Shape-As-Points (SAP): Theory & Applications

• Shape-As-Points (SAP) is a framework representing shapes as finite sets of points, defining their structure through set relations and topology.
• SAP approaches use quotient operations to remove redundancies like translation and rotation, yielding intrinsic shape spaces with practical computational advantages.
• Applications of SAP include analytic signatures for shape matching and differentiable Poisson solvers for GPU-accelerated, robust mesh reconstruction.

A shape-as-points (SAP) model represents shapes as explicit finite sets of points, with the structure, relations, and equivalence classes of these point sets defining the topology or geometry of shape spaces. This approach underpins a broad spectrum of mathematical and applied shape analysis, ranging from the categorical topology of finite sets to highly-structured representations in image analysis and mesh reconstruction. The SAP paradigm can be realized algebraically as finite spaces, topologically as spaces built via quotient operations, and computationally through analytic and differentiable operators on point clouds (Haridis, 2020, Rodrigues et al., 2010, Peng et al., 2021, Anderson, 2017, Anderson, 2017).

1. Algebraic and Topological Foundations of Shape-As-Points

Let $U_0$ denote the algebra of shapes whose only basic elements are points. Any "shape made with points" is a finite arrangement $S \in U_0$, represented concretely as a finite set $S = \{p_1, p_2, ..., p_n\}$. The part-relation “$\leq$” on shapes coincides with set-inclusion, $A \leq S \iff A \subseteq S$, and algebraic operations reduce to union ($A+B = A \cup B$) and intersection ($A \cdot B = A \cap B$). Each point $p_i$ is atomistic: its only part is itself (Haridis, 2020).

A crucial structural insight is the categorical equivalence between SAPs and finite topological spaces: any topology $\mathcal{T}$ on $S$ is specified by a family of open sets—subsets of $S$—and this is equivalent to endowing $S$ with a preorder relation. For every topology $\mathcal{T}$, the minimal open set containing $p$ is $U_p = \bigcap\{U \in \mathcal{T} : p \in U\}$; defining $q \preceq p$ iff $q \in U_p$ yields a reflexive, transitive preorder. Conversely, given a preorder $\preceq$, $U_p = \{q : q \preceq p\}$ forms a basis for a topology. Thus, every SAP topological structure is equivalently captured by the theory of finite preorders (Haridis, 2020).

Example Table: SAP—Topological Constructions

Algebraic Object	SAP Topology	Preorder Description
$S = \{a, b, c\}$	$\{\varnothing, \{a\}, \{a,b\}, S\}$	$q \preceq p \iff q \in U_p$

The table illustrates the correspondence among the set-theoretic, topological, and order-theoretic presentations.

2. SAP Shape Spaces and Configuration Theory

In shape theory, SAP formalizes the process of isolating shape as the intrinsic relational structure of point sets, independent of embedding, transformations, labeling, or scale. Start with $N$ labeled points in $\mathbb{R}^d$; the configuration space is $Q = \mathbb{R}^{Nd}$. By successively quotienting out physical redundancies—translations ($\text{Tr}(d)$), global scale (dilations, $\text{Dil}$), and rotations ($\text{SO}(d)$)—one obtains preshape and pure shape spaces:

• Relative space: $Q/\text{Tr}(d)$, parameterized by $N-1$ independent relative vectors;
• Preshape space: unit sphere $S^{(N-1)d-1}$ via normalization;
• Pure shape space: quotienting out rotations, leading to spaces such as $S(N,d) = P(N,d)/\text{SO}(d)$;
• Discrete quotients: further reduction under permutations ($S_N$) and mirror identification ($\mathbb{Z}_2$) yields the so-called Leibniz shape space $S(N,d)/(S_N\times \mathbb{Z}_2)$ (Anderson, 2017, Anderson, 2017).

For $N=3$, $d=1$, the SAP pure shape space is a circle $S^1$, with distinguished points corresponding to collision configurations and uniform (equally spaced) states. With $N=4$, $d=1$, the shape space is the 2-sphere $S^2$, tessellated into domains associated with different clustering patterns and symmetries. The quotienting by automorphism groups produces nontrivial graphs (e.g., the 74-vertex cubic graph for fully labeled 4-point shapes), polytopal complexes, and stratified orbifolds (Anderson, 2017, Anderson, 2017).

3. SAP in Computational and Applied Shape Analysis

SAP representations naturally extend to computational frameworks for shape matching, analysis, and reconstruction, particularly with point cloud data. Notably, ANSIG ("Analytic Signature") realizes SAP by representing a two-dimensional shape as an unlabeled set of points $z = [z_1, ..., z_n] \in \mathbb{C}^n$, mapped to an analytic function

$$a(z, \xi) = \frac{1}{n}\sum_{m=1}^n \exp(z_m \xi), \quad \xi \in \mathbb{C}.$$

This analytic signature is invariant under permutations and forms a maximal invariant for the action of the symmetric group $S_n$. Centering and normalization ensure invariance with respect to translation and scale; rotation acts equivariantly on the argument of $\xi$. Shape comparison is efficiently implemented via FFT-based alignment of sampled signatures on the unit circle, and robust similarity measures (such as cosine similarity after optimal circular shift) distinguish shape classes even under substantial noise or outlier perturbation (Rodrigues et al., 2010).

