A triangle-based logic for affine-invariant querying of spatial and spatio-temporal data

In spatial databases, incompatibilities often arise due to different choices of origin or unit of measurement (e.g., centimeters versus inches). By representing and querying the data in an affine-invariant manner, we can avoid these incompatibilities. In practice, spatial (resp., spatio-temporal) data is often represented as a finite union of triangles (resp., moving triangles). As two arbitrary triangles are equal up to a unique affinity of the plane, they seem perfect candidates as basic units for an affine-invariant query language. We propose a so-called “triangle logic”, a query language that is affine-generic and has triangles as basic elements. We show that this language has the same expressive power as the affine-generic fragment of first-order logic over the reals on triangle databases. We illustrate that the proposed language is simple and intuitive. It can also serve as a first step towards a “moving-triangle logic” for spatio-temporal data.

💡 Research Summary

The paper addresses a fundamental problem in spatial and spatio‑temporal databases: queries often depend on arbitrary choices of origin or measurement unit, which makes results non‑robust under affine transformations (scaling, translation, rotation, shear). To obtain a query language that is invariant under any affine map of the plane, the authors propose a “triangle logic” in which the basic data objects are triangles (or moving triangles for the temporal case) rather than points or real‑valued coordinates.

The authors begin by recalling that in practice spatial data are frequently stored as a finite union of triangles—e.g., Triangulated Irregular Networks (TIN) in GIS or triangular meshes in computer graphics. A key geometric fact is that any two non‑degenerate triangles are related by a unique affine transformation. This motivates treating a triangle itself as an atomic entity that already embodies affine invariance.

A formal framework is built on top of the well‑known semi‑algebraic (constraint) database model. A triangle variable Δ is a triple of points in ℝ², and a triangle relation of arity k is a set of k‑tuples of such triples. By means of a canonical bijection can_tr the authors map a triangle relation to a relation over ℝ⁶ᵏ, thus embedding it into the usual first‑order logic FO(+ ,× ,< ,0,1). They also define a “moving‑triangle” as a triple of points in ℝ²×ℝ, allowing the representation of temporally evolving geometric objects.

The core contribution is the definition of a first‑order language FO({Triangle}) whose atomic predicates are expressed directly on triangles (e.g., “point‑between”, “triangle‑inside”, “triangles intersect”). Using the canonical bijections, the paper proves two crucial equivalences:

1. FO({Triangle}) is expressively equivalent to FO({Between})—the point‑based language that uses only the betweenness predicate. This result leverages earlier work showing that FO({Between}) captures exactly the affine‑generic fragment of FO(+ ,× ,< ,0,1).

2. Consequently, FO({Triangle}) has the same expressive power as the affine‑generic fragment of FO(+ ,× ,< ,0,1) on triangle databases. In other words, any query that is invariant under all affine transformations can be written in triangle logic, and vice versa.

The authors then investigate safety (finite‑output) issues. They show that, as in ordinary FO, it is undecidable whether an arbitrary triangle query yields a finite result on all finite inputs. However, for a given concrete database D, it is decidable whether the result can be represented as a finite union of triangles; they provide an algorithm to test this property and to construct the finite representation when it exists. This is practically important for visualisation and storage of query results.

The paper extends the framework to spatio‑temporal data by introducing moving triangles (triples of points each equipped with a timestamp). The same canonical embeddings map moving‑triangle relations to ℝ⁹ᵏ, and the authors prove that the moving‑triangle logic remains expressively equivalent to the affine‑generic fragment of FO on spatio‑temporal databases. This opens the way to query evolving geometric objects (e.g., moving objects, changing terrains) in an affine‑invariant manner.

Throughout the text, illustrative examples demonstrate how natural spatial queries—such as “do two triangles intersect?”, “list all triangles contained in a given region”, or “find moving triangles that pass through a region during a time interval”—are succinctly expressed in the proposed language. The authors also discuss how the triangle‑based approach eliminates the need for pre‑normalising coordinates, since the language itself respects affine transformations.

In conclusion, the paper delivers a theoretically solid and practically motivated query language that treats triangles as first‑class citizens, achieves full affine invariance, matches the expressive power of the affine‑generic fragment of FO over the reals, and provides decidable criteria for finite representability of query results. The work paves the way for future extensions to higher dimensions (e.g., tetrahedra), optimization techniques for triangle‑based query processing, and integration into real GIS and spatio‑temporal systems.