A Python extension to provide a Quaternion type and some associated math functions together with a QuaternionArray object.
general
mathematical operations
mixed mode arithmetic
construction
attributes
instance functions
static functions
magic functions
maths functions
module variables
hash function
rotation matrices
quarternion arrays
background
references
credits
Within this module, a Quaternion q is defined to be:
q = w + x.i + y.j + z.k
(through out this document = means 'is equal to', as opposed to assignment)
where the coefficients w, x, y and z are real; and i, j and k three imaginary
numbers such that:
i.i = j.j = k.k = i.j.k = -1
i.j = +k, j.k = +i, k.i = +j
j.i = -k, k.j = -i, i.k = -j
The Quaternion type has four member attributes to access these coefficients. These instance attributes are w, x, y and z respectively.
A Quaternion may also be considered to be a real scalar part plus a vector (with 3 real components). The real part accessible via the real attribute. Both q.w and q.real return the real or scalar part of q.
The vector part is accessible via both the vector and imag attributes which provide a tuple of floats. The following Python expressions are equivalent:
q.vector
q.imag
(q.x, q.y, q.z)
q.real and q.imag provide a "complex" like view of a Quaternion at the expense of providing an un-Pythonic duplication of q.w and q.vector respectively.
The following mathematical operations are provided.
unary: +, -, abs
binary: +, -, *, /
power: **
There is no mod (%) or integer division (//) operation available. Therefore only the two argument version of the pow() function is available - see below.
The Quaternion type is associative under both addition and multiplication, i.e.:
(p + q) + r = p + (q + r)
(p * q) * r = p * (q * r)
The Quaternion type is also distributive:
p * (q + r) = p*q + p*r
The Quaternion type is non-commutative with respect to multiplication and division, i.e. in general p * q and q * p result in different values. To divide one Quaternion by another, there are two possible options:
p * q.inverse() ; or
q.inverse() * p.
The quotient function uses p * q.inverse(), therefore:
(p / q) * q = p
This non-commutative nature also explains why p ** q is undefined, as this could implemented as:
exp (q * log (p)) ; or
exp (log (p) * q)
However, mixed-mode ** is possible - please see below.
Quaternions numbers and scalar numbers, i.e. int or float, are inter-operable. int and float numbers are treated as Quaternions with zero imaginary components. Note: float numbers (a) and Quaternion numbers (q) do commute under multiplication and division:
q * a = a * q
q / a = (1/a) * q
a / q = q.inverse() * a
Mixed mode with complex numbers is also allowed. A complex number, z, is treated as a Quaternions, q, such that q.w = z.real, q.y = z.imag, and q.x and q.z are zero.
The choice of aligning the imaginary part of a complex number to the j imaginary part as opposed to i or k imaginary parts is mathematically arbitrary. However for Python, j is the natural choice and then the following, bar any rounding errors, will hold true for any complex value z:
Quaternion(z) = Quaternion(str(z))
The complex part of a Quaternion may be obtained using the complex attribute, such that:
q.complex = complex(q.w, q.y).
There is no complementary attribute to obtain q.x and q.z as a single item.
