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# Object Oriented CGA¶

This is a shelled out demo for a object-oriented approach to CGA with clifford. The CGA object holds the original layout for an arbitrary geometric algebra , and the conformalized version. It provides up/down projections, as well as easy ways to generate objects and operators.

## Quick Use Demo¶

:

from clifford.cga import CGA, Round, Translation
from clifford import Cl

cga = CGA(g3)  # make cga from existing ga
# or
cga = CGA(3)   # generate cga from dimension of 'base space'

C = cga.round(e1,e2,e3,-e2)      # generate unit sphere from points
C

/home/docs/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.8/site-packages/pyganja/__init__.py:2: UserWarning: Failed to import cef_gui, cef functions will be unavailable
from .script_api import *

:

Sphere

:

## Objects
cga.round()              # from None
cga.round(3)             # from dim of space
cga.round(e1,e2,e3,-e2)  # from points
cga.round(e1,e2,e3)      # from points
cga.round(e1,e2)         # from points
cga.round(cga.round().mv)# from existing multivector

cga.flat()               # from None
cga.flat(2)              # from dim of space
cga.flat(e1,e2)          # from points
cga.flat(cga.flat().mv)  # from existing multivector

## Operations
cga.dilation()          # from from None
cga.dilation(.4)        # from int

cga.translation()       # from None
cga.translation(e1+e2)  # from vector
cga.translation(cga.down(cga.null_vector()))

cga.rotation()          # from None
cga.rotation(e12+e23)   # from bivector

cga.transversion(e1+e2).mv

/home/docs/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.8/site-packages/clifford/_multivector.py:262: RuntimeWarning: divide by zero encountered in true_divide
newValue = self.value / other
/home/docs/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.8/site-packages/clifford/_multivector.py:262: RuntimeWarning: invalid value encountered in true_divide
newValue = self.value / other

:

1.0 + (0.5^e14) - (0.5^e15) + (0.5^e24) - (0.5^e25)

:

cga.round().inverted()

:

-(0.47846^e1234) + (0.12837^e1235) + (1.05451^e1245) - (0.12634^e1345) + (0.29067^e2345)

:

D = cga.dilation(5)
cga.down(D(e1))

:

(5.0^e1)

:

C.mv # any CGA object/operator has a multivector

:

(1.0^e1235)

:

C.center_down,C.radius # some properties of spheres

:

(0, 1.0)

:

T = cga.translation(e1+e2) # make a translation
C_ = T(C)                  # translate the sphere
cga.down(C_.center)        # compute center again

:

(1.0^e1) + (1.0^e2)

:

cga.round()       #  no args == random sphere
cga.translation() #             random translation

:

Translation

:

if 1 in map(int, [1,2]):
print(3)

3


## Objects¶

### Vectors¶

:

a = cga.base_vector()  # random vector with components in base space only
a

:

(0.51296^e1) + (0.95581^e2) + (0.65308^e3)

:

cga.up(a)

:

(0.51296^e1) + (0.95581^e2) + (0.65308^e3) + (0.30161^e4) + (1.30161^e5)

:

cga.null_vector()  # create null vector directly

:

(1.02882^e1) + (1.81476^e2) - (0.41951^e3) + (1.7639^e4) + (2.7639^e5)


### Sphere (point pair, circles)¶

:

C = cga.round(e1, e2, -e1, e3) # generates sphere from points
C = cga.round(e1, e2, -e1)     # generates circle from points
C = cga.round(e1, e2)          # generates point-pair from points
#or
C2 = cga.round(2)            # random 2-sphere  (sphere)
C1 = cga.round(1)            # random 1-sphere, (circle)
C0 = cga.round(0)            # random 0-sphere, (point pair)

C1.mv                        # access the multivector

:

(0.25644^e123) + (0.78834^e124) + (1.09508^e125) + (0.51988^e134) + (0.75719^e135) + (0.10771^e145) + (0.4448^e234) + (0.57059^e235) - (0.14535^e245) - (0.15662^e345)

:

C = cga.round(e1, e2, -e1, e3)

:

(-(1.0^e4) + (1.0^e5), 1.0)

:

cga.down(C.center) == C.center_down

:

True

:

C_ = cga.round().from_center_radius(C.center,C.radius)

:

(-(2.0^e4) + (2.0^e5), 0.9999999999999999)


### Operators¶

:

T = cga.translation(e1) # generate translation
T.mv

:

1.0 - (0.5^e14) - (0.5^e15)

:

C = cga.round(e1, e2, -e1)
T.mv*C.mv*~T.mv         # translate a sphere

:

-(0.5^e124) + (0.5^e125) - (1.0^e245)

:

T(C)                # shorthand call, same as above. returns type of arg

:

Circle

:

T(C).center

:

(2.0^e1) + (2.0^e5)

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