Plane-based Geometric Algebra (PGA)

Updated 18 November 2025

Plane-based Geometric Algebra is a coordinate-free framework that represents planes, lines, and points as multivectors with explicit algebraic operations.
It uses a degenerate quadratic form to simultaneously encode affine and Euclidean structures, enabling uniform operations like intersections, joins, and metric measurements.
PGA underpins advances in computational geometry, kinematics, and geometric deep learning by offering a compact, efficient alternative to traditional vector and quaternion methods.

Plane-based Geometric Algebra (PGA) is a coordinate-free, projective Clifford algebra in which Euclidean geometric primitives—planes, lines, and points—are represented by algebraic objects of varying grade, and all classical geometric constructions (intersections, joins, distances, angles, rigid motions) are realized as explicit algebraic operations. PGA’s hallmark is its use of a degenerate quadratic form, encoding both affine and Euclidean structure in a single algebra, and its capacity to provide a uniform framework for Euclidean, elliptic, and hyperbolic geometries via signature change. This paradigm has led to major advances in computational geometry, kinematics, geometric deep learning, and mesh processing, supplanting traditional vector and quaternion approaches in numerous applications.

1. Algebraic Foundations and Structure

Plane-based Geometric Algebra in dimension $n$ is constructed as the Clifford algebra $\mathrm{Cl}(n,0,1)$ over a $(n+1)$ -dimensional real vector space $V$ with basis $\{e_0, e_1,\dots, e_n\}$ and degenerate signature $(n,0,1)$ :

$e_0^2=0$ (the null direction, representing the ideal hyperplane at infinity),
$e_i^2=+1$ for $i=1...n$ ,
$e_i\cdot e_j=0$ for $i\ne j$ .

The geometric product is defined so that for $a, b \in V$ , $ab = a\cdot b + a\wedge b$ , with $\cdot$ the symmetric inner product and $\wedge$ the antisymmetric exterior (wedge) product (Bamberg et al., 24 Aug 2024, Gunn, 2019, Gunn, 2014). The resulting algebra is graded, with scalars (grade 0), vectors (grade 1, representing planes), bivectors (grade 2, lines), up to $(n+1)$ -blades (the pseudoscalar).

Owing to $e_0^2=0$ , the algebra is split by the direct sum $\mathrm{Cl}(V) \cong \mathrm{Cl}(V/\mathbb{R}e_0) \ltimes_\alpha \mathrm{Cl}(V/\mathbb{R}e_0)$ , where the first term captures "Euclidean" parts and the second term (multiplied by $e_0$ ) captures ideal or at-infinity elements. This split extension corresponds at the geometric level to the decomposition between finite and ideal features, and algebraically allows direct encoding of affine parallelism and Playfair's axiom (Bamberg et al., 24 Aug 2024).

2. Representation of Geometric Primitives

In the plane-based (dual) model, geometric objects are encoded as multivectors:

Planes (or lines in 2D): 1-vectors— $a = a_0 e_0 + a_1 e_1 + ... + a_n e_n$ —corresponding to hyperplanes $a_1 x_1 + ... + a_n x_n + a_0 = 0$ (Gunn, 2014, Gunn, 2015, Gunn, 2020).
Lines (or points in 2D): Wedge product of two 1-vectors—a bivector—represents their intersection.
Points: Intersection of $n$ planes (or $n$ -vectors)—for 3D, a trivector—encodes a point. E.g., in 3D, a point $(x,y,z)$ is represented as $P = x e_{032} + y e_{013} + z e_{021} + e_{123}$ (Ruhe et al., 2023, Haan et al., 2023, Brehmer et al., 2023).

Each geometric object also possesses an ideal part (associated with $e_0$ ), representing its at-infinity limit: for example, in 3D, a plane $n_x e_1 + n_y e_2 + n_z e_3 + \delta e_0$ has Euclidean normal $(n_x,n_y,n_z)$ and offset $\delta$ from the origin (Keninck et al., 11 Nov 2025).

Incidence relations, such as point-on-plane or line-in-plane, become simple algebraic statements: e.g., $P \vee \Pi = 0$ if and only if point $P$ lies on plane $\Pi$ .

3. Duality, Meet and Join Operations

PGA natively encodes both meet (intersection) and join (span) in a manner that is uniform and naturally extends to degenerate metrics:

Meet ( $\wedge$ ): Outer product, corresponding to geometric intersections. For instance, the intersection of two planes $\Pi_1,\Pi_2$ yields a line: $L = \Pi_1 \wedge \Pi_2$ .
Join ( $\vee$ ): Regessive (dual) product, defined via the algebra’s dual structure. For two points $P,Q$ (grade- $n$ trivectors in 3D), their join (the line through them) is: $L = P \vee Q = (J(P) \wedge J(Q))$ pulled back to the original algebra via the canonical duality map $J$ (Gunn, 2022, Gunn, 2015).

Duality in degenerate Clifford algebras requires care: the double algebra dual coordinate map $J$ is used in favor of Hodge duality, yielding coordinate-free formulas and ensuring every geometric primitive appears twice (as both a point-based and plane-based object) (Gunn, 2022, Navrat et al., 2020). This dual view is essential for geometric computations, such as extracting coordinates or constructing the regressive product; numerically, join and meet coincide in implementation but differ in interpretation.