Mixed mode is also available for the ** operator. If a is a float or integer number, then with some restrictions the following are both provided:
q ** a
a ** q -- a must be > 0.0
A Quaternion number may be constructed using one of the following forms:
- Quaternion () -> quaternion zero
- Quaternion (w[, x[, y[, z]]]) -> quaternion number
- Quaternion (real=float,imag=(float,float,float)) -> quaternion number
- Quaternion (angle=float,axis=(float,float,float)) -> quaternion rotation number
- Quaternion (number) -> quaternion number
- Quaternion ("string representation") -> quaternion number
- Quaternion (matrix=3x3 nested iterator of numerics) -> quaternion number
A Quaternion number may be created from:
a) the real part and an optional imaginary parts. w, x, y and z must be float or number types which can be converted to float;
b) from a real number and a 3-tuple vector;
c) from an angle (radians) and a 3-tuple axis of rotation (which is automatically normalised), which generates a rotator Quaternion, which can then be used in conjunction with the rotate method;
d) from a single number parameter: int, float, complex or another Quaternion. When the number is complex, the imaginary part of the complex number is assigned to the j imaginary part;
e) from the string representation of a Quaternion (modeled on the complex type). The following are valid:
Quaternion("1.2")
Quaternion("1.2+2i")
Quaternion("1.22_33+4.11_22i") -- 3.6 or later allows underscores
Quaternion("1.2+3i-1j")
Quaternion("(1.2+3i-1j)")
Quaternion("1.2i+0.3k")
Quaternion(" ( 1.2+0.3j ) ")
Quaternion("1.2+3.4i+2.6j-2k")
The following are invalid:
Quaternion("1.2 + 3.4i+ 2.6j- 2k") -- spaces within actual number
Quaternion("1.2+3.4i+2.6k-2j") -- out of order
Quaternion("(1.2+3.4i+2.6j-2k") -- unmatched parenthesis
f) from a 3x3 matrix of floats (and/or float-able objects). The matrix must be a triplet iterator with each item itself being a triplet iterator of float or floatable objects. The matrix should ideally be a rotation matrix, i.e. the determinent should be 1, however no attempt is made to check this nor is any attempt made to normalise the matrix. The resulting quaternion may be subsequently normalised or reconstructed from the rotation angle and axis.
- w - float - real/scalar part
- x - float - i imaginary part
- y - float - j imaginary part
- z - float - k imaginary part
- vector - tuple - the tuple (x, y, z)
- complex - complex - the complex number w + y.j
- real - float - real/scalar part
- imag - tuple - the imaginary part, the same as vector.
- data - tuple - the raw data as a tuple (w, x, y, z).
q.conjugate() returns the Quaternion conjugate of q.
q.inverse () returns s such that: s * q = q * s = 1
q.normalise () returns s such that: s = q / abs (q)
q.quadrance () returns s such that s = q.w*q.w + q.x*q.x + q.y*q.y + q.z*q.z
q.matrix () returns a 3-tuple of 3-tuple of floats representing the 3x3 rotation matrix equivalent of q. q should be a rotation quaternion.
q.angle () returns the angle (float, in radians) of q. q should be a rotation quaternion. This method may raise a ValueError if q is not a rotation quaternion.
This should not be confused with the polor co-ordinate form phase (aka argument) angle.
q.axis () returns the normalised axis of q, where q should be a rotation quaternion. This is essentially identical to the maths function Quaternion.axis (q)
q.rotate (point, origin=None) -> point, where q is a rotation number, i.e. q = Quaternion (angle=a,axis=(x,y,z)). The returned value is rotated by an angle a radians about the axis (x,y,z).
These are the equivilent of "@staticmethod" functions
Quaternion.brief() modifies the behaviour of the __repr__ function. Within interactive python/ipython we have:
In [2]: q = quaternion.Quaternion (1,0,1,0)
In [3]: q
Out[3]: quaternion.Quaternion(1, +0, +1, +0)
and after invoking brief we have:
In [4]: quaternion.Quaternion.brief()
In [5]: q
Out[5]: (1+0i+1j+0k)
This also impacts how lists ,tuples, dictionarys etc. that contain quaternions are converted to str and printed.
Quaternion.reset() un-does the __repr__ function behaviour modification instigated by the call to brief(). I might change this name.
q.__format__ (fmtstr) -> str
Format to a string according to format_spec. This allows, for example:
q = Quaternion(...)
s1 = "... {p:20.2f} ...".format(p=q)
s2 = f"... {q:20.2f} ..."
q.__getnewargs__ () returns a 4-tuple s, such that s = (q.w, q.x, q.y, q.z)
This allows Quaternion numbers to be pickled and unpickled, and hence used with jsonpickle.
q.__round__ (ndigits=0) returns a Quaternion with each component rounded, e.g. round (q.w, ndigits). While the method can be called directly, one would normally invoke
round(q)
round(q, n)
This is the equivalent of round (float, [ndigits]), and ndigits may be either positive of negative. This is perhaps most useful as an alternative to using format when printing Quaternion, e.g.
print("result : %s" % round(q,2))
A number of maths functions that operate on Quaternions are also provided. Where these function have the same name as a function out of the math and/or cmath modules, the function provides the equivalent operation on a quaternion or pair of quaternions.
