BasisPGA3 — Projective Geometric Algebra in 3D¶

This document describes the mathematical foundation of the BasisPGA3 class and its implementation of the Gunn/Dorst plane‑based PGA model within TANGA's 5D null‑vector embedding.

Related reading: - pga_null_embedding.md — the general null‑vector embedding technique - bases.md — brief overview of all four basis classes

1. The Gunn/Dorst 4D PGA Model¶

BasisPGA3 implements the plane‑based projective geometric algebra (PGA) described by Charles Gunn and Leo Dorst. In this model:

Planes are grade‑1 vectors: a plane with unit normal n and signed distance d from the origin is n + d·e₀.
Lines are grade‑2 bivectors: the intersection of two planes.
Points are grade‑3 trivectors: the intersection of three planes. The dual of a point trivector is a grade‑1 vector x·e₁ + y·e₂ + z·e₃ + e₀.

Model	Dim	Point	Plane	Line	Null vector
Gunn/Dorst PGA	4	Grade 3 (or dual grade 1)	Grade 1	Grade 2	`e₀` (true null, `e₀² = 0`)

The algebra is G(3, 0, 1) — three basis vectors squaring to +1, and one null basis vector e₀ with e₀² = 0.

1.1 Key References¶

Charles Gunn — Geometric algebras for Euclidean geometry (arXiv:1411.6502, 2016). The definitive modern treatment of 4D PGA (also called "rigid geometric algebra" or RGA).
Leo Dorst — A Guided Tour to the Plane‑Based Geometric Algebra PGA (bivector.net/PGA4CS.html, 2020). An accessible tutorial for CS practitioners.

2. Our Implementation — 5D Null‑Vector Embedding¶

TANGA's Clifford algebra only supports metric signatures where every basis vector squares to ±1. A true null vector (squaring to 0) cannot be represented natively. We therefore embed the 4D PGA into a 5D algebra via the null‑vector embedding:

\[ e₀ → e_p + e_m \qquad e_p² = +1,\; e_m² = -1 \]

The pair (e_p, e_m) generates the 5‑dimensional algebra G(5, 0b10000). The subspace {e₁, e₂, e₃, e₀} where e₀ = e_p + e_m is algebraically isomorphic to the Gunn/Dorst 4D PGA.

The reciprocal of the null vector is:

\[ e₀^{\text{recip}} = \tfrac{1}{2}\,e_p - \tfrac{1}{2}\,e_m \qquad \langle e₀ \cdot e₀^{\text{recip}} \rangle_0 = 1 \]

This embedding technique is fully documented in pga_null_embedding.md.

2.1 Class Hierarchy¶

BasisPGA3 extends Algebra directly (not BasisN3). It is a standalone algebra class that does not inherit the N3 conformal model. The names einf and eo (which belong to N3) are not exposed on BasisPGA3.

from pytanga.basis import BasisPGA3
pga = BasisPGA3()

# Basis vectors
pga.e1      # e₁, blade ID 1
pga.e2      # e₂, blade ID 2
pga.e3      # e₃, blade ID 4
pga.e0      # e₀ = ep + em, the Gunn/Dorst null vector
pga.e0_recip  # reciprocal of e₀ = 0.5·ep − 0.5·em

# Internal embedding vectors (private; prefer e0)
pga.ep      # e₄, blade ID 8, ep² = +1
pga.em      # e₅, blade ID 16, em² = -1

2.2 Point Representation¶

A finite point at position (x, y, z) is represented in IPNS (dual) form as a grade‑1 vector:

\[ P_{\text{IPNS}} = x \cdot e₁ + y \cdot e₂ + z \cdot e₃ + e₀ \]

which in raw blade terms is:

\[ P_{\text{IPNS}} = x \cdot e₁ + y \cdot e₂ + z \cdot e₃ + e_p + e_m \]

The OPNS form (grade‑3 trivector) is obtained by dualizing:

\[ P_{\text{OPNS}} = \text{dual}(P_{\text{IPNS}}) = (e₁ - x·e₀) \wedge (e₂ - y·e₀) \wedge (e₃ - z·e₀) \]

p = pga("3 e1 + 4 e2 + 5 e3 + e0")
# IPNS:  x·e1 + y·e2 + z·e3 + e0
# OPNS:  dual → grade-3 trivector

