What are spatial objects?: a critical data infrastructure paradigm shift Digital Business Symposium 2025 University of Westminster, London - 8 th January 2026 Chris Partridge 1, 2 Andrew Mitchell 1, 2 Sergio de Cesare 2 Oscar Xiberta Soto 1 1 BORO Solutions 2 University of Westminster Overview What are spatial objects?: a critical data infrastructure paradigm shift Overall goal To showcase an example of a critical data infrastructure, ontological, paradigm shift to show what this kind of shift is, and how the shift can radically change both our understanding, and the associated data infrastructure foundation Example chosen: ‘geospatial coordinate reference systems’ this links to a broader pattern that solves the puzzle of what a spatial object is and how it is dependent upon a spatial reference system also, stepping back, it gives us an idea of what critical data infrastructure, ontological, paradigm shifts are 3 Some ISO Geospatial Standards Structure Framing ontological, data infrastructure, paradigm shifts Current geospatial coordinate reference systems paradigm Shifting to the new paradigm constructing a physical basis for the coordinate system adding the coordinate reference system gauge freedoms Summary 4 Framing critical data infrastructure, ontological, paradigm shifts Context Framing ‘critical data infrastructure’ Framing critical data infrastructure, ontological, paradigm shifts Critical data infrastructure (Computer) Data Infrastructure the structures that underlie the (computer) data we use these enable (or inhibit) interoperability they are critical for computers to use and share data Data Infrastructure is often hidden (not just data) so, needs to be ‘surfaced’ – a kind of ethnographical enterprise as we shall see, ontologising is a natural way to surface “ This article is in a way a call to study boring things. Many aspects of infrastructure are singularly unexciting. They appear as lists of numbers and technical specifications, or as hidden mechanisms subtending those processes more familiar to social scientists. It takes some digging to unearth the dramas inherent in system design creating, to restore narrative to what appears to be dead lists . … Struggles with infrastructure are built into the very fabric of technical work.” Star, S. L. (1999). The Ethnography of Infrastructure. built on archaic foundations from a different era (pre-computing) – and familiar in other domains ripe for reengineering as we shall see, ontologising can be a fruitful way to radically evolve the foundations to the new technologies 7 style.visibility Framing ‘ontological’ Framing critical data infrastructure, ontological, paradigm shifts What is an ontology? “the set of things whose existence is acknowledged by a particular theory or system of thought.” Jonathon Lowe in The Oxford Companion to Philosophy in the data, these ‘things’ are usually unclear, often obscured or even hidden especially the underlying infrastructure – which is typically largely hidden revealing (‘surfacing’) these ‘things’ provides a less obscured, clear(er) picture the upper levels of the ontology ‘surface’ the infrastructure so, using a top level ontology helps to ensure the (hidden) infrastructure are exposed why important this clear(er) picture is effectively the (conceptually) ideal data structure the upper levels are effectively the (conceptually) ideal data infrastructure 9 a cartoon ‘bad’ ontologisation style.visibility How does ontologising find its ‘things’? Often, ontological re-engineering turns out to be a ‘janitorial’ activity starts with the existing conceptual scheme, the existing data structures tidies it up – and improves it ontology provides the tools for a ‘rigorous forensic investigation’ , and sometimes this reveals a new underlying radical data structure “Here is the task of making explicit what had been tacit, and precise what had been vague; of exposing and resolving paradoxes, smoothing kinks, lopping off vestigial growths, clearing ontological slums . … There is no such cosmic exile. He cannot study and revise the fundamental conceptual scheme of science and common sense without having some conceptual scheme, whether the same or another no less in need of philosophical scrutiny, in which to work. He can scrutinize and improve the system from within, appealing to coherence and simplicity; but this is the theoretician’s method generally.” Quine WVO, Word and Object, 1960 “One comes to philosophy already endowed with a stock of opinions. It is not the business of philosophy either to undermine or to justify these preexisting opinions, to any great extent, but only to try to discover ways of expanding them into an orderly system. … It is not to be demanded that a philosophical theory should agree with anything that the man on the street would insist on offhand, uninformed and therefore uninfluenced by any theoretical gains to be had by changing his mind .” Lewis DK, Counterfactuals , 1973 10 Framing paradigm shifts Framing critical data infrastructure, ontological, paradigm shifts Classical paradigm shifts (Kuhn/Butterfield) ‘Classical’ paradigm shifts follow the ontologising pattern look at the existing data in a radically new way by “placing them in a new system of relations with one another” “by giving them a different framework” one perceptive historian, viewing a classic case of a science’s reorientation by paradigm change, recently described it as “picking up the other end of the stick,” a process that involves “handling the same bundle of data as before, but placing them in a new system of relations with one another by giving them a different framework.” 