BasisE2 — Euclidean 2D¶

BasisE2 provides the Euclidean 2D geometric algebra \(G(2, 0)\) with named blade attributes. It is the simplest tangent-space algebra, supporting vectors, bivectors, and rotors in the plane.

from pytanga.basis import BasisE2

E2 = BasisE2()            # default dtype='float64'

Algebra Properties¶

Property	Value
Algebra	\(G(2, 0)\)
Dimension	2
Signature	0 (all positive)
Num blades	\(2^2 = 4\)

Named Blades¶

Attribute	Blade	Bitmask
`e1`	\(e_1\)	`0b01`
`e2`	\(e_2\)	`0b10`
`e12`	\(e_1 \wedge e_2\)	`0b11`
`I`	Pseudoscalar (\(e_1 \wedge e_2\))	`0b11`

Constructing Multivectors¶

Vectors are built from strings; rotors are created through the geometry submodule:

v = E2("3 e1 + 4 e2")                   # 3·e1 + 4·e2

# Rotor: rotation by angle θ in the e12 plane (the only rotation plane in 2D)
from pytanga.geometry import Direction, Rotor, create_operator
r = create_operator(E2, Rotor(angle=1.57, axis=Direction(0, 0, 1)))  # 90° CCW

Display¶

E2.show(v, "v")          # print in grade order
E2.show(v, "v", ".6f")   # with format specifier

Example: Vectors and Rotors¶

from pytanga.basis import BasisE2
import math

E2 = BasisE2()

# Create vectors
a = E2("e1")
b = E2("e2")

# Geometric product
ab = a * b
E2.show(ab, "a*b")       # e12 (bivector)

# Rotor: rotate a by 90° CCW
from pytanga.geometry import Direction, Rotor, create_operator
R = create_operator(E2, Rotor(angle=math.pi / 2, axis=Direction(0, 0, 1)))
a_rotated = R * a * ~R
E2.show(a_rotated, "R·a·R⁻¹")  # -e2 (e₁ → -e₂ clockwise by default convention)

E2 has no points

E2 can only represent directions (vectors) and rotors. To work with points, use BasisP2 (projective) or BasisN2 (conformal).

Three Patterns for Accessing Blades¶

The same three patterns described in Bases apply to BasisE2:

Pattern 1 — Explicit assignment (recommended)

E2 = BasisE2()
e1 = E2.e1
e2 = E2.e2
e12 = E2.e12
I = E2.I

Pattern 2 — Attribute access

v = E2.e1 * E2.e2    # → e12

Pattern 3 — Namespace injection

globals().update(E2.blades())