An ideal point (direction at infinity) has no e₀ component:

\[ D_{\text{IPNS}} = x \cdot e₁ + y \cdot e₂ + z \cdot e₃ \]

d = pga("e1")  # x·e1 + y·e2 + z·e3 (no e0 component)

2.3 Plane Representation¶

A plane with unit normal n = (n_x, n_y, n_z) and signed distance d from the origin is a grade‑1 vector:

\[ \Pi = n_x \cdot e₁ + n_y \cdot e₂ + n_z \cdot e₃ + d \cdot e₀ \]

# Plane at z = 3, normal pointing in +z direction
plane = pga("e3 + 3 e0")

3. Entity Representations (Gunn/Dorst OPNS Grades)¶

Entity	Grade	OPNS Representation	Description
Plane	1	`n + d·e₀`	Direct plane vector
Line	2	Intersection of 2 planes	Bivector: `Π₁ ∧ Π₂`
Point	3	Intersection of 3 planes	Trivector: `Π₁ ∧ Π₂ ∧ Π₃`
Direction	3	Dual has no e₀ component	Ideal point at infinity
Space	4	`e₁ ∧ e₂ ∧ e₃ ∧ e₀`	4D pseudoscalar `I₄`

3.1 IPNS Interpretation¶

In IPNS (dual) form, the grades are swapped:

Entity	IPNS Grade	Description
Point	1	`x·e₁ + y·e₂ + z·e₃ + e₀`
Direction	1	`x·e₁ + y·e₂ + z·e₃` (no e₀)
Line	2	Grade-2 bivector (self‑dual in 4D sense)
Plane	3	Grade-3 trivector, dual of a plane vector
Space	5	Pseudoscalar `I₅`

4. Operator Representations (Versors)¶

Operator	Factor Count	Structure	Description
Reflection	1	`R = n`	Single reflector (plane normal), no null
Rotor	2	`R = n₁·n₂`	Two Euclidean reflectors → 3D rotation
Translator	2	`T = 1 − ½∑ d_i·(e_i∧e₀)`	Two null reflectors → translation
GeneralRotor	2	`G = T·R·T̃`	Rotation about displaced axis (`angle`, `axis`, `origin`), grades {0, 2}
Motor	4	`M = T·R`	Rotation + translation (rigid body motion)
ReflectionLine	2	`d∧e₀`	Reflection about a line through origin
ReflectionPoint	3	`e₁∧e₂∧e₃`	Reflection about the origin

4.1 Rotor¶

Same Euclidean bivector basis as in E3/P3:

\[ R = \cos\frac{\theta}{2} + \sin\frac{\theta}{2} \big( a_x·e_{23} + a_y·e_{31} + a_z·e_{12} \big) \]

Blade basis: {1, e23(6), e31(5), e12(3)} — 4 blades, grades 0 and 2.

4.2 Translator¶

\[ T = 1 - \frac{1}{2} \big( d_x·(e_1∧e₀) + d_y·(e_2∧e₀) + d_z·(e_3∧e₀) \big) \]

where e_1∧e₀ = e_1∧e_p + e_1∧e_m (blades 9 and 17).

Extraction: d_x = -2·coeff[9], d_y = -2·coeff[10], d_z = -2·coeff[12] (read directly from bivector coefficients).

4.3 GeneralRotor¶

A general rotor applies a rotation about an axis that does not pass through the origin. It is constructed from an angle, axis (Direction), and origin (Point on the axis):

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

Internally, this is built by conjugating a Rotor with a Translator: G = T·R·T̃. The result has grades {0, 2} (scalar + bivector), distinguishing it from a Motor which also has a grade‑4 term.

Analysis: The Euclidean bivector part yields the rotation angle and axis; the null bivector part encodes the axis displacement.

4.4 Motor¶

A motor is the geometric product of a translator and a rotor:

\[ M = T \cdot R \]

In versor factorization, a motor produces 4 reflector factors (2 Euclidean + 2 null). The analysis separates these into rotor (from Euclidean factors) and translator (from versor coefficients).

