BasisPGA2 — Plane‑Based PGA 2D¶

BasisPGA2 implements the Gunn/Dorst plane‑based projective geometric algebra (Gunn 2016, Dorst 2020) for 2D Euclidean geometry. It extends Algebra in 4 dimensions via the null-vector embedding $e_0 = e_p + e_m$ where $e_p^2 = +1$ and $e_m^2 = -1$.

In plane‑based PGA, lines are the fundamental primitives (grade‑1 vectors), and points are formed by intersecting two lines (grade‑2 bivectors).

from pytanga.basis import BasisPGA2

pga2 = BasisPGA2()        # default dtype='float64'

Algebra Properties¶

Property	Value
Algebra	$G(4, 0\text{b}1000)$ (Gunn/Dorst model)
Dimension	4
Signature	0b1000 (one negative square for embedding)
Num blades	$2^4 = 16$
Model	Plane‑based (lines are grade‑1 vectors)

Null Vector Embedding¶

BasisPGA2 uses the Gunn/Dorst null vector convention:

\[e_0 = e_p + e_m, \quad e_0^2 = 0$$ $$e_0^{\text{recip}} = \tfrac{1}{2}e_p - \tfrac{1}{2}e_m, \quad \langle e_0 \cdot e_0^{\text{recip}} \rangle_0 = 1\]

The names einf and eo (which belong to the N2 conformal model) are not exposed on this class. Use e0 and e0_recip instead.

Background: pga_null_embedding.md.

Named Blades¶

Attribute	Blade	Description
`e1`, `e2`	Euclidean basis vectors	$e_1$, $e_2$
`e0`	$e_p + e_m$	Gunn/Dorst null vector, $e_0^2 = 0$
`e0_recip`	$0.5 \cdot e_p - 0.5 \cdot e_m$	Reciprocal of $e_0$
`ep`	$e_3$ ($e_p^2 = +1$)	Internal embedding (prefer `e0`)
`em`	$e_4$ ($e_m^2 = -1$)	Internal embedding (prefer `e0`)

Constructing Multivectors¶

p = pga2("3 e1 + 4 e2 + e0")     # IPNS point
d = pga2("e1")                   # ideal point / direction (no e0 component)
ℓ = pga2("e1 + 2 e0")            # line: nx·e1 + ny·e2 + d·e0 (grade‑1 vector)

Geometric entities are also available through the geometry submodule, e.g. create_entity(pga2, Point(3, 4, 0)).

Entity Grades (Gunn/Dorst Convention)¶

Entity	OPNS Grade	IPNS Grade
Line	1	3
Point	2	2 (self‑dual)
Direction	2	2 ($e_0 = 0$)
Space	4	0 (scalar)

Example: Lines and Points¶

from pytanga.basis import BasisPGA2

pga2 = BasisPGA2()

# A line: grade-1 vector in OPNS
line_x = pga2("e1")                     # line through origin along y-axis
line_y = pga2("e2")                     # line through origin along x-axis

# A point is the intersection (meet) of two lines
# OPNS: line_x ∨ line_y = bivector
origin = pga2.op(line_x, line_y)        # point at (0, 0)
pga2.show(origin, "origin (OPNS)")

# Point in IPNS (string conversion)
p = pga2("2 e1 + 3 e2 + e0")
pga2.show(p, "point (2,3) IPNS")        # e1·2 + e2·3 + e₀·1

Display¶

show() prints in the $\{e_1, e_2, e_0\}$ display basis:

pga2.show(mv, "label")       # print in display basis
pga2.show(mv, "label", ".6f")  # with format specifier

Differences from BasisN2¶

Although both BasisPGA2 and BasisN2 are built on $G(4, 0\text{b}1000)$, they are different models:

Aspect	BasisPGA2	BasisN2
Model	Plane‑based (Gunn/Dorst)	Conformal
Null vector name	`e0`	`einf` / `eo`
Lines	Grade‑1 vectors (OPNS)	Grade‑3 blades
Points	Grade‑2 bivectors (OPNS)	Grade‑1 vectors (IPNS)
Sphere/Circle	Not available	Grade‑4 blade (IPNS)
Translations	Known limitation	Full support

Meet / Join Convention (Gunn/Dorst)¶

For BasisPGA2/BasisPGA3 the user-facing MV.meet/MV.join follow the Gunn/Dorst convention, which is the opposite of the Hestenes/DFM07 convention used by the other algebras (E2/E3/P2/P3/N2/N3):

Operation	PGA2/3 (Gunn/Dorst)	Other algebras
`meet`	intersection (progressive/outer product `∧`)	regressive (largest blade contained in both)
`join`	union/span (regressive product `∨`)	progressive (smallest blade containing both)

The outer (^/op) and inner (|/ip) products are unchanged; only the meet/join names swap for the PGA models.

from pytanga.basis import BasisPGA2
from pytanga.geometry import Geometry, Point, Line, Direction

pga2 = BasisPGA2()
geo = Geometry(pga2)
a = geo(Point(1, 0, 0))
b = geo(Point(0, 1, 0))

line = a.join(b)        # the connecting line (grade 1) — the *join* of two points

# meet of two lines is their intersection point
l1 = geo(Line(Point(0, 0, 0), Direction(1, 0, 0)))
l2 = geo(Line(Point(0, 0, 0), Direction(0, 1, 0)))
l1.meet(l2)             # grade-2 point (the origin)

Incidence¶

Incidence in PGA is tested with the complement dual (J‑map / Hodge star ⋆): ⋆A ∧ ⋆B == 0, equivalently A.dual() ^ B.dual() == 0. For example, a point P lies on a line L iff:

P.dual().op(L.dual()).is_zero   # True iff P is on L

This follows from the join identity A ∨ B = ⋆(⋆A ∧ ⋆B) (PGA4CS §9.2).

Note: the metric-contraction form A.dual() | B is not valid in PGA: the PGA pseudoscalar I₃ = e₀∧e₁∧e₂ is null (I₃² = 0), so dualization is a complement map, not the metric dual (PGA4CS §3.2, §9.1).

Three Patterns for Accessing Blades¶

The same three patterns described in Bases apply to BasisPGA2.

Property	Value
Algebra	\(G(4, 0\text{b}1000)\) (Gunn/Dorst model)
Dimension	4
Signature	0b1000 (one negative square for embedding)
Num blades	\(2^4 = 16\)
Model	Plane‑based (lines are grade‑1 vectors)

Attribute	Blade	Description
`e1`, `e2`	Euclidean basis vectors	\(e_1\), \(e_2\)
`e0`	\(e_p + e_m\)	Gunn/Dorst null vector, \(e_0^2 = 0\)
`e0_recip`	\(0.5 \cdot e_p - 0.5 \cdot e_m\)	Reciprocal of \(e_0\)
`ep`	\(e_3\) (\(e_p^2 = +1\))	Internal embedding (prefer `e0`)
`em`	\(e_4\) (\(e_m^2 = -1\))	Internal embedding (prefer `e0`)