clifford.MultiVector

class clifford.MultiVector(layout, value=None, string=None, *, dtype: numpy.dtype = <class 'numpy.float64'>)

An element of the algebra

Parameters

• layout (instance of clifford.Layout) – The layout of the algebra

• value (sequence of length layout.gaDims) – The coefficients of the base blades

• dtype (numpy.dtype) –

  The datatype to use for the multivector, if no value was passed.

  New in version 1.1.0.

Notes

The following operators are overloaded:

• A * B : geometric product

• A ^ B : outer product

• A | B : inner product

• A << B : left contraction

• ~M : reversion

• M(N) : grade or subspace projection

• M[N] : blade projection

exp() → clifford._multivector.MultiVector

vee(other) → clifford._multivector.MultiVector

Vee product \(A \vee B\).

This is often defined as:

\[\begin{split}(A \vee B)^* &= A^* \wedge B^* \\ \implies A \vee B &= (A^* \wedge B^*)^{-*}\end{split}\]

This is very similar to the meet() function, but always uses the dual in the full space .

Internally, this is actually implemented using the complement functions instead, as these work in degenerate metrics like PGA too, and are equivalent but faster in other metrics.

__and__(other) → clifford._multivector.MultiVector

self & other, an alias for vee()

__mul__(other) → clifford._multivector.MultiVector

self * other, the geometric product \(MN\)

__xor__(other) → clifford._multivector.MultiVector

self ^ other, the Outer product \(M \wedge N\)

__or__(other) → clifford._multivector.MultiVector

self | other, the inner product \(M \cdot N\)

__add__(other) → clifford._multivector.MultiVector

self + other, addition

__sub__(other) → clifford._multivector.MultiVector

self - other, Subtraction

right_complement() → clifford._multivector.MultiVector

left_complement() → clifford._multivector.MultiVector

as_array() → numpy.ndarray

mag2() → numbers.Number

Magnitude (modulus) squared, \({|M|}^2\)

Note in mixed signature spaces this may be negative

adjoint() → clifford._multivector.MultiVector

Adjoint / reversion, \(\tilde M\)

Aliased as ~M to reflect \(\tilde M\), one of several conflicting notations.

Note that ~(N * M) == ~M * ~N.

__invert__() → clifford._multivector.MultiVector

Adjoint / reversion, \(\tilde M\)

Aliased as ~M to reflect \(\tilde M\), one of several conflicting notations.

Note that ~(N * M) == ~M * ~N.

__getitem__(key: Union[MultiVector, tuple, int]) → numbers.Number

value = self[key].

If key is a blade tuple (e.g. (0, 1) or (1, 3)), or a blade, (e.g. e12), then return the (real) value of that blade’s coefficient. Otherwise, treat key as an index into the list of coefficients.

__call__(other, *others) → clifford._multivector.MultiVector

Return a new multi-vector projected onto a grade or another MultiVector

M(g1, ... gn) gives \(\left<M\right>_{g1} + \cdots + \left<M\right>_{gn}\)

M(N) calls project() as N.project(M).

Examples

>>> from clifford.g2 import *
>>> M = 1 + 2*e1 + 3*e12
>>> M(0)
1
>>> M(0, 2)
1 + (3^e12)

clean(eps=None) → clifford._multivector.MultiVector

Sets coefficients whose absolute value is < eps to exactly 0.

eps defaults to the current value of the global _settings._eps.

round(eps=None) → clifford._multivector.MultiVector

Rounds all coefficients according to Python’s rounding rules.

eps defaults to the current value of the global _settings._eps.

lc(other) → clifford._multivector.MultiVector

The left-contraction of two multivectors, \(M\rfloor N\)

property pseudoScalar

Returns a MultiVector that is the pseudoscalar of this space.

property I

Returns a MultiVector that is the pseudoscalar of this space.

invPS() → clifford._multivector.MultiVector

Returns the inverse of the pseudoscalar of the algebra.

isScalar() → bool

Returns true iff self is a scalar.

isBlade() → bool

Returns true if multivector is a blade.

isVersor() → bool

Returns true if multivector is a versor. From Leo Dorsts GA for computer science section 21.5, definition from 7.6.4

grades(eps=None) → Set[int]

Return the grades contained in the multivector.

Changed in version 1.1.0: Now returns a set instead of a list

Changed in version 1.3.0: Accepts an eps argument

property blades_list

ordered list of blades present in this MV

normal() → clifford._multivector.MultiVector

Return the (mostly) normalized multivector.

The _mostly_ comes from the fact that some multivectors have a negative squared-magnitude. So, without introducing formally imaginary numbers, we can only fix the normalized multivector’s magnitude to +-1.

\(\frac{M}{|M|}\) up to a sign

leftLaInv() → clifford._multivector.MultiVector

Return left-inverse using a computational linear algebra method proposed by Christian Perwass.

normalInv(check=True) → clifford._multivector.MultiVector

The inverse of itself if \(M \tilde M = |M|^2\).

\[M^{-1} = \tilde M / (M \tilde M)\]

Parameters

check (bool) – When true, the default, validate that it is appropriate to use this method of inversion.

inv() → clifford._multivector.MultiVector

leftInv() → clifford._multivector.MultiVector

Return left-inverse using a computational linear algebra method proposed by Christian Perwass.

rightInv() → clifford._multivector.MultiVector

Return left-inverse using a computational linear algebra method proposed by Christian Perwass.

dual(I=None) → clifford._multivector.MultiVector

The dual of the multivector against the given subspace I, \(\tilde M = MI^{-1}\)

I defaults to the pseudoscalar.

commutator(other) → clifford._multivector.MultiVector

The commutator product of two multivectors.

\([M, N] = M \times N = (MN + NM)/2\)

x(other) → clifford._multivector.MultiVector

The commutator product of two multivectors.

\([M, N] = M \times N = (MN + NM)/2\)

anticommutator(other) → clifford._multivector.MultiVector

The anti-commutator product of two multivectors, \((MN + NM)/2\)

gradeInvol() → clifford._multivector.MultiVector

The grade involution of the multivector.

\[M^* = \sum_{i=0}^{\text{dims}} {(-1)^i \left<M\right>_i}\]

property even

Even part of this multivector

defined as M + M.gradInvol()

property odd

Odd part of this mulitvector

defined as M +- M.gradInvol()

conjugate() → clifford._multivector.MultiVector

The Clifford conjugate (reversion and grade involution).

\(M^*\) = (~M).gradeInvol()

project(other) → clifford._multivector.MultiVector

Projects the multivector onto the subspace represented by this blade.

\(P_A(M) = (M \rfloor A) A^{-1}\)

factorise() → Tuple[List[clifford._multivector.MultiVector], numbers.Number]

Factorises a blade into basis vectors and an overall scale.

Uses Leo Dorsts algorithm from 21.6 of GA for Computer Science

basis() → List[clifford._multivector.MultiVector]

Finds a vector basis of this subspace.

join(other) → clifford._multivector.MultiVector

The join of two blades, \(J = A \cup B\)

Similar to the wedge, \(W = A \wedge B\), but without decaying to 0 for blades which share a vector.

meet(other, subspace=None) → clifford._multivector.MultiVector

The meet of two blades, \(A \cap B\).

Computation is done with respect to a subspace that defaults to the join() if none is given.

Similar to the vee(), \(V = A \vee B\), but without decaying to 0 for blades lying in the same subspace.

astype(*args, **kwargs)

Change the underlying scalar type of this vector.

Can be used to force lower-precision floats or integers

See np.ndarray.astype for argument descriptions.