The Algebra Of Space (G3)¶

In this notebook, we give a more detailed look at how to use clifford, using the algebra of three dimensional space as a context.

Setup¶

First, we import clifford as cf, and instantiate a three dimensional geometric algebra using Cl()

:
from numpy import e,pi
import clifford as cf

layout, blades = cf.Cl(3) # creates a 3-dimensional clifford algebra

Given a three dimensional GA with the orthonormal basis,

$e_{i}\cdot e_{j}=\delta_{ij}$

The basis consists of scalars, three vectors, three bivectors, and a trivector.

$\{\underbrace{\alpha}_{\mbox{scalar}},\qquad\underbrace{e_{1},e_{2},e_{3}}_{\mbox{vectors}},\qquad\underbrace{e_{12},e_{23},e_{13}}_{\mbox{bivectors}},\qquad\underbrace{e_{123}}_{\mbox{trivector}}\}$

Cl() creates the algebra and returns a layout and blades. The layout holds information and functions related this instance of G3, and the blades is a dictionary which contains the basis blades, indexed by their string representations,

:
:
{'e1': (1^e1),
'e2': (1^e2),
'e3': (1^e3),
'e12': (1^e12),
'e13': (1^e13),
'e23': (1^e23),
'e123': (1^e123)}

You may wish to explicitly assign the blades to variables like so,

:
# etc ...

Or, if you’re lazy and just working in an interactive session you can use locals() to update your namespace with all of the blades at once.

:

Now, all the blades have been defined in the local namespace

:
e3, e123
:
((1^e3), (1^e123))

Basics¶

Products¶

The basic products are available

:
e1*e2 # geometric product
:
(1.0^e12)
:
e1|e2 # inner product
:
0
:
e1^e2 # outer product
:
(1.0^e12)
:
e1^e2^e3 # even more outer products
:
(1.0^e123)

Defects in Precedence¶

Python’s operator precedence makes the outer product evaluate after addition. This requires the use of parentheses when using outer products. For example

:
e1^e2+e2^e3 # fail
:
(2.0^e123)
:
(e1^e2) + (e2^e3) # correct
:
(1.0^e12) + (1.0^e23)

Also the inner product of a scalar and a Multivector is 0,

:
4|e1
:
0

So for scalars, use the outer product or geometric product instead

:
4*e1
:
(4^e1)

Multivectors¶

Multivectors can be defined in terms of the basis blades. For example you can construct a rotor as a sum of a scalar and bivector, like so

:
from scipy import cos, sin

theta = pi/4
R = cos(theta) - sin(theta)*e23
R
:
0.70711 - (0.70711^e23)

You can also mix grades without any reason

:
A = 1 + 2*e1 + 3*e12 + 4*e123
A
:
1.0 + (2.0^e1) + (3.0^e12) + (4.0^e123)

Reversion¶

The reversion operator is accomplished with the tilde ~ in front of the Multivector on which it acts

:
~A
:
1.0 + (2.0^e1) - (3.0^e12) - (4.0^e123)

Taking a projection onto a specific grade $$n$$ of a Multivector is usually written

$\langle A \rangle _n$

can be done by using soft brackets, like so

:
A(0) # get grade-0 elements of R
:
1.0
:
A(1) # get grade-1 elements of R
:
(2.0^e1)
:
A(2)  #  you get it
:
(3.0^e12)

Magnitude¶

Using the reversion and grade projection operators, we can define the magnitude of $$A$$

$|A|^2 = \langle A\tilde{A}\rangle$
:
(A*~A)(0)
:
30.0

This is done in the abs() operator

:
abs(A)**2
:
30.0

Inverse¶

The inverse of a Multivector is defined as $$A^{-1}A=1$$

:
A.inv()*A
:
1.0
:
A.inv()
:
0.13415 + (0.12195^e1) - (0.14634^e3) + (0.18293^e12) + (0.09756^e23) - (0.29268^e123)

Dual¶

The dual of a multivector $$A$$ can be defined as

$AI^{-1}$

Where, $$I$$ is the pseudoscalar for the GA. In $$G_3$$, the dual of a vector is a bivector,

:
a = 1*e1 + 2*e2 + 3*e3
a.dual()
:
-(3.0^e12) + (2.0^e13) - (1.0^e23)

Pretty, Ugly, and Display Precision¶

You can toggle pretty printing with with pretty() or ugly(). ugly returns an eval-able string.

:
cf.ugly()
A.inv()
:
MultiVector(Layout([1, 1, 1], [(), (1,), (2,), (3,), (1, 2), (1, 3), (2, 3), (1, 2, 3)], firstIdx=1, names=['', 'e1', 'e2', 'e3', 'e12', 'e13', 'e23', 'e123']), value=[0.13414634146341464, 0.12195121951219513, -0.0, -0.14634146341463417, 0.1829268292682927, -7.80625564189563e-18, 0.09756097560975611, -0.29268292682926833])

You can also change the displayed precision

:
cf.pretty(precision=2)

A.inv()
:
0.13 + (0.12^e1) - (0.15^e3) + (0.18^e12) + (0.1^e23) - (0.29^e123)

This does not effect the internal precision used for computations.