8 Others who have noted this aspect of scientific advance have emphasized its similarity to a change in visual gestalt : the marks on paper that were first seen as a bird are now seen as an antelope, or vice versa. 8 Herbert Butterfield, The Origins of Modern Science, 1300-1800 (London, 1949), pp. 1-7. Kuhn, T. S. (1970). The structure of scientific revolutions. 12 Visualising the dynamics of the paradigm shift 13 Figure 7.1: Shifting views Partridge, C. (1996). Business Objects: Re-Engineering for Re-Use . Butterworth-Heinemann. “That parallel can be misleading. Scientists do not see something as something else; instead, they simply see it. ... In addition, the scientist does not preserve the gestalt subject’s freedom to switch back and forth between ways of seeing. Nevertheless, the switch of gestalt, particularly because it is today so familiar, is a useful elementary prototype for what occurs in full-scale paradigm shift.” Kuhn, T. S. (1970). The structure of scientific revolutions. Visualising the radical shift in the ontologies 14 Figure 1.3: Map of the semantic structure of the two views Partridge, C. (1996). Business Objects: Re-Engineering for Re-Use . Butterworth-Heinemann. If you look at the ‘things’ in the views’ ontologies, the shift is obvious Many fundamental examples 15 Galilei, G. Dialogues concerning two new sciences. (1954 [1632]). Galilean relativity Einstein, A. Relativity: The Special and General Theory (1920) Einsteinian relativity Train and embankment simultaneity thought experiment Ship, ball and fish relative motion thought experiment When does the new paradigm emerge? Is there a historical pattern? Kuhn notes that: in the history of science, work often starts on the new paradigm well before the current paradigm is seen as ‘in crisis’ Often a new paradigm emerges, at least in embryo, before a crisis has developed far or been explicitly recognized. Lavoisier’s work provides a case in point. His sealed note was deposited with the French Academy less than a year after the first thorough study of weight relations in the phlogiston theory and before Priestley’s publications had revealed the full extent of the crisis in pneumatic chemistry. Or again, Thomas Young’s first accounts of the wave theory of light appeared at a very early stage of a developing crisis in optics, one that would be almost unnoticeable except that, with no assistance from Young, it had grown to an international scientific scandal within a decade of the time he first wrote. In cases like these one can say only that a minor breakdown of the paradigm and the very first blurring of its rules for normal science were sufficient to induce in someone a new way of looking at the field. What intervened between the first sense of trouble and the recognition of an available alternate must have been largely unconscious. Kuhn, T. S. (1970). The structure of scientific revolutions 16 Surfacing the ‘geospatial coordinate reference systems’, ontological, paradigm shift How do we surface the shift? The basic process start with the infrastructure of the current ‘geospatial coordinate reference systems’ paradigm surface/identify the relevant objects in its ontology use ontological tools to ‘tidy’ them up surfacing/identifying the objects in the tidied ontology Divide the paradigm shift into more ‘digestible’ chunks divide the overall shift into two major sub-shifts, with a further sub-divisions physicalising the coordinate system gauge setting 18 Current geospatial coordinate reference systems paradigm The starting pre-shift paradigm structure Coordinates – essentially time invariant 20 ‘ ’ marks the position the coordinate labels. the label is a sequence of numbers – in this case, <51°31'18" N 0°9'19" W> (Full British National Grid <528082 181918>) The label marks the same position both now and tomorrow and yesterday. That is what makes the map useful. Essentially time invariant => essentially spatial As we shall see, this time invariance characterises the spatial The starting, pre-shift paradigm structure Current geospatial coordinate reference systems paradigm Mathematical spaces ℝ 3 ℝ 2 Features Space and time t 1 t 2 t 3 22 The starting paradigm abc <x, y, z> def <e, n> UoW point at t 1 UoW Point UoW point at t 2 UoW point at t 3 simplified datum realization map projection time slicing An operational perspective Geodetic Datum Features Geometry (Mathematical) Objects Space and time The starting paradigm: conceptual data model view 23 space-time universe Earth Earth time-slice at t 1 UoW point UoW point at t 1 Geometry Objects ℝ 3 Geometry Objects ℝ 2 Geometry Objects abc <x, y, z> def <e, n> UoW point at t 2 Earth time-slice at t 2 Earth Bound Points Earth Bound Points at Times Ellipsoids abc reference ellipsoid ℝ 3 Coordinates ℝ 2 Coordinates Features Feature time-slices ABC Realisations ABC-DEF Projections time slicing simplified See this division appearing in many standards 24 ISO/IEC 18026:2009: Information technology – Spatial Reference Model (SRM) 0.1 Purpose Spatial information processing requires a robust capability to describe geometric properties such as position, direction and distance. The starting paradigm: Euler diagram view 25 Features Geometry (Mathematical) Objects ℝ 3 ℝ 2 Space and time UoW point UoW point at t 1 abc <x, y, z> def <e, n> UoW point at t 2 space-time universe Earth Architectural theme: Different objects for different purposes in different distinct domains Domains are different in nature. Use domains/objects to separate the concerns. objects in different domains cannot touch or overlap one another objects in the same domain can touch or overlap one another As we shall see this then raises questions about how these domains are connected. simplified Shifting to the new paradigm Two stages in the paradigm shift Shifting to the new paradigm Two stages in the paradigm shift constructing a physical basis for the coordinate system adding the coordinate system gauge freedoms 27 Constructing a physical basis for the coordinate system Stage one: the first major sub-shift The first major stage of the shift In this stage, two strong motivations explaining the relation between spacetime and space providing a physicalisation of the mathematics (a common theme, as we shall see, in relativity) For ease of explanation, we divide the first stage into these more ‘digestible’ chunks space-time unifying physicalising coordinate mathematics physicalising coordinate system bases After the final shift, coordinate systems have a firm physical basis 29 Space-time unifying paradigm shift Unifying space and time into spacetime Mathematical spaces ℝ 3 ℝ 2 Space-time 31 Space-time unifying paradigm shift abc <x, y, z> def <e, n> UoW Worldline Introduce worldlines in space-time This unifies space and time into space-time: space and time (time-slices) disappear. Datum realization is now a direct mapping. simplified (4D) Earth datum realization map projection Geodetic Datum Space-time unifying paradigm shift: Euler diagram view 32 Features Worldlines space-time universe Earth Geometry Objects ℝ 3 Geometry Objects ℝ 2 Geometry Objects UoW worldline ℝ 3 Coordinates abc reference ellipsoid abc <x, y, z> ℝ 2 Coordinates def <e, n> Architectural theme: Use worldlines to unify the space and time elements – and so bring these into the space-time domain. Eliminates the need for (physically) separate space and time (and space-time) domains. (Can recover these as perspectives over space-time.) Explains how space, time and space-time can overlap and touch each other parts of the same physical world However, geometry is still distinct. simplified Physicalising coordinate mathematics paradigm shift Shifting from Platonism to Structuralism Mathematics – starting paradigm Platonist ontology The UoW coordinate <51.522, -0.155> is an abstract (Platonic) point in ℝ 2 . the geospatial literature seems to adopt a Platonist stance Platonism abstract mathematical objects exist outside time and space only one ℝ 3 and ℝ 2 space for each coordinate, <c 1 , …>, only one such coordinate. 34 Mathematics – alternative structuralist ontology However, this is not the only ontology for mathematical objects Another option is structuralism structuralism mathematics is about the structure of relations Where this structure can appear in multiple instances The UoW coordinate <51.522, -0.155> is a physical object (the worldline) in a structure that is isomorphic to (and so an instance of) ℝ 2 . this option enables us to physicalise the mathematical objects recognised by the starting ontology 35 simplified Space-time ℝ 3 Structure Perspective 36 Physicalising coordinate mathematics abc reference hyper-ellipsoid AKA Geodetic Datum UoW Worldline abc <x, y, z> ℝ 2 Structure Perspective abc <x, y, z> def <e, n> (4D) Earth instance: abc reference ellipsoid The ℝ 2 space’s atoms are the worldlines of the relevant surface of the reference ellipsoid these are a subset of the ‘coordinate’ worldlines. def <e, n> datum realization map projection Mathematics now provides the structuring framework, rather than the coordinate objects themselves. The reference ‘geodetic datum’ becomes an object in the 4D world in a mathematical structure, and Coordinate points become worldlines in a mathematical structure. Coordinates become a perspective over the worldlines, a way of ‘seeing’ them. Physicalising coordinate mathematics: Euler diagram view 37 Features Worldlines space-time universe Earth ABC ℝ 3 Objects abc reference ellipsoid ABC Worldlines DEF Worldlines UoW worldline simplified Architectural theme: Worldlines are the ‘atoms’ of the ℝ 3 and ℝ 2 structures. This both unifies and separates the mathematical coordinate structures with the physical world. Worldlines now unify the space-time and mathematical domains. Eliminates the need for (physically) separate domains. Can recover the separation through the use of worldline ‘atoms’. Explains how space-time and mathematical objects can overlap and touch each other parts of the same physical world