This page was generated from docs/tutorials/apollonius-cga-augmented.ipynb. Interactive online version: .

# Working with custom algebras¶

This notebook explores the algebra defined in The Lie Model for Euclidean Geometry (Hongbo Li), and its application to solving Apollonius’ Problem. It also shows

The algebra is constructed with basis elements \(e_{-2}, e_{-1}, e_1, \cdots, e_n, e_{n+1}\), where \(e_{-2}^2 = -e_{-1}^2 = -e_{n+1}^2 = 1\). This is an extension of a standard conformal algebra, with an extra \(e_{n+1}\) basis vector.

Note that we permuted the order in the source code below to make `ConformalLayout`

happy.

```
[1]:
```

```
from clifford import ConformalLayout, BasisVectorIds, MultiVector, transformations
class OurCustomLayout(ConformalLayout):
def __init__(self, ndims):
self.ndims = ndims
euclidean_vectors = [str(i + 1) for i in range(ndims)]
conformal_vectors = ['m2', 'm1']
# Construct our custom algebra. Note that ConformalLayout requires the e- and e+ basis vectors to be last.
ConformalLayout.__init__(
self,
[1]*ndims + [-1] + [1, -1],
ids=BasisVectorIds(euclidean_vectors + ['np1'] + conformal_vectors)
)
self.enp1 = self.basis_vectors_lst[ndims]
# Construct a base algebra without the extra `enp1`, which would not be understood by pyganja.
self.conformal_base = ConformalLayout(
[1]*ndims + [1, -1],
ids=BasisVectorIds(euclidean_vectors + conformal_vectors)
)
# this lets us convert between the two layouts
self.to_conformal = transformations.between_basis_vectors(self, self.conformal_base)
```

The code above also defines a stardard conformal \(\mathbb{R}^{N+1,1}\) layout without this new basis vector. This is primarily to support rendering with `pyganja`

, which doesn’t support the presence of this extra vector. `BasisVectorMap`

defaults to preserving vectors by name between one algebra and another, while throwing away blades containing vectors missing from the destination algebra.

We define an `ups`

function which maps conformal dual-spheres into this algebra, as \(s^\prime = s + \left|s\right|e_{n+1}\), and a `downs`

that applies the correct sign. The `s`

suffix here is chosen to mean `sphere`

.

```
[2]:
```

```
def ups(self, s):
return s + self.enp1*abs(s)
OurCustomLayout.ups = ups; del ups
def downs(self, mv):
if (mv | self.enp1)[()] > 0:
mv = -mv
return mv
OurCustomLayout.downs = downs; del downs
```

Before we start looking at specified dimensions of euclidean space, we build a helper to construct conformal dual circles and spheres, with the word `round`

being a general term intended to cover both circles and spheres.

```
[3]:
```

```
def dual_round(at, r):
l = at.layout
return l.up(at) - 0.5*l.einf*r*r
```

In order to render with `pyganja`

, we’ll need a helper to convert from our custom \(\mathbb{R}^{N+1,2}\) layout into a standard conformal \(\mathbb{R}^{N+1,1}\) layout. `clifford`

maps indices in `.value`

to basis blades via `layout._basis_blade_order.index_to_bitmap`

, which we can use to convert the indices in one layout to the indices in another.

## Visualization¶

Finally, we’ll define a plotting function, which plots the problem and solution circles in suitable colors via `pyganja`

. Note that this all works because of our definition of the `to_conformal`

`BasisVectorMap`

.

```
[4]:
```

```
import itertools
from pyganja import GanjaScene, draw
def plot_rounds(in_rounds, out_rounds, scale=1):
colors = itertools.cycle([
(255, 0, 0),
(0, 255, 0),
(0, 0, 255),
(0, 255, 255),
])
# note: .dual() neede here because we're passing in dual rounds, but ganja expects direct rounds
s = GanjaScene()
for r, color in zip(in_rounds, colors):
s.add_object(r.layout.to_conformal(r).dual(), color=color)
for r in out_rounds:
s.add_object(r.layout.to_conformal(r).dual(), color=(64, 64, 64))
draw(s, sig=r.layout.conformal_base.sig, scale=scale)
```

```
/home/docs/checkouts/readthedocs.org/user_builds/clifford/envs/v1.3.0/lib/python3.8/site-packages/pyganja/__init__.py:2: UserWarning: Failed to import cef_gui, cef functions will be unavailable
from .script_api import *
```

