions as a class for general calculations.
Classes:
BiQuaternion
Misc variables:
II JJ KK EE
- class biquaternion_py.biquaternion.BiQuaternion(*args)[source]
Bases:
ExprClass implementing Bi-Quaternions.
Bi-Quaternions are represented as $a + II b + JJ c + KK d + EE (w + II x + JJ y + KK z)$.
- coeffs
coefficients of the quaternion as a list in the canonical order.
- Type:
list
- scal
scalar value of the Bi-Quaternion
- Type:
sympy.Expr, numeric
- i
II value of the Bi-Quaternion
- Type:
sympy.Expr, numeric
- j
JJ value of the Bi-Quaternion
- Type:
sympy.Expr, numeric
- k
KK value of the Bi-Quaternion
- Type:
sympy.Expr, numeric
- eps
dual scalar value of the Bi-Quaternion
- Type:
sympy.Expr, numeric
- ei
dual II value of the Bi-Quaternion
- Type:
sympy.Expr, numeric
- ej
dual JJ value of the Bi-Quaternion
- Type:
sympy.Expr, numeric
- ek
dual KK value of the Bi-Quaternion
- Type:
sympy.Expr, numeric
- __new__(cls, \*args):
Create new instance of the BiQuaternion class.
- __mul__(other):
Multiply BiQuaternion with other.
- __pos__(self):
Positive of itsself.
- __neg__(self):
Negative of itsself.
- __add__(other):
Add BiQuaternion to other.
- __sub__(other):
Subtract other from BiQuaternion.
- __rsub__(other):
Subtract BiQuaternion from other.
- __radd__ = __add__
- __rmul__ = __mul__
- __eq__(other):
Test equality of two biquaternions.
- __hash__ = super.__hash__
- __repr__():
Convert BiQuaternion to a readable format in shell.
- __str__():
Converst BiQuaternion to string.
- __pow__(other):
Power function of BiQuaternion.
- __invert__():
(Bi)-Quaternion conjugate of the quaternion.
- __truediv__(other):
Division of BiQuaternion by other.
- __rtruediv__(other):
Divide other by BiQuaternion.
- coeff(var, power):
Rewriting of Expr.coeff to work for BiQuaternions
- apply_elementwise(func, *args)[source]
Apply a function with specified arguments elementwise.
- Parameters:
func (function) – Function to be applied elementwise
*args (unknown) – Arguments to be passed to the function
- Returns:
Biquaternion with function applied to each coefficient individually.
- Return type:
- coeff(var, power=1, right=False, _first=True)[source]
Rewriting of Expr.coeff to work for BiQuaternions.
- property coeffs
Coefficients describing an instance of BiQuaternion.
- conjugate()[source]
Conjugate of this instance of BiQuaternion.
Conjugation of a quaternion inverts the sign of the non-scalar part of a (bi-)quaternion. This happens in the same fashion as for complex numbers.
- default_assumptions = {'algebraic': False, 'commutative': False, 'complex': False, 'composite': False, 'even': False, 'extended_negative': False, 'extended_nonnegative': False, 'extended_nonpositive': False, 'extended_nonzero': False, 'extended_positive': False, 'extended_real': False, 'imaginary': False, 'integer': False, 'irrational': False, 'negative': False, 'noninteger': False, 'nonnegative': False, 'nonpositive': False, 'nonzero': False, 'odd': False, 'positive': False, 'prime': False, 'rational': False, 'real': False, 'transcendental': False, 'zero': False}
- dual()[source]
Dual part of the dual quaternion.
- Return type:
Notes
The dual part of a dual quaternion is defined as the part only containing factors epsilon.
- property ei
Value of the eps*II part of the instance of BiQuaternion.
- property ej
Value of the eps*JJ part of the instance of BiQuaternion.
- property ek
Value of the eps*KK part of the instance of BiQuaternion.
- property eps
Value of the eps part of the instance of BiQuaternion.
- eps_conjugate()[source]
Epsilon conjugation of the biquaternion.
Epsilon conjugation inverts the sign of the dual part of a quaternion
- property i
Value of the II part of the instance of BiQuaternion.
- is_algebraic = False
- is_commutative: bool | None = False
- is_complex = False
- is_composite = False
- is_even = False
- is_extended_negative = False
- is_extended_nonnegative = False
- is_extended_nonpositive = False
- is_extended_nonzero = False
- is_extended_positive = False
- is_extended_real: bool | None = False
- is_imaginary = False
- is_integer = False
- is_irrational = False
- is_negative: bool | None = False
- is_noninteger = False
- is_nonnegative = False
- is_nonpositive = False
- is_nonzero = False
- is_odd = False
- is_positive = False
- is_prime = False
- is_rational = False
- is_real: bool | None = False
- is_transcendental = False
- is_zero: bool | None = False
- property j
Value of the JJ part of the instance of BiQuaternion.
- property k
Value of the KK part of the instance of BiQuaternion.
- primal()[source]
Primal part of the dual quaternion.
- Return type:
Notes
The primal part of a dual quaternion is defined as the part not containing a factor epsilon.