4. Metric Structure: Distances, Angles, and Rational Trigonometry

PGA’s degenerate signature encodes not only affine but also Euclidean metric properties:

The squared norm of a join or meet encodes fundamental quantities. For two normalized points $A,B$ , the quadrance (squared distance) is $Q(A,B) = (A\vee B)^2$ . For two normalized lines $\ell,m$ , the spread (squared sine of angle) is $s(\ell,m) = -(\ell\wedge m)^2$ (Gunn, 2014).
This symmetry reflects an intrinsic duality: up to sign, quadrance and spread are mirror images under point-line duality.
Parallelism is absorbed algebraically: for parallel lines, the meet vanishes, so $(\ell\wedge m)^2=0$ (Gunn, 2019).
Formulas unify metric and projective aspects: e.g., in 3D, the norm $||P_0 \vee ... \vee P_k||$ yields the volume of the simplex defined by $k+1$ points, applicable for all $k$ and reducing to known expressions for lines (length), triangles (area), tetrahedra (volume) (Keninck et al., 11 Nov 2025).

These metric features extend seamlessly beyond the Euclidean case. Changing the inner product signature yields PGA models of elliptic or hyperbolic geometry; the same algebraic formulas hold, distinguishing geometry only by the underlying quadratic form (Gunn, 2014).

5. Isometries and Kinematics: Sandwich Operators and Motors

All Euclidean isometries—reflections, rotations, translations, and screws—are represented in PGA by versors (products of 1-vectors), with action implemented as sandwich operators:

Reflections: For a normalized plane $a$ , $X' = a X a$ reflects any object $X$ in that plane.
Rotations and translations: Rotors (even versors) encode rotations via $R(\theta,B) = e^{-B \theta/2}$ , with $B$ a unit bivector aligned with the axis, and translations via products of parallel plane reflections, i.e. $T(\mathbf t) = 1 + 1/2 e_0 \mathbf t$ (Gunn, 2014, Haan et al., 2023).
Screw motions: Compositions of rotation and translation encoded compactly as exponentials of bivectors.
Kinematics and rigid-body mechanics: Motions and the corresponding ODEs (e.g., body angular velocity, momentum) are compactly formulated in PGA, with mechanical quantities realized as multivectors and dynamics equations (Euler–Poincaré) taking their Lie algebraic form automatically (Gunn, 2020, Ruhe et al., 2023).

PGA motors (even subalgebra elements) are isomorphic to dual quaternions; thus, classical screw theory and rigid motion formulas are subsumed (Gunn, 2014). Automatic differentiation arises natively using the pseudoscalar, which plays the role of dual number generator (Gunn, 2019).

6. Application Domains: Geometry, Deep Learning, and Mesh Algorithms

PGA’s unification of affine, Euclidean, and projective geometry has catalyzed progress in several areas:

Geometric computation: Representation of geometric primitives, construction of intersection and join, calculation of projections, and explicit computation of volumes, centroids, and inertias for meshes and $k$ -complexes all follow from direct manipulation of PGA multivectors (Keninck et al., 11 Nov 2025).
Geometric deep learning: Neural architectures such as Geometric Clifford Algebra Networks (GCANs) and Geometric Algebra Transformers (GATr) encode multivectors as network features, naturally enforcing Euclidean equivariance via sandwich actions. These architectures maintain grade-awareness, enforce equivariance, and outperform non-geometric and pure-Euclidean models on a range of physics, simulation, and robotics tasks (Ruhe et al., 2023, Brehmer et al., 2023, Haan et al., 2023).
Software and symbolic computation: PGA serves as a minimal (dimension $2^{n+1}$ ) yet complete model for flat geometry, supporting both symbolic manipulation and high-performance numerical implementation. Its structure allows software systems to toggle between plane- and point-based representations, implement geometric duality, and promote maximal code generality by embedding vector, quaternion, and dual quaternion subalgebras (Gunn, 2022, Navrat et al., 2020).

Empirical studies show that while "vanilla" PGA-based neural architectures lack certain distance-sensitive expressivity compared to conformal models (CGA), augmenting PGA with join operations or mapping into CGA subspaces restores universality and performance (Haan et al., 2023).

7. Comparison with Other Geometric Algebras and Limitations

PGA (Clifford algebra of signature $(n,0,1)$ ) emphasizes flat geometry. In this respect:

It is isomorphic to the subalgebra of flat objects inside Conformal Geometric Algebra (CGA, signature $(n+1,1,0)$ ), but with lower embedding dimension (e.g., $n=3$ : PGA has dimension 16, CGA has 32) (Navrat et al., 2020).
For any calculation not requiring spheres, circles, or conformal transformations, PGA is more economical and numerically stable than CGA (Gunn, 2014).
Classical vector and quaternion algebras are embedded as natural subalgebras of PGA, allowing a smooth transition and code reuse (Gunn, 2019).
The degeneracy of the inner product ( $e_0^2=0$ ) means care must be taken with duality, norm definitions, and handling of ideal elements, but it also encodes affine parallelism and ideal elements algebraically (Bamberg et al., 24 Aug 2024, Gunn, 2022).
Purely within PGA, distance- or angle-sensitive attention functions (as in deep learning) are constrained: certain invariants are constant across translations, limiting expressivity. Augmenting with join or passing through CGA regains universality (Haan et al., 2023, Brehmer et al., 2023).

PGA’s conceptual clarity and algebraic economy have established it as the preferred model for flat Euclidean, affine, and Cayley-Klein geometries in contemporary geometric computing, mesh processing, and 3D learning applications. Its broader adoption continues to influence the design of algorithms, data representations, and machine learning strategies for geometric information.