## Apollonius’ problem in \(\mathbb{R}^2\) with circles¶

```
[5]:
```

```
l2 = OurCustomLayout(ndims=2)
e1, e2 = l2.basis_vectors_lst[:2]
```

This gives us the `Layout`

`l2`

with the desired metric,

```
[6]:
```

```
import pandas as pd # convenient but somewhat slow trick for showing tables
pd.DataFrame(l2.metric, index=l2.basis_names, columns=l2.basis_names)
```

```
[6]:
```

e1 | e2 | enp1 | em2 | em1 | |
---|---|---|---|---|---|

e1 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 |

e2 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 |

enp1 | 0.0 | 0.0 | -1.0 | 0.0 | 0.0 |

em2 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 |

em1 | 0.0 | 0.0 | 0.0 | 0.0 | -1.0 |

Now we can build some dual circles:

```
[7]:
```

```
# add minus signs before `dual_round` to flip circle directions
c1 = dual_round(-e1-e2, 1)
c2 = dual_round(e1-e2, 0.75)
c3 = dual_round(e2, 0.5)
```

Compute the space orthogonal to all of them, which is an object of grade 2:

```
[8]:
```

```
pp = (l2.ups(c1) ^ l2.ups(c2) ^ l2.ups(c3)).dual()
pp.grades()
```

```
[8]:
```

```
{2}
```

We hypothesize that this object is of the form `l2.ups(c4) ^ l2.ups(c5)`

. Taking a step not mentioned in the original paper, we decide to treat this as a regular conformal point pair, which allows us to project out the two factors with the approach taken in A Covariant Approach to Geometry using Geometric Algebra. Here, we normalize with \(e_{n+1}\) instead of the usual \(n_\infty\):

```
[9]:
```

```
def pp_ends(pp):
P = (1 + pp.normal()) / 2
return P * (pp | pp.layout.enp1), ~P * (pp | pp.layout.enp1)
c4u, c5u = pp_ends(pp)
```

And finally, plot our circles:

```
[10]:
```

```
plot_rounds([c1, c2, c3], [l2.downs(c4u), l2.downs(c5u)], scale=0.75)
```

This works for colinear circles too:

```
[11]:
```

```
c1 = dual_round(-1.5*e1, 0.5)
c2 = dual_round(e1*0, 0.5)
c3 = dual_round(1.5*e1, 0.5)
c4u, c5u = pp_ends((l2.ups(c1) ^ l2.ups(c2) ^ l2.ups(c3)).dual())
plot_rounds([c1, c2, c3], [l2.downs(c4u), l2.downs(c5u)])
```

```
[12]:
```

```
c1 = dual_round(-3*e1, 1.5)
c2 = dual_round(-2*e1, 1)
c3 = -dual_round(2*e1, 1)
c4u, c5u = pp_ends((l2.ups(c1) ^ l2.ups(c2) ^ l2.ups(c3)).dual())
plot_rounds([c1, c2, c3], [l2.downs(c4u), l2.downs(c5u)])
```

## Apollonius’ problem in \(\mathbb{R}^3\) with spheres¶

```
[13]:
```

```
l3 = OurCustomLayout(ndims=3)
e1, e2, e3 = l3.basis_vectors_lst[:3]
```

Again, we can check the metric:

```
[14]:
```

```
pd.DataFrame(l3.metric, index=l3.basis_names, columns=l3.basis_names)
```

```
[14]:
```

e1 | e2 | e3 | enp1 | em2 | em1 | |
---|---|---|---|---|---|---|

e1 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

e2 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 |

e3 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 |

enp1 | 0.0 | 0.0 | 0.0 | -1.0 | 0.0 | 0.0 |

em2 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 |

em1 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | -1.0 |

And apply the solution to some spheres, noting that we now need 4 in order to constrain our solution

```
[15]:
```

```
c1 = dual_round(e1+e2+e3, 1)
c2 = dual_round(-e1+e2-e3, 0.25)
c3 = dual_round(e1-e2-e3, 0.5)
c4 = dual_round(-e1-e2+e3, 1)
c5u, c6u = pp_ends((l3.ups(c1) ^ l3.ups(c2) ^ l3.ups(c3) ^ l3.ups(c4)).dual())
plot_rounds([c1, c2, c3, c4], [l3.downs(c6u), l3.downs(c5u)], scale=0.25)
```

Note that the figure above can be rotated!