An Affine-invariant Time-dependent Triangulation of Spatio-temporal Data

TL;DR

This paper introduces a polynomial-time algorithm for creating an affine-invariant, time-dependent triangulation of spatio-temporal data, enabling consistent querying and animation regardless of coordinate transformations.

Contribution

It presents a novel normal form and triangulation algorithm that partition spatio-temporal data invariant under affine transformations, improving data consistency and processing.

Findings

Algorithm is correct and polynomial-time

Triangulation is invariant under time-dependent affine transformations

Supports applications in querying and animation of spatio-temporal data

Abstract

In the geometric data model for spatio-temporal data, introduced by Chomicki and Revesz, spatio-temporal data are modelled as a finite collection of triangles that are transformed by time-dependent affinities of the plane. To facilitate querying and animation of spatio-temporal data, we present a normal form for data in the geometric data model. We propose an algorithm for constructing this normal form via a spatio-temporal triangulation of geometric data objects. This triangulation algorithm generates new geometric data objects that partition the given objects both in space and in time. A particular property of the proposed partition is that it is invariant under time-dependent affine transformations, and hence independent of the particular choice of coordinate system used to describe he spatio-temporal data in. We can show that our algorithm works correctly and has a polynomial time complexity (of reasonably low degree in the number of input triangles and the maximal degree of the polynomial functions that describe the transformation functions). We also discuss several possible applications of this spatio-temporal triangulation.