This notebook is part of the clifford documentation: https://clifford.readthedocs.io/.
In this notebook, we give a more detailed look at how to use clifford, using the algebra of three dimensional space as a context.
First, we import clifford as cf, and instantiate a three dimensional geometric algebra using Cl() (docs).
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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,
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blades
You may wish to explicitly assign the blades to variables like so,
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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.
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locals().update(blades)
Now, all the blades have been defined in the local namespace
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e3, e123
The basic products are available
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e1*e2 # geometric product
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e1|e2 # inner product
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e1^e2 # outer product
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e1^e2^e3 # even more outer products
Python's operator precedence makes the outer product evaluate after addition. This requires the use of parentheses when using outer products. For example
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e1^e2 + e2^e3 # fail, evaluates as
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(e1^e2) + (e2^e3) # correct
Also the inner product of a scalar and a Multivector is 0,
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4|e1
So for scalars, use the outer product or geometric product instead
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4*e1
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
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from scipy import cos, sin
theta = pi/4
R = cos(theta) - sin(theta)*e23
R
You can also mix grades without any reason
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A = 1 + 2*e1 + 3*e12 + 4*e123
A
The reversion operator is accomplished with the tilde ~ in front of the Multivector on which it acts
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~A
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
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A(0) # get grade-0 elements of R
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A(1) # get grade-1 elements of R
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A(2) # you get it
Using the reversion and grade projection operators, we can define the magnitude of $A$
$$|A|^2 = \langle A\tilde{A}\rangle$$
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(A*~A)(0)
This is done in the abs() operator
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abs(A)**2
The inverse of a Multivector is defined as $A^{-1}A=1$
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A.inv()*A
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A.inv()
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,
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a = 1*e1 + 2*e2 + 3*e3
a.dual()
You can toggle pretty printing with with pretty() or ugly(). ugly returns an eval-able string.
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cf.ugly()
A.inv()
You can also change the displayed precision
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cf.pretty(precision=2)
A.inv()
This does not effect the internal precision used for computations.
Reflecting a vector $c$ about a normalized vector $n$ is pretty simple,
$$ c \rightarrow ncn$$
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c = e1+e2+e3 # a vector
n = e1 # the reflector
n*c*n # reflect `a` in hyperplane normal to `n`
Because we have the inv() available, we can equally well reflect in un-normalized vectors using,
$$ a \rightarrow nan^{-1}$$
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a = e1+e2+e3 # the vector
n = 3*e1 # the reflector
n*a*n.inv()
Reflections can also be made with respect to the a 'hyperplane normal to the vector $n$', in this case the formula is negated $$c \rightarrow -ncn^{-1}$$
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
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import math
R = math.e**(-math.pi/4*e12) # enacts rotation by pi/2
R
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R*e1*~R # rotate e1 by pi/2 in the e12-plane
Maybe we want to define a function which can return rotor of some angle $\theta$ in the $e_{12}$-plane,
$$ R_{12} = e^{-\frac{\theta}{2}e_{12}} $$
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R12 = lambda theta: e**(-theta/2*e12)
R12(pi/2)
And use it like this
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a = e1+e2+e3
R = R12(math.pi/2)
R*a*~R
You might as well make the angle argument a bivector, so that you can control the plane of rotation as well as the angle
$$ R_B = e^{-\frac{B}{2}}$$
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R_B = lambda B: math.e**(-B/2)
Then you could do
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R12 = R_B(math.pi/4*e12)
R23 = R_B(math.pi/5*e23)
or
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R_B(math.pi/6*(e23+e12)) # rotor enacting a pi/6-rotation in the e23+e12-plane
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 $$ \underline{R_C}( \underline{R_B}( \underline{R_A}(a)))$$
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def R_factory(B):
def apply_rotation(a):
R = math.e**(-B/2)
return R*a*~R
return apply_rotation
R = R_factory(pi/6*(e23+e12)) # this returns a function
R(a) # which acts on a vector
Then you can do things like
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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)))
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 $x$.
$$f(A,B,C,x) = \underline{R_A} (\underline{R_B} (\underline{R_C}(x)))$$
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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(pi/2*e23, pi/2*e12, e1)
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,
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layout,blades = cf.Cl(2, names=['','x','y','i'])
blades
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locals().update(blades)
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1*x + 2*y
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1 + 4*i