In 3D, SAP is instantiated in reconstruction pipelines via explicit oriented point clouds $\{(c_i, n_i)\}$, where $c_i \in \mathbb{R}^3$ are points and $n_i$ are normals. Meshes are recovered by solving a differentiable Poisson equation for the indicator function, enabling GPU-accelerated, topology-agnostic, and watertight mesh generation. The SAP-based Poisson formulation

$$\Delta \chi(x) = \nabla \cdot v(x); \quad v(x) = \sum_{i=1}^N n_i \delta(x-c_i)$$

maps oriented point data to indicator fields on a volumetric grid. Differentiable implementation provides end-to-end gradients with respect to point positions and normals, supporting both direct optimization and deep learning pipelines for shape prediction. Experimental benchmarks demonstrate that SAP methods converge faster and produce more accurate mesh reconstructions, compared to neural implicit and other point/mesh-based alternatives (Peng et al., 2021).

4. SAP, Symmetries, Quotients, and Shape Space Graphs

A defining feature of SAP is its systematic treatment of symmetries and equivalence. The process of quotienting by group actions (continuous and discrete) is essential to the reduction from raw data to intrinsic shape. The combinatorics and topology of the resulting shape space graphs elucidate the full taxonomy of collision types, merger events, and uniform states.

In the $(N,d)=(3,1)$ and $(4,1)$ models, vertices in the shape space graph correspond to distinct topological types (generic, binary, ternary coincidences, etc.), with adjacency defined by minimal allowed moves (e.g., separating coincident points). Automorphism groups encode the residual isometries; for $(4,1)$, the automorphism group of the unlabeled, mirror-identified ("eib") 5-vertex graph is trivial, reflecting the full reduction of symmetry in the most abstract shape space.

The shape-theoretic Aufbau Principle prescribes a hierarchical approach to SAP: structure discovered at lower $N$, $d$, and small group quotients serves as a template for understanding higher-dimensional and -cardinality shape spaces (Anderson, 2017, Anderson, 2017).

5. SAP Versus Point-Free and Continuum Models

A central contrast arises between SAP (point-built, $i=0$) and point-free ($i>0$) approaches. In the SAP regime, the underlying set of points is predetermined and fixed; topology and shape structure are governed by the combinatorics of open subsets and finite preorders. There are only finitely many possible topologies, and the structure is inherently discrete.

In point-free formalisms, by contrast, shapes are continua in which open parts and their lattice of relations are an open-ended matter of choice. The topology is not imposed on an underlying point set, but rather constructs its own atoms through the recognition of open parts. This distinction leads to fundamentally different mathematical structures in the theory of shapes, with the SAP framework reducing shape-topology to that of finite spaces, while point-free models admit indefinite refinement and a richer spectrum of topological phenomena (Haridis, 2020).

6. Quantitative and Qualitative Structure in SAP Spaces

Quantitative measures in SAP spaces track clustering, collisions, mergers, and degrees of uniformity. For example, the inhomogeneity of a configuration $S$ can be quantified combinatorially as $C(S) = \ln \prod_k n_b!$, where $\{n_b\}$ are counts of equal coincidences or separations (Anderson, 2017). Metic shape spaces, such as the 2-sphere for 4-points in 1D, support tests of geodesic distance, tessellation into domains corresponding to equivalence classes, and the analysis of singularities, such as swallowtail and butterfly catastrophes arising in the stratification of shape spaces under symmetry reductions.

Automorphism groups and Killing vector structure on shape spaces, induced by residual isometries, generate conserved momenta in relational dynamical or quantum mechanical models formulated on these SAP spaces. For instance, in the $(3,1)$ model, the full $S^1$ preshape has a $U(1)$ isometry, which is broken by quotienting under $S_3$ and $\mathbb{Z}_2$ to lower symmetry residuals, corresponding to observable consequences in quantized models (Anderson, 2017).

7. Applications and Empirical Performance

The SAP paradigm is manifested in diverse applications, including shape matching, classification, and 3D surface reconstruction. In the ANSIG framework, robust shape signatures enable perfect or near-perfect shape discrimination even under high levels of geometric noise and point permutation (Rodrigues et al., 2010). SAP-based differentiable Poisson solvers provide superior performance in Chamfer distance and F-Score metrics relative to competing surface reconstruction pipelines, achieving faster inference and producing watertight, manifold meshes even in the presence of data incompleteness, outliers, and nontrivial topology (Peng et al., 2021). The explicit mapping between SAP representations and classic mathematical invariants makes these methods interpretable, lightweight, and scalable.

SAP modeling furnishes a set of canonical, mathematically grounded techniques for both the pure and applied analysis of shape, offering powerful tools for invariance, quotienting, and computation in a finite (and often combinatorially rich) landscape. The continuing development of SAP techniques in theory and application underscores their centrality to the modern understanding of shape spaces.