The functions provided are:
acos
acosh
asin
asinh
atan
atanh
axis
cos
cosh
dot
exp
isclose
isfinite
isinf
isnan
log
log10
phase
polar
rect
sin
sinh
slerp - since 1.3.4
sqrt
tan
tanh
Note: there is no separate qmath module.
- one = Quaternion (1.0, 0.0, 0.0, 0.0)
- i = Quaternion (0.0, 1.0, 0.0, 0.0)
- j = Quaternion (0.0, 0.0, 1.0, 0.0)
- k = Quaternion (0.0, 0.0, 0.0, 1.0)
- e = 2.718281828459045 - float
- pi = 3.141592653589793 - float
- tau = 6.283185307179586 - float
- __version__ = the version number as str.
The hash of a quaternion follows the ideas used in the complex hash function such that if q = Quaternion (q.complex) then hash(q) = hash (q.complex), and if q = Quaternion (q.real) then hash(q) = hash (q.real)
If quaternion number, q, is a rotation quaternion, then q.matrix() function can be used to obtain the equivilent 3x3 rotation matrix. A rotation quaternion is one who's length is 1, and is typically created using:
q = Quaternion(angle=..., axis=(..., ..., ...))
If abs(q) is not 1, there is no obvious interpretation of the generated matrix. Likewise a 3x3 rotation matrix, A, may be used to create the equivilent quaternion:
q = Quaternion(matix=((A11,A12,A13), (A21,A22,A23), (A31,A32,A33)))
However, if A is not a rotation matrix, i.e. det(A) != 1, then the quaternion will have no obvious interpretation with respect to the matrix.
Matrices have 9 degrees of freedom while quaternions have only 4, and rotation quaternions have effectivly only 3. This is why only rotation matrices can be sensibly converted to a meaningfull quaternion number.
Note: neither the Quaternion(matrix=...) constructor nor the matrix() method attempt to validate or normalise the input values. They just run the algorithms "AS IS".
Release 1.3 and later sees the introduction of a QuaternionArray type. The API and behaviour of this class has been implemented to mimic as far as resonabley possbile the API and behaviour of the inbuilt array.array type, save that it only holds quaternions.
The QuaternionArray class provides a number of additonal methods:
- clear() - this is equivilent to the list class clear() method;
- reserve(int) - this method allows the minimum space allocated for the array to be specified (expressed in number of quaterions, not number of bytes).
The QuaternionArray class also provides two additional attributes:
- allocated - provides the allocated memory size (expressed in quaterions);
- reserved - provides the minimum allocated memory size (expressed in quaterions)
There is (currently) no equivilent of the fromlist and tolist methods. The extend method provides essentially the same functionality as the fromlist method (with no argument restriction). The fuctionality of array.tolist() can be acheived by calling list(array).
This was initially developed more or less as an experiment to create a Python extension written in C that was a bit more challenging than just a simple "hello world" extension.
Although there are already a number of Quaternion Python implementations out there, this has the advantage of speed over the pure Python implementations and the advantage of no dependencies on any other modules such as numpy.
- http://onlinelibrary.wiley.com/doi/10.1002/9780470682906.app4/pdf
- https://www.geometrictools.com/Documentation/Quaternions.pdf
- https://en.wikipedia.org/wiki/Quaternion
Guidance from https://docs.python.org/3.5/extending/newtypes.html together with cribbing many code-snippets and ideas from the complex type and the array.array type; and last but not least Sir William R. Hamilton.
Last updated: Thu Sep 22 14:58:35 2022