This notebook is part of the clifford
documentation: https://clifford.readthedocs.io/.
Conformal Geometric Algebra (CGA) is a projective geometry tool which allows conformal transformations to be implemented with rotations. To do this, the original geometric algebra is extended by two dimensions, one of positive signature $e_+$ and one of negative signature $e_-$. Thus, if we started with $G_p$, the conformal algebra is $G_{p+1,1}$.
It is convenient to define a null basis given by
$$e_{o} = \frac{1}{2}(e_{-} -e_{+})\\e_{\infty} = e_{-}+e_{+}$$A vector in the original space $x$ is up-projected into a conformal vector $X$ by
$$X = x + \frac{1}{2} x^2 e_{\infty} +e_o $$To map a conformal vector back into a vector from the original space, the vector is first normalized, then rejected from the minkowski plane $E_0$,
$$ X = \frac{X}{X \cdot e_{\infty}}$$then
$$x = X \wedge E_0\, E_0^{-1}$$To implement this in clifford
we could create a CGA by instantiating the it directly, like Cl(3,1)
for example, and then making the definitions and maps described above relating the various subspaces. Or, we you can use the helper function conformalize()
.
conformalize()
The purpose of conformalize()
is to remove the redundancy associated with creating a conformal geometric algebras. conformalize()
takes an existing geometric algebra layout and conformalizes it by adding two dimensions, as described above. Additionally, this function returns a new layout for the CGA, a dict of blades for the CGA, and dictionary containing the added basis vectors and up/down projection functions.
To demonstrate we will conformalize $G_2$, producing a CGA of $G_{3,1}$.
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from numpy import pi,e
from clifford import Cl, conformalize
G2, blades_g2 = Cl(2)
blades_g2 # inspect the G2 blades
Now, conformalize it
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G2c, blades_g2c, stuff = conformalize(G2)
blades_g2c #inspect the CGA blades
Additionally lets inspect stuff
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stuff
It contains the following:
ep
- positive basis vector addeden
- negative basis vector addedeo
- zero vector of null basis (=.5*(en-ep))einf
- infinity vector of null basis (=en+ep)E0
- minkowski bivector (=einf^eo)up()
- function to up-project a vector from GA to CGAdown()
- function to down-project a vector from CGA to GAhomo()
- function to homogenize a CGA vectorWe can put the blades
and the stuff
into the local namespace,
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locals().update(blades_g2c)
locals().update(stuff)
Now we can use the up()
and down()
functions to go in and out of CGA
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x = e1+e2
X = up(x)
X
In [ ]:
down(X)
Conformal transformations in $G_n$ are achieved through versors in the conformal space $G_{n+1,1}$. These versors can be categorized by their relation to the added minkowski plane, $E_0$. There are three categories,
A three dimensional projection for conformal space with the relevant subspaces labeled is shown below.
First we generate some vectors in G2, which we can operate on
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a= 1*e1 + 2*e2
b= 3*e1 + 4*e2
Inversion is a reflection in $e_+$, this swaps $e_o$ and $e_{\infty}$, as can be seen from the model above.
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assert(down(ep*up(a)*ep) == a.inv())
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assert(down(E0*up(a)*E0) == -a)
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from scipy import rand,log
D = lambda alpha: e**((-log(alpha)/2.)*(E0))
alpha = rand()
assert(down( D(alpha)*up(a)*~D(alpha)) == (alpha*a))
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T = lambda x: e**(1/2.*(einf*x))
assert(down( T(a)*up(b)*~T(a)) == b+a)
A transversion is an inversion, followed by a translation, followed by a inversion. The versor is
$$V= e_+ T_a e_+$$which is recognised as the translation bivector reflected in the $e_+$ vector. From the diagram, it is seen that this is equivalent to the bivector in $x\wedge e_o$,
$$ e_+ (1+e_{\infty}a)e_+ $$$$ e_+^2 + e_+e_{\infty}a e_+$$$$2 +2e_o a$$the factor of 2 may be dropped, because the conformal vectors are null
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V = ep * T(a) * ep
assert ( V == 1+(eo*a))
K = lambda x: 1+(eo*a)
B= up(b)
assert( down(K(a)*B*~K(a)) == 1/(a+1/b) )
Versors that are out of $E_0$ are made up of the versors within the original space. These include reflections and rotations, and their conformal representation is identical to their form in $G^n$, except the minus sign is dropped for reflections,
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m = 5*e1 + 6*e2
n = 7*e1 + 8*e2
assert(down(m*up(a)*m) == -m*a*m.inv())
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R = lambda theta: e**((-.5*theta)*(e12))
theta = pi/2
assert(down( R(theta)*up(a)*~R(theta)) == R(theta)*a*~R(theta))
As a simple example consider the combination operations of translation,scaling, and inversion.
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A = up(a)
V = T(e1)*E0*D(2)
B = V*A*~V
assert(down(B) == (-2*a)+e1 )
A transversion may be built from a inversion, translation, and inversion.
$$c = (a^{-1}+b)^{-1}$$In conformal GA, this is accomplished by
$$C = VA\tilde{V}$$$$V= e_+ T_b e_+$$
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A = up(a)
V = ep*T(b)*ep
C = V*A*~V
assert(down(C) ==1/(1/a +b))
Rotation about a point $a$ can be achieved by translating the origin to $a$, then rotating, then translating back. Just like the transversion can be thought of as translating the involution operator, rotation about a point can also be thought of as translating the Rotor itself. Covariance.