The following experimenting is based on https://github.com/pygae/clifford/blob/master/docs/ConformalGeometricAlgebra.ipynb ( see it at https://nbviewer.jupyter.org/github/pygae/clifford/blob/master/docs/ConformalGeometricAlgebra.ipynb )
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!pip install clifford
!pip install matplotlib
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from IPython.display import display, Math, Latex, Markdown
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%matplotlib inline
%config InlineBackend.figure_format = 'svg'
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
import numpy as np
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def str_ga(*gas, expand_wedge=False):
rets = []
for ga in gas:
ret = []
first = True
for i, v in enumerate(ga.value):
if not first:
if v > 0:
ret.append(' + ')
if v < 0:
ret.append(' - ')
if v != 0:
v = abs(v)
blade_tuple = ga.layout.bladeTupList[i]
if blade_tuple != ():
if v != 1:
ret.append('%g ' % v)
if expand_wedge:
ret.append(' \\wedge '.join([ ('e_%d' % base) for base in blade_tuple ]))
else:
ret.append('e_{%s}' % ''.join([ ('%d' % base) for base in blade_tuple ]))
else:
ret.append('%g ' % v)
first = False
if ret == []:
ret.append('0')
rets.append('%s' % ''.join(ret))
return rets
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def print_ga(*gas, expand_wedge=False):
for ga_latex in str_ga(*gas, expand_wedge=False):
display(Math(ga_latex))
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().
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
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G2c, blades_g2c, stuff = conformalize(G2)
blades_g2c #inspect the CGA blades
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print_ga(blades_g2c['e4'])
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stuff
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It contains the following:
ep - postive basis vector addeden - negative basis vector addedeo - zero vecror 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 ot homogenize a CGA vector
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locals().update(blades_g2c)
locals().update(stuff)
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x = e1+e2
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print_ga(x)
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print_ga(up(x))
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print_ga(down(up(x)))
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a= 1*e1 + 2*e2
b= 3*e1 + 4*e2
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print_ga(a, b)
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print_ga(down(ep*up(a)*ep), a.inv())
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print_ga(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()
print_ga(down( D(alpha)*up(a)*~D(alpha)), (alpha*a))
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T = lambda x: e**(1/2.*(einf*x))
print_ga(down( T(a)*up(b)*~T(a)), b+a)
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from pprint import pprint
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pprint(vars(einf))
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print_ga(ep, en, eo)
A transversion is an inversion, followed by a translation, followed by a inversion. The verser 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)
print_ga( down(K(a)*B*~K(a)) , 1/(a+1/b) )
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print_ga(a, 1/a)
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print_ga(e1,e2, e1 | e2, e1^e2, e1 * e2)
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print_ga(a,b, a | b, a^b, a * b)
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soa = np.array([[0, 0, 1, 1, -2, 0], [0, 0, 2, 1, 1, 0],
[0, 0, 3, 2, 1, 0], [0, 0, 4, 0.5, 0.7, 0]])
X, Y, Z, U, V, W = zip(*soa)
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.quiver(X, Y, Z, U, V, W)
ax.set_xlim([-1, 0.5])
ax.set_ylim([-1, 1.5])
ax.set_zlim([-1, 8])
plt.show()
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print_ga(a)
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def to_vector(ga, max=-1):
vec = [ ga.value[i] for i, t in enumerate(ga.layout.bladeTupList) if len(t) == 1 ]
max = len(vec) if max == -1 else max
return vec[0:max]
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[ a.value[i] for i, t in enumerate(a.layout.bladeTupList) if len(t) == 1 ]
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to_vector(a)
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to_vector(a, max=2)
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def plot_as_vector(*gas):
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
l = len(gas)
# print_ga(*gas)
arr = [to_vector(ga, max=3) for ga in gas]
soa = np.array(arr)
ga_latexes = str_ga(*gas)
for i, v in enumerate(arr):
ax.text(v[0], v[1], v[2], ('$ %s $' % ga_latexes[i]))
# print(soa)
X, Y, Z = np.zeros(l), np.zeros(l), np.zeros(l)
U, V, W = [list(a) for a in zip(*soa)]
# print(X, Y, Z, U, V, W)
ax.quiver(X, Y, Z, U, V, W)
xlim = max(abs(min(U)), abs(max(U)), 2)
ylim = max(abs(min(V)), abs(max(V)), 2)
zlim = max(abs(min(W)), abs(max(W)), 2)
ax.set_xlim([-xlim, xlim])
ax.set_ylim([-ylim, ylim])
ax.set_zlim([-zlim, zlim])
# ax.set_xlim([-1, 0.5])
# ax.set_ylim([-1, 1.5])
# ax.set_zlim([-1, 8])
plt.show()
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plot_as_vector(a)
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m = 5*e1 + 6*e2
n = 7*e1 + 8*e2
print_ga(down(m*up(a)*m), -m*a*m.inv())
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str_ga(a, m, down(m*up(a)*m))
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print_ga(a, m, down(m*up(a)*m))
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plot_as_vector(a)
plot_as_vector(m)
plot_as_vector(down(m*up(a)*m))
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R = lambda theta: e**((-.5*theta)*(e12))
theta = pi/2
print_ga(down( R(theta)*up(a)*~R(theta)))
print_ga(R(theta)*a*~R(theta))
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plot_as_vector(a, down( R(theta)*up(a)*~R(theta)))
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from ipywidgets import interact, interactive, fixed, interact_manual
import ipywidgets as widgets
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def plot_rotate(origin, theta=pi/2):
R = lambda theta: e**((-.5*theta)*(e12))
rotated = (down( R(theta)*up(origin)*~R(theta)))
plot_as_vector(origin, rotated)
interactive_plot = interactive(plot_rotate, origin=fixed(a), theta=(0, 2 * pi, 0.1))
output = interactive_plot.children[-1]
output.layout.height = '350px'
interactive_plot
As a simple example consider the combination operations of translation,scaling, and inversion.
$$b=-2a+e_0 \quad \rightarrow \quad B= (T_{e_0}E_0 D_2) A \tilde{ (D_2 E_0 T_{e_0})} $$
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A = up(a)
V = T(e1)*E0*D(2)
B = V*A*~V
assert(down(B) == (-2*a)+e1 )
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plot_as_vector(a, down(B))
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))
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plot_as_vector(a, down(C))
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