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# 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() (docs).

:

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.

$\{\hspace{0.5em} \underbrace{\hspace{0.5em}\alpha,\hspace{0.5em}}_{\mbox{scalar}}\hspace{0.5em} \underbrace{\hspace{0.5em}e_{1},\hspace{1.5em}e_{2},\hspace{1.5em}e_{3},\hspace{0.5em}}_{\mbox{vectors}}\hspace{0.5em} \underbrace{\hspace{0.5em}e_{12},\hspace{1.5em}e_{23},\hspace{1.5em}e_{13},\hspace{0.5em}}_{\mbox{bivectors}}\hspace{0.5em} \underbrace{\hspace{0.5em}e_{123}\hspace{0.5em}}_{\text{trivector}} \hspace{0.5em} \}$

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,

:

blades

:

{'': 1,
'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,

:

e1 = blades['e1']
# 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.

:

locals().update(blades)


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^e12)

:

e1|e2  # inner product

:

0

:

e1^e2  # outer product

:

(1^e12)

:

e1^e2^e3  # even more outer products

:

(1^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, evaluates as

:

(2^e123)

:

(e1^e2) + (e2^e3)  # correct

:

(1^e12) + (1^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

:

import math

theta = math.pi/4
R = math.cos(theta) - math.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 + (2^e1) + (3^e12) + (4^e123)


### Reversion¶

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

:

~A

:

1 + (2^e1) - (3^e12) - (4^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

:

A(1)  # get grade-1 elements of R

:

(2^e1)

:

A(2)  # you get it

:

(3^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


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^e12) + (2^e13) - (1^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],
ids=BasisVectorIds.ordered_integers(3),
names=['', 'e1', 'e2', 'e3', 'e12', 'e13', 'e23', 'e123']),
[0.13414634146341464, 0.12195121951219512, 0.0, -0.14634146341463414, 0.18292682926829268, 0.0, 0.0975609756097561, -0.2926829268292683])


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¶ 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^e1) - (1^e2) - (1^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

:

R = math.e**(-math.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: math.e**(-theta/2*e12)
R12(math.pi/2)

:

0.71 - (0.71^e12)


And use it like this

:

a = e1+e2+e3
R = R12(math.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: math.e**(-B/2)


Then you could do

:

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


or

:

R_B(math.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 apply_rotation(a):
R = math.e**(-B/2)
return R*a*~R
return apply_rotation

R = R_factory(math.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(math.pi/3*e12)
R23 = R_factory(math.pi/3*e23)
R13 = R_factory(math.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, x = args
R_lst = [math.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 R*x*~R

# rotation sequence by  pi/2-in-e12 THEN pi/2-in-e23
R_seq(math.pi/2*e23, math.pi/2*e12, 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'])


:

{'': 1, 'x': (1^x), 'y': (1^y), 'i': (1^i)}

:

locals().update(blades)

:

1*x + 2*y

:

(1^x) + (2^y)

:

1 + 4*i

:

1 + (4^i)