Operator Data Classes¶

Operator data classes are algebra-independent @dataclass types that represent versors (geometric transformations). They can be used as input to Geometry.create() (or the plain create() function) and as output from Geometry.which_operator() (or the plain analyze_operator() function).

All classes are imported from pytanga.geometry (defined in pytanga.geometry.operators).

2D usage

2D operators use the same classes as 3D. Rotor(angle, Direction(0, 0, 1)) always rotates in the XY plane — the only rotation plane in 2D. For translations in 2D, use Translator(vector=Direction(dx, dy, 0)).

Reflection (ReflectionPlane)¶

Reflection in a plane through the origin. Reflection is an alias for ReflectionPlane.

from pytanga.geometry import Reflection, Direction

refl = Reflection(normal=Direction(0, 0, 1))

Algebra	Supported
E3, P3, PGA3, N3	✓

ReflectionLine¶

Reflection about a line through the origin (grade‑2 bivector d∧e₀).

from pytanga.geometry import ReflectionLine, Direction

refl_line = ReflectionLine(direction=Direction(0, 0, 1))

Algebra	Supported
E3, P3, PGA3, N3	✓

ReflectionPoint¶

Reflection about the origin (trivector e₁∧e₂∧e₃) — equivalent to ReflectionPoint(Point(0, 0, 0)).

from pytanga.geometry import ReflectionPoint, Point

refl_origin = ReflectionPoint(Point(0, 0, 0))

Algebra	Supported
P3, PGA3, N3	✓

Note

ReflectionPoint is the general point-reflection operator — pass any Point to reflect about an arbitrary center.

Inversion (N3 only)¶

Inversion in a sphere centered at center with radius radius (default 1.0).

from pytanga.geometry import Inversion, Point

inv = Inversion(center=Point(0, 0, 0))                 # unit sphere (radius 1.0)
inv2 = Inversion(center=Point(1, 0, 0), radius=2.0)    # sphere of radius 2.0

Algebra	Supported
N3	✓ (requires eo null vector)

Rotor¶

A 3D rotation (even-grade versor: scalar + bivector).

from pytanga.geometry import Rotor, Direction
import math

r = Rotor(angle=math.pi / 2, axis=Direction(0, 0, 1))

The MV representation is cos(θ/2) + sin(θ/2)·B where B is the unit bivector.

Algebra	Supported
E3, P3, PGA3, N3	✓

Translator (PGA3, N3)¶

A translation in 3D space.

from pytanga.geometry import Translator, Direction

t = Translator(vector=Direction(1, 2, 0))

Algebra	Supported
PGA3, N3	✓

Dilator (N3 only)¶

A uniform dilation (scaling) about an origin point.

from pytanga.geometry import Dilator, Point

d = Dilator(factor=2.0)  # scale about origin (0,0,0)
d2 = Dilator(factor=3.0, origin=Point(1, 0, 0))  # scale about point (1,0,0)

The MV representation is D = 1 + (1−d)/(1+d)·E where d is the dilation factor and E = einf∧eo (Perwass formula). For non-origin dilators, D_t = T · D · T̃ where T translates from global origin to the dilation center.

Algebra	Supported
N3	✓ (requires E = einf∧eo)

Motor (PGA3, N3)¶

A rigid body motion: rotation followed by translation.

from pytanga.geometry import Motor, Rotor, Translator, Direction
import math

m = Motor(
    rotor=Rotor(angle=0.5, axis=Direction(0, 0, 1)),
    translator=Translator(vector=Direction(1, 0, 0)),
)

Algebra	Supported
PGA3, N3	✓

TwistBivector (N3 only)¶

The grade-2 twist bivector of a motor: the rotation bivectors plus the translation bivectors, obtained by building the motor and projecting it onto the motor's grade-2 blade mask. It is not visualizable, but Geometry.create_var and Geometry.mask_for work. mask_for(TwistBivector) returns the 9 raw twist blades with a 6-direction named basis (e12, e13, e23, e1∧e∞, e2∧e∞, e3∧e∞), so a variable built from it collapses via get_tensor().get_array() to the 6 physical degrees of freedom.

from pytanga.geometry import TwistBivector, Rotor, Translator, Direction
import math

twist = TwistBivector(
    rotor=Rotor(angle=0.5, axis=Direction(0, 0, 1)),
    translator=Translator(vector=Direction(1, 0, 0)),
)

Algebra	Supported
N3	✓ (N3 only; other algebras raise `TypeError`)

GeneralRotor¶

A rotation about an axis that does not pass through the origin. Takes the rotation angle, axis direction, and an origin point on the axis.

from pytanga.geometry import GeneralRotor, Direction, Point
import math

gr = GeneralRotor(
    angle=0.5,
    axis=Direction(0, 0, 1),
    origin=Point(1, 0, 0),
)

Algebra	Supported
PGA3, N3	✓

Operator Coverage Matrix¶

Operator	E3	P3	PGA3	N3	E2	P2	PGA2	N2
ReflectionPlane	✓	✓	✓	✓	—	✓	✓	✓
ReflectionLine	✓	✓	✓	✓	✓	✓	✓	✓
ReflectionPoint	—	✓	✓	✓	—	✓	✓	✓
Inversion	—	—	—	✓	—	—	—	✓
Rotor	✓	✓	✓	✓	✓	✓	✓	✓
Translator	—	—	✓	✓	—	—	—	✓
Dilator	—	—	—	✓	—	—	—	✓
Motor	—	—	✓	✓	—	—	—	✓
TwistBivector	—	—	—	✓	—	—	—	—
GeneralRotor	—	—	✓	✓	—	—	—	✓