- quadrance()[source]
Quadrance of a quaternion.
The quadrance of a biquaternion is its norm. It is defined as the product of a biquaternion and its conjugate.
- property scal
Value of the scalar part of the instance of BiQuaternion.
- scalar_part()[source]
Scalar part of the dual quaternion.
- Return type:
Notes
The scalar part of a dual quaternion is defined as the part only not containing any of the numbers i, j, or k
- biquaternion_py.biquaternion.define_algebra(i_square=-1, j_square=-1, e_square=0)[source]
Define the algebra.
- Parameters:
i_square (float) – value of II**2
j_square (float) – value of JJ**2
e_square (float) – value of EE**2
- Return type:
None
Notes
The algebra of bi-quaternions is fully defined by the following conditions: .. math:: i^2 = a, j^2 = b, e^2 = c,\ ij = k, ji = -k,\ ei = ie, ej = je, ek = ke All objects commute with elements of the chosen base field. Mostly this is \(\mathbb{R}\) or \(\mathbb{C}\).
biquaternion_py.lines module
Module for line creation, representation and manipulation.
- biquaternion_py.lines.line_to_pluecker(direction, point)[source]
Generate Pluecker coordinates for a line with given direction through a point.
biquaternion_py.poly_tools module
Extra functions for polynomials.
- biquaternion_py.poly_tools.factorize_bq_poly(poly, domain=None)[source]
Factorize Biquaternion polynomial into linear factors.
- Parameters:
poly (Poly) – Polynomial which to factorize
domain (string, optional) – Domain over which to calculate the irreducible factors. (Default None lets sympy decide which domain to use.)
- Returns:
factors (array of Poly) – Array of linear factors of poly associated to the order of the factors
given by irreducible_factors.
- biquaternion_py.poly_tools.factorize_from_list(poly, factors)[source]
Factorize polynomial given a list of factors of the norm polynomial.
- biquaternion_py.poly_tools.gcd_conj_pd(poly)[source]
Real gcd of c, primal*dual.conjugate(), primal.conjugate()*dual.
- biquaternion_py.poly_tools.irreducible_factors(poly, domain=None)[source]
Calculate the irreducible factors of a polynomial.
- Parameters:
poly (Poly) – Polynomial of which to calculate the irreducible factors.
domain (string, optional) – Domain over which to calculate the irreducible factors. (Default None lets sympy decide which domain to use.)
- Returns:
out – List of irreducible factors.
- Return type:
array of Poly
Notes
If the irreducible factors are not calculated correctly this might be an issue of sympy assuming to much about the domain. Chaning this to “QQ” or “RR”
- biquaternion_py.poly_tools.is_poly_reduced(poly)[source]
Check if polynomial is reduced.
- Parameters:
poly (Poly) – Polynomial which to check for reducedness.
- Returns:
True if the polynomial is reduced.
- Return type:
bool
Notes
A polynomial is called reduced, if the primal and dual part have no common real factor. [1]
[1]Z. Li, J. Schicho, H.-P. Schröcker, The rational motion of minimal dual quaternion degree with prescribed trajectory, Computer Aided Geometric Design, Volume 41, 2016, Pages 1-9, ISSN 0167-8396, https://doi.org/10.1016/j.cagd.2015.10.002.
- biquaternion_py.poly_tools.max_real_poly_fact(poly)[source]
Calculate maximal real polynomial factor of the BiQuaternionpolynomial poly.
biquaternion_py.polynomials module
Implementation of polynomial class and associated functions.
- class biquaternion_py.polynomials.Poly(*args)[source]
Bases:
ExprClass implementing arbitrary polynomials.
- all_coeffs()[source]
Compute all coefficients in ascending order of all variables in given order as nested list.
- all_indet_coeffs(indet)[source]
Compute all coefficients with respect to var.
- Parameters:
indet (sympy.Symbol) – Variable with respect which to compute all coefficients.
- Returns:
List of coefficients for powers of indet in ascending order.
- Return type:
list of BiQuaternions
- coeff(var, power=1, right=False, _first=True)[source]
Coefficient of polynomial with respect to var**(power).
- default_assumptions = {}
- eval(vals, right=True)[source]
Evaluate polynomial for variables set as in val.
- Parameters:
vals (list) – List of values which the indeterminates should take.
right (bool (optional, defaulf = True)) – Should polynomial be evaluated assuming variables are left,
coefficients. (or right of) –
- Return type:
type(val)
- property indets
- property poly
- biquaternion_py.polynomials.poly_div(poly_1, poly_2, var, right=True)[source]
Polynomial division with remainder of poly_1 and poly_2 with respect to var.
- Parameters:
- Returns:
quotient (Poly) – Result of division
remainder (Poly) – Remainder of division
Notes
This function produces polynomials quotient and remainder such that poly_1 = poly_2 * quotient + remainder for right = True poly_1 = quotient * poly_2 + remainder for right = False
biquaternion_py.random_gen module
Module for generation of random objects
- biquaternion_py.random_gen.rand_bq(maximum=10)[source]
Random biquaternion with rational coefficients.