5. Entity & Operator Coverage¶

Entity	PGA3	N3	Note
Point	✓	✓	Gunn/Dorst grade 3 (OPNS)
Direction	✓	✓	Ideal point, no e₀ component
Line	✓	✓	Grade 2 (intersection of 2 planes)
Plane	✓	✓	Grade 1 vector
Space	✓	✓	Grade 4 pseudoscalar
PointPair	✗	✓	Requires full conformal structure
Circle	✗	✓	Requires full conformal structure
Sphere	✗	✓	Requires full conformal structure

Operator	PGA3	N3	Note
Reflection	✓	✓
ReflectionLine	✓	✓
ReflectionPoint	✓	✓
Rotor	✓	✓
Translator	✓	✓
GeneralRotor	✓	✓	Rotation about displaced axis
Motor	✓	✓
Inversion	✗	✓	Requires eo as independent element
Dilator	✗	✓	Requires `E = e₀ ∧ e₀^{\text{recip}}`

6. Design Notes¶

6.1 Weight Normalization¶

Point extraction in analysis uses algebraic weight normalization. For a grade‑1 IPNS point vector P = x·e₁ + y·e₂ + z·e₃ + α·e₀, the homogeneous weight α is extracted via:

\[ \alpha = \langle P \cdot e₀^{\text{recip}} \rangle_0 \]

The Euclidean coordinates are then (x/α, y/α, z/α). This handles arbitrary scaling (centroids, interpolated points, versor‑transformed points) correctly.

6.2 Blade‑ness Validation¶

Before factorizing a bivector as a line, the analysis checks that the bivector is a simple blade (B ∧ B = 0). A non‑simple bivector (a screw/motor bivector) raises ValueError with guidance to use analyze_operator instead.

6.3 Limitations¶

5D embedding overhead. The 4D Gunn/Dorst algebra is embedded in a 5D algebra, which means there are extra degrees of freedom (ep and em blades) that must be handled consistently.
Motor decomposition is non‑unique — the same motor can be factored in multiple ways.
No native null vector. A true G(3, 0, 1) implementation with a proper null basis vector may be considered in a future release.

6.4 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 BasisPGA3
from pytanga.geometry import Geometry, Point, Plane, Direction

pga = BasisPGA3()
geo = Geometry(pga)
a = geo(Point(1, 0, 0))
b = geo(Point(0, 1, 0))

line = a.join(b)      # the connecting line (grade 2) — the *join* of two points
a.meet(b)             # empty intersection (two points do not meet)

# meet of two planes is their intersection line
p1 = geo(Plane(Point(0, 0, 0), Direction(1, 0, 0)))
p2 = geo(Plane(Point(0, 0, 0), Direction(0, 1, 0)))
p1.meet(p2)           # grade-2 line (the x/y-plane intersection)

6.5 Incidence¶

Incidence in PGA is tested with the complement dual (J‑map / Hodge star ⋆):

\[\star A \wedge \star B = 0 \quad\Longleftrightarrow\quad A \text{ and } B \text{ are incident}\]

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 works because the join satisfies A ∨ B = ⋆(⋆A ∧ ⋆B) (PGA4CS §9.2, eq. 133): the join degenerates exactly when the dual outer product vanishes.

Note: the metric-contraction form A.dual() | B (used in algebras with an invertible pseudoscalar, e.g. CGA) is not valid in PGA. The PGA pseudoscalar I₄ = e₀∧e₁∧e₂∧e₃ is null (I₄² = 0), so there is no I₄⁻¹ and dualization is a complement map, not the metric dual (PGA4CS §3.2, §9.1).

7. References¶

Gunn/Dorst 4D PGA (our implementation target)¶

Publication	Authors	Year
Geometric algebras for Euclidean geometry	Charles Gunn	2016
On the Homogeneous Model of Euclidean Geometry (in Guide to GA in Practice)	Charles Gunn	2011
A Guided Tour to the Plane‑Based Geometric Algebra PGA	Leo Dorst	2020

Null‑vector embedding technique¶

Publication	Authors	Year
Geometric Algebra with Applications in Engineering (esp. §4.2–4.3)	Christian Perwass	2009
pga_null_embedding.md	—	—

General GA theory¶

Publication	Authors	Year
Geometric Algebra for Computer Science	Dorst, Fontijne, Mann	2007
Clifford algebras and spinors (esp. §17.3)	Pertti Lounesto	2001