clifford.tools.classify

Tools for interpreting conformal blades

clifford.tools.classify.classify(x) → Blade

Classify a conformal multivector into a parameterized geometric description.

The multivector should be from a ConformalLayout, such as the one returned by clifford.conformalize().

Implemented based on the approach described in table 14.1 of Geometric Algebra for Computer Science (Revised Edition).

Example usage:

>>> from clifford.g3c import *

>>> classify(e1)
DualFlat[1](flat=Plane(direction=-(1.0^e23), location=0))

>>> classify(einf)
InfinitePoint(direction=1.0)

>>> classify(up(e1))
Point(direction=1.0, location=(1.0^e1))

>>> classify(up(3*e1)^up(4*e2))
PointPair(direction=-(3.0^e1) + (4.0^e2), location=(1.5^e1) + (2.0^e2), radius=2.5)

>>> classify(up(e1)^up(e2)^up(e1+2*e2))
Circle(direction=-(2.0^e12), location=(1.0^e1) + (1.0^e2), radius=1.0)

>>> classify(up(e1)^up(e2)^up(e1+2*e2)^einf)
Plane(direction=-(2.0^e12), location=0)

>>> classify(up(e1)^e2)
Tangent[2](direction=(1.0^e2), location=(1.0^e1))

# how the inheritance works
>>> Point.mro()  
[Point, Tangent[1], Round[1], Blade[1], Tangent, Round, Blade, object]

The reverse of this operation is Blade.mv.

The return type is a Blade:

class clifford.tools.classify.Blade(*args, **kwargs)

Base class for providing interpretation of blades.

Note that thanks to the unual metaclass, this class and its subclasses are have grade-specific specializations, eg Blade[2] is a type for blades of grade 2.

layout

The layout to which this blade belongs

Type:

Layout

property mv: MultiVector

Convert this back into its GA representation

The subclasses below are the four categories to which all blades belong, where \(E\) is a euclidean blade, and \(T_p[X]\) represents a translation of the conformal blade \(X\) by the euclidean vector \(p\).

class clifford.tools.classify.Direction(*args, **kwargs)

\(En_\infty\)

direction

The euclidean direction, \(E\)

Type:

MultiVector

class clifford.tools.classify.Flat(*args, **kwargs)

\(T_p[n_o \wedge (En_\infty)]\)

direction

The euclidean direction, \(E\)

Type:

MultiVector

location

The closest point on this flat to the origin, \(p\), as a euclidean vector.

Type:

MultiVector

class clifford.tools.classify.DualFlat(*args, **kwargs)

Dual of Flat

flat

The flat this is the dual of

Type:

Flat

class clifford.tools.classify.Round(*args, **kwargs)

\(T_p[(n_o + \frac{1}{2}\rho^2 n_\infty)E]\)

direction

The euclidean direction, \(E\)

Type:

MultiVector

location

The euclidean center, \(p\)

Type:

MultiVector

radius

The radius, \(\rho\), which may be imaginary

Type:

float or complex

These can be constructed directly, and will attempt to show a grade-specific interpretation:

>>> from clifford.g3c import *
>>> Round(location=e1, direction=e1^e2, radius=1)
Circle(direction=(1^e12), location=(1^e1), radius=1)
>>> Round(direction=e1^e2^e3, location=e1, radius=1)
Sphere(direction=(1^e123), location=(1^e1), radius=1)

Aliased types

In addition, aliases are created for specific grades of the above types, with more convenient names:

class clifford.tools.classify.Tangent(*args, **kwargs)

Bases: Round

A Round of radius 0, \(T_p[n_o E]\)

class clifford.tools.classify.Point(*args, **kwargs)

Bases: Tangent[1]

A conformal point, \(A\)

class clifford.tools.classify.PointFlat(*args, **kwargs)

Bases: Flat[2]

A flat point, \(A \wedge n_\infty\)

class clifford.tools.classify.Line(*args, **kwargs)

Bases: Flat[3]

A line, \(A \wedge B \wedge n_\infty\)

class clifford.tools.classify.Plane(*args, **kwargs)

Bases: Flat[4]

A line, \(A \wedge B \wedge C \wedge n_\infty\)

class clifford.tools.classify.PointPair(*args, **kwargs)

Bases: Round[2]

A point pair, \(A \wedge B\)

class clifford.tools.classify.Circle(*args, **kwargs)

Bases: Round[3]

A circle, \(A \wedge B \wedge C\)

class clifford.tools.classify.Sphere(*args, **kwargs)

Bases: Round[4]

A sphere, \(A \wedge B \wedge C \wedge D\)

class clifford.tools.classify.InfinitePoint(*args, **kwargs)

Bases: Direction[1]

A scalar multiple of \(n_\infty\)