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()
In [1]:
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,
The basis consists of scalars, three vectors, three bivectors, and a 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,
In [2]:
blades
Out[2]:
{'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,
In [3]:
e1 = blades['e1']
e2 = blades['e2']
# 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.
In [4]:
locals().update(blades)
Now, all the blades have been defined in the local namespace
In [5]:
e3, e123
Out[5]:
((1^e3), (1^e123))
Basics¶
Products¶
The basic products are available
In [6]:
e1*e2 # geometric product
Out[6]:
(1.0^e12)
In [7]:
e1|e2 # inner product
Out[7]:
0
In [8]:
e1^e2 # outer product
Out[8]:
(1.0^e12)
In [9]:
e1^e2^e3 # even more outer products
Out[9]:
(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
In [10]:
e1^e2+e2^e3 # fail
Out[10]:
(2.0^e123)
In [11]:
(e1^e2) + (e2^e3) # correct
Out[11]:
(1.0^e12) + (1.0^e23)
Also the inner product of a scalar and a Multivector is 0,
In [12]:
4|e1
Out[12]:
0
So for scalars, use the outer product or geometric product instead
In [13]:
4*e1
Out[13]:
(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
In [14]:
from scipy import cos, sin
theta = pi/4
R = cos(theta) - sin(theta)*e23
R
Out[14]:
0.70711 - (0.70711^e23)
You can also mix grades without any reason
In [15]:
A = 1 + 2*e1 + 3*e12 + 4*e123
A
Out[15]:
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
In [16]:
~A
Out[16]:
1.0 + (2.0^e1) - (3.0^e12) - (4.0^e123)
Grade Projection¶
Taking a projection onto a specific grade \(n\) of a Multivector is usually written
can be done by using soft brackets, like so
In [17]:
A(0) # get grade-0 elements of R
Out[17]:
1.0
In [18]:
A(1) # get grade-1 elements of R
Out[18]:
(2.0^e1)
In [19]:
A(2) # you get it
Out[19]:
(3.0^e12)
Magnitude¶
Using the reversion and grade projection operators, we can define the magnitude of \(A\)
In [20]:
(A*~A)(0)
Out[20]:
30.0
This is done in the abs()
operator
In [21]:
abs(A)**2
Out[21]:
30.0
Inverse¶
The inverse of a Multivector is defined as \(A^{-1}A=1\)
In [22]:
A.inv()*A
Out[22]:
1.0
In [23]:
A.inv()
Out[23]:
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
Where, \(I\) is the pseudoscalar for the GA. In \(G_3\), the dual of a vector is a bivector,
In [24]:
a = 1*e1 + 2*e2 + 3*e3
a.dual()
Out[24]:
-(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.
In [25]:
cf.ugly()
A.inv()
Out[25]:
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
In [26]:
cf.pretty(precision=2)
A.inv()
Out[26]:
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¶
In [27]:
from IPython.display import Image
Image(url='_static/reflection_on_vector.svg')
Out[27]:
Reflecting a vector \(c\) about a normalized vector \(n\) is pretty simple,
In [28]:
c = e1+e2+e3 # a vector
n = e1 # the reflector
n*c*n # reflect `a` in hyperplane normal to `n`
Out[28]:
(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,
In [29]:
a = e1+e2+e3 # the vector
n = 3*e1 # the reflector
n*a*n.inv()
Out[29]:
(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
Rotations¶
A vector can be rotated using the formula
Where \(R\) is a rotor. A rotor can be defined by multiple reflections,
or by a plane and an angle,
For example
In [30]:
from numpy import pi
R = e**(-pi/4*e12) # enacts rotation by pi/2
R
Out[30]:
0.71 - (0.71^e12)
In [31]:
R*e1*~R # rotate e1 by pi/2 in the e12-plane
Out[31]:
(0.0^e1) + (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,
In [32]:
R12 = lambda theta: e**(-theta/2*e12)
R12(pi/2)
Out[32]:
0.71 - (0.71^e12)
And use it like this
In [33]:
a = e1+e2+e3
R = R12(pi/2)
R*a*~R
Out[33]:
-(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
In [34]:
R_B = lambda B: e**(-B/2.)
Then you could do
In [35]:
R12 = R_B(pi/4*e12)
R23 = R_B(pi/5*e23)
or
In [36]:
R_B(pi/6*(e23+e12)) # rotor enacting a pi/6-rotation in the e23+e12-plane
Out[36]:
0.93 - (0.26^e12) - (0.26^e23)
Maybe you want to define a function which returns a function that enacts a specified rotation,
This just saves you having to write out the sandwich product, which is nice if you are cascading a bunch of rotors, like so
In [37]:
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
Out[37]:
(0.52^e1) + (0.74^e2) + (1.48^e3)
Then you can do things like
In [38]:
R12 = R_factory(pi/3*e12)
R23 = R_factory(pi/3*e23)
R13 = R_factory(pi/3*e13)
R12(R23(R13(a)))
Out[38]:
(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 \(x\).
In [39]:
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)
Out[39]:
(0.0^e1) + (0.0^e2) + (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,
In [40]:
layout,blades = cf.Cl(2, names = ['','x','y','i'])
blades
Out[40]:
{'x': (1^x), 'y': (1^y), 'i': (1^i)}
In [41]:
locals().update(blades)
In [42]:
1*x+2*y
Out[42]:
(1^x) + (2^y)
In [43]:
(1+4*i)
Out[43]:
1.0 + (4.0^i)
In [44]: