quaternionic.algebra
Source: quaternionic/algebra.py
Essential functions for quaternion algebra.
These functions form the basic algebraic behavior of quaternions — addition, multiplication, exp, log, etc. Each function takes one or two array-like inputs — depending on whether it is unary or binary — as the first parameter or two, and another array-like object for output as the final parameter. Even for functions that return a single float or bool, the output must be array-like so that it can be modified inside the function. These functions are JIT-compiled by numba's guvectorize function, meaning that they can also act on arbitrary arrays just like standard numpy arrays, as long as the final dimension of any quaternion-valued input has size 4 to represent the components of the quaternion. These functions are generic, meaning that they can be used without the quaternionic.array object. However, these functions are implemented as the "ufunc"s of those arrays, meaning that we can perform algebra directly on them in natural ways — as in q1+q2, q1*q2, etc. — and using the standard numpy functions — as in np.exp(q), np.log(q), etc. For this purpose, we implement as many of the standard ufuncs as make sense for quaternions. For the most part, this means ignoring operations related to integers, remainders, ordering, or trigonometric functions. The bit-twiddling functions are re-interpreted as they usually are in geometric algebra to denote geometric operations. All functions in this module are magically compiled to ufuncs in __init__.py and placed attached to the algebra_ufuncs object.
absolute
absolute(q, qout)
Source: quaternionic/algebra.py
Return absolute value of quaternion |q|
add
add(q1, q2, qout)
Source: quaternionic/algebra.py
Add two quaternions q1+q2
angle
angle(q, qout)
Source: quaternionic/algebra.py
Return angle (in radians) through which input quaternion rotates a vector
If q = np.exp(v̂ * θ/2) for some unit vector v̂ and an angle θ ∈[-2π,2π], then this function returns abs(θ). This equals 2*abs(log(q)), but is more efficient.
bitwise_or
bitwise_or(q1, q2, qout)
Source: quaternionic/algebra.py
Return scalar product of quaternions q1|q2.
If we denote by ⟨a⟩ₛ the grade-s component of the general multivector a, we can express this product as a | b = Σₛ,ᵣ ⟨⟨a⟩ₛ ⟨b⟩ᵣ⟩₀ Note that this is different from the "Hestenes dot" product where the sum runs over s≠0 and r≠0; that is the product returned by galgebra using this operator.
bitwise_xor
bitwise_xor(q1, q2, qout)
Source: quaternionic/algebra.py
Return outer product of quaternions q1^q2.
This is the generalized outer product of geometric algebra. If we denote by ⟨a⟩ₛ the grade-s component of the general multivector a, we can express this product as a ^ b = Σₛ,ᵣ ⟨⟨a⟩ₛ ⟨b⟩ᵣ⟩ₛ₊ᵣ Note that the result may seem surprising because we sometimes think of quaternions as
conj
conj(q, qout)
Source: quaternionic/algebra.py
Return quaternion-conjugate of quaternion q̄
conjugate
conjugate(q, qout)
Source: quaternionic/algebra.py
Return quaternion-conjugate of quaternion q̄
divide
divide(q1, q2, qout)
Source: quaternionic/algebra.py
Divide quaternions q1/q2 = q1 * q2.inverse
divide_scalar
divide_scalar(s, q, qout)
Source: quaternionic/algebra.py
Divide scalar by quaternion s/q = s * q.inverse
equal
equal(q1, q2, bout)
Source: quaternionic/algebra.py
exp
exp(q, qout)
Source: quaternionic/algebra.py
Return exponential of input quaternion exp(q)
float_power
float_power(q, s, qout)
Source: quaternionic/algebra.py
Raise quaternion to scalar power exp(log(q)*s)
invert
invert(q, qout)
Source: quaternionic/algebra.py
Return quaternion-conjugate of quaternion q̄
isfinite
isfinite(qin, bout)
Source: quaternionic/algebra.py
isinf
isinf(qin, bout)
Source: quaternionic/algebra.py
isnan
isnan(qin, bout)
Source: quaternionic/algebra.py
left_shift
left_shift(q1, q2, qout)
Source: quaternionic/algebra.py
Return left-contraction of quaternions q1<<q2 = q1⌋q1.
For all quaternions a, b, c, we have (a ^ b) * c = a * (b ⌋ c)
log
log(q, qout)
Source: quaternionic/algebra.py
Return logarithm of input quaternion log(q)
logical_and
logical_and(q1, q2, bout)
Source: quaternionic/algebra.py
logical_or
logical_or(q1, q2, bout)
Source: quaternionic/algebra.py
multiply
multiply(q1, q2, qout)
Source: quaternionic/algebra.py
Multiply quaternions q1*q2
multiply_scalar
multiply_scalar(s, q, qout)
Source: quaternionic/algebra.py
Multiply scalar by quaternion s*q
negative
negative(q, qout)
Source: quaternionic/algebra.py
Return negative quaternion -q
not_equal
not_equal(q1, q2, bout)
Source: quaternionic/algebra.py
positive
positive(q, qout)
Source: quaternionic/algebra.py
Return input quaternion q
reciprocal
reciprocal(q, qout)
Source: quaternionic/algebra.py
Return reciprocal (inverse) of quaternion q.inverse
right_shift
right_shift(q1, q2, qout)
Source: quaternionic/algebra.py
Return right-contraction of quaternions q1>>q2 = q1⌊q2.
For all quaternions a, b, c, we have c * (b ^ a) = (c ⌊ b) * a
scalar_divide
scalar_divide(q, s, qout)
Source: quaternionic/algebra.py
Divide quaternion by scalar q/s
scalar_multiply
scalar_multiply(q, s, qout)
Source: quaternionic/algebra.py
Multiply quaternion by scalar q*s
scalar_true_divide
scalar_true_divide(q, s, qout)
Source: quaternionic/algebra.py
Divide quaternion by scalar q/s
sqrt
sqrt(q, qout)
Source: quaternionic/algebra.py
Return square-root of input quaternion √q.
The general formula whenever the denominator is nonzero is √q = (|q| + q) / √(2|q| + 2q.w) This can be proven by expanding q as q.w + q.vec and multiplying the expression above by itself. When the denominator is zero, the quaternion is a pure-real negative number. It is not clear what the appropriate square-root is in this case (because the quaternions come with infinitely many elements that square to -1), so we arbitrarily choose the result to be proportional to the x quaternion.
square
square(q, qout)
Source: quaternionic/algebra.py
Return square of quaternion q*q
subtract
subtract(q1, q2, qout)
Source: quaternionic/algebra.py
Subtract quaternion q1-q2
true_divide
true_divide(q1, q2, qout)
Source: quaternionic/algebra.py
Divide quaternions q1/q2 = q1 * q2.inverse
true_divide_scalar
true_divide_scalar(s, q, qout)
Source: quaternionic/algebra.py
Divide scalar by quaternion s/q = s * q.inverse