Applications¶

Reflections¶

:
from IPython.display import Image
Image(url='_static/reflection_on_vector.svg')
: Reflecting a vector $$c$$ about a normalized vector $$n$$ is pretty simple,

$c \rightarrow ncn$
:
c = e1+e2+e3    # a vector
n = e1          # the reflector
n*c*n          # reflect a in hyperplane normal to n
:
(1.0^e1) - (1.0^e2) - (1.0^e3)

Because we have the inv() available, we can equally well reflect in un-normalized vectors using,

$a \rightarrow nan^{-1}$
:
a = e1+e2+e3    # the vector
n = 3*e1          # the reflector
n*a*n.inv()
:
(1.0^e1) - (1.0^e2) - (1.0^e3)

Reflections can also be made with respect to the a ‘hyperplane normal to the vector $$n$$’, in this case the formula is negated

$c \rightarrow -ncn^{-1}$

Rotations¶

A vector can be rotated using the formula

$a \rightarrow Ra\tilde{R}$

Where $$R$$ is a rotor. A rotor can be defined by multiple reflections,

$R=mn$

or by a plane and an angle,

$R = e^{-\frac{\theta}{2}\hat{B}}$

For example

:
from numpy import pi

R = e**(-pi/4*e12) # enacts rotation by pi/2
R
:
0.71 - (0.71^e12)
:
R*e1*~R    # rotate e1 by pi/2 in the e12-plane
:
(1.0^e2)

Some Ways to use Functions¶

Maybe we want to define a function which can return rotor of some angle $$\theta$$ in the $$e_{12}$$-plane,

$R_{12} = e^{-\frac{\theta}{2}e_{12}}$
:
R12 = lambda theta: e**(-theta/2*e12)
R12(pi/2)
:
0.71 - (0.71^e12)

And use it like this

:
a = e1+e2+e3
R = R12(pi/2)
R*a*~R

:
-(1.0^e1) + (1.0^e2) + (1.0^e3)

You might as well make the angle argument a bivector, so that you can control the plane of rotation as well as the angle

$R_B = e^{-\frac{B}{2}}$
:
R_B = lambda B: e**(-B/2.)

Then you could do

:
R12 = R_B(pi/4*e12)
R23 = R_B(pi/5*e23)

or

:
R_B(pi/6*(e23+e12))  # rotor enacting a pi/6-rotation in the e23+e12-plane
:
0.93 - (0.26^e12) - (0.26^e23)

Maybe you want to define a function which returns a function that enacts a specified rotation,

$f(B) \rightarrow \underline{R_B}(a) = R_Ba\tilde{R_B}$

This just saves you having to write out the sandwich product, which is nice if you are cascading a bunch of rotors, like so

$\underline{R_C}( \underline{R_B}( \underline{R_A}(a)))$
:
def R_factory( B):
def dummy_f(a):
R = e**(-B/2)
return R*a*~R
return dummy_f

R = R_factory(pi/6*(e23+e12)) # this returns a function
R(a) # which acts on a vector
:
(0.52^e1) + (0.74^e2) + (1.48^e3)

Then you can do things like

:
R12 = R_factory(pi/3*e12)
R23 = R_factory(pi/3*e23)
R13 = R_factory(pi/3*e13)

R12(R23(R13(a)))
:
(0.41^e1) - (0.66^e2) + (1.55^e3)

To make cascading a sequence of rotations as concise as possible, we could define a function which takes a list of bivectors $$A,B,C,..$$ , and enacts the sequence of rotations which they represent on a some vector $$x$$.

$f(A,B,C,x) = \underline{R_A} (\underline{R_B} (\underline{R_C}(x)))$
:
from functools import reduce

# a sequence of rotations
def R_seq(*args):
Bs,a = args[:-1],args[-1]
R_lst =  [e**(-B/2) for B in Bs]  # create list of Rotors from list of Bivectors
R = reduce(cf.gp, R_lst)          # apply the geometric product to list of Rotors
return lambda a: R*a*~R

# rotation sequence by  pi/2-in-e12 THEN pi/2-in-e23
R = R_seq(pi/2*e23, pi/2*e12, e1)

R(e1)
:
(1.0^e3)

Changing Basis Names¶

If you want to use different names for your basis as opposed to e’s with numbers, supply the Cl() with a list of names. For example for a two dimensional GA,

:
layout,blades = cf.Cl(2, names = ['','x','y','i'])