BasisP2 — Projective 2D¶

BasisP2 provides the projective 2D geometric algebra \(G(3, 0)\) with named blade attributes. It extends the Euclidean 2D basis with a homogeneous coordinate, enabling point and line representations.

from pytanga.basis import BasisP2

P2 = BasisP2()            # default dtype='float64'

Algebra Properties¶

Property	Value
Algebra	\(G(3, 0)\)
Dimension	3
Signature	0 (all positive)
Num blades	\(2^3 = 8\)

Named Blades¶

Attribute	Blade	Bitmask
`e1`	\(e_1\)	`0b001`
`e2`	\(e_2\)	`0b010`
`e3`	\(e_3\) (homogeneous direction)	`0b100`
`e12`	\(e_1 \wedge e_2\)	`0b011`
`e13`	\(e_1 \wedge e_3\)	`0b101`
`e23`	\(e_2 \wedge e_3\)	`0b110`
`e123`	\(e_1 \wedge e_2 \wedge e_3\)	`0b111`
`I`	Pseudoscalar (\(e_1 \wedge e_2 \wedge e_3\))	`0b111`

Constructing Multivectors¶

Points and directions are built from strings, or through the geometry submodule:

p = P2("3 e1 + 4 e2 + e3")   # x·e1 + y·e2 + e3  (homogeneous point)
d = P2("e1")                 # x·e1 + y·e2         (ideal point, at infinity)

# Equivalent, via the geometry submodule:
from pytanga.geometry import Direction, Point, create_entity
p = create_entity(P2, Point(3, 4, 0))
d = create_entity(P2, Direction(1, 0, 0))

Display¶

P2.show(p, "p")          # print in grade order
P2.show(p, "p", ".6f")   # with format specifier

Example: Points and Lines¶

from pytanga.basis import BasisP2

P2 = BasisP2()

# Create a point
p = P2("2 e1 + 3 e2 + e3")
P2.show(p, "point")               # e1 · 2 + e2 · 3 + e3 · 1

# Create a direction (ideal point at infinity)
d = P2("e1 + e2")
P2.show(d, "direction")           # e1 · 1 + e2 · 1 (no e3 component)

# A line through two points p and q is their outer product
q = P2("5 e1 + e2 + e3")
line = P2.op(p, q)                # p ∧ q = bivector in P2
P2.show(line, "line p∧q")

Homogeneous coordinate

The third basis vector \(e_3\) serves as the homogeneous coordinate. A point has coefficient 1 for \(e_3\); a direction (ideal point) has coefficient 0. This distinction is encoded in the multivector coefficients (or handled automatically by the geometry submodule).

Three Patterns for Accessing Blades¶

The same three patterns described in Bases apply to BasisP2.