Surfacing physical reference ellipsoid identity Different coordinate systems can share the ‘same’ worldline points. changing the unit of measure (unit conversion - e.g. kilometres to miles) same points, different coordinate labels. many transformations between coordinate systems preserve the coordinate points while changing the labels. examples shifting the axis order/direction shifting the origin Different reference ellipsoids can share the ‘same’ component worldline points so, are the same underlying physical thing The new paradigm recognises a physical reference ellipsoid one built from physical ‘coordinate’ worldlines Where the starting paradigm has multiple ellipsoids (with different coordinate labels) the new paradigm recognises a single physical (4D) ellipsoid which is the basis for multiple different coordinate labelling systems 38 Physicalising coordinate system bases paradigm shift Giving the whole coordinate system a physical basis Surfacing the reference frame’s physical basis The datum realisation typically creates a geodetic reference frame within this the reference ellipsoid is defined and used for map projections In the starting paradigm, the geodetic reference frame identity is linked to the way it is constructed – and so, the coordinate labels this generates. Physically, however, two geodetic reference frames can have the same physical worldlines as their physical basis the current infrastructure does not have the resources to characterise this physical basis how can we characterise it? 40 Objects’ relative spaces t 1 t 2 t 3 <x, y, z> time Objects trace out worldlines in spacetime these can be extended beyond their physical boundaries The full (extended) set of worldlines called (timelike) congruences do not intersect, and cover the space (at the times they exist) are the basis for Einsteinian reference frames called ‘relative spaces’ Norton, J. (1985). What was Einstein’s principle of equivalence? https://doi.org/10.1016/0039-3681(85)90002-0 In each relative space, each worldline can be uniquely labelled with a coordinate. Being in the same ‘spatial’ place is relative to a space of worldlines: remaining on the same ‘coordinate’ reference worldlines for all spatial points in the relative space 41 Worldlines: different reference worldframes t 1 t 2 t 3 f 2 f 1 f 2 worldframe t 1 t 2 t 3 f 2 f 1 f 1 worldframe Every object traces out worldlines relative to itself – creating a relative space. Galileo’s falling ball and stationary fish: have different relative spaces (reference worldframes) and so have different ‘same’ places These are sets of ‘mutually at rest’ worldlines: relative spaces or worldframes time Galilean relativity Galilei, G. (1954 [1632]). Dialogues concerning two new sciences . 42 Shift to relative spaces This shift introduces time-like congruences (or worldframes ) these are sets of worldlines that cover a region of space without intersecting coordinate points are a clear candidate for this. these time-like congruences are a basis for a spatial frame – a relative space This is not a new idea it was at the centre of Einstein’s relativistic insight every extended object traces out worldlines that can be extended into time-like congruences, which is a relativistic reference frame) objects that are relatively at rest, trace out the same relativistic reference frame the time-like congruence can be expanded into a relative space with the worldlines as atoms these relative spaces form the physical basis for coordinate systems each of these relative spaces can then be structured into an infinite variety of dependent coordinate reference systems one can divide space-time into three-dimensional space in an infinite number of ways. NB: Only interested in one aspect of relativity the recognition of relativistic reference frames and their relative spaces AKA spatial projections other aspects, such as space-time curvature, are independent Explaining why this (specific) idea is not in modern geographic information systems historically, geodesy’s major foundational development happened in the 19th century prior to relativity the (hidden) infrastructure may be archaic, but worked 43 Features Physicalising coordinate system bases: Euler diagram view 44 Worldlines ABC Relative Space ABC Relative Space Worldlines DEF Relative Space DEF Relative Space Worldlines UoW worldline Architectural theme: Relative spaces provide a single framework based upon worldlines. This enables a (worldline-based) physical basis for relative space-based reference frames Spatial – mereotopological – invariance The foundational component Spatial mereotopological relations between geometric objects There is a requirement to measure spatial mereotopological relations between ‘geometric’ objects this only makes exact sense in the same relative space 46 ISO. (2004). ISO 19125-1 Geographic information—Simple feature access—Part 1: Common architecture (No. ISO 19125-1:2004(E)). Spatial mereotopology The relative space is composed of the timelike congruent worldlines these define a spatial mereotopology one that is invariant over time this captures a coordinate systems’ time invariance which the dependent coordinate reference systems share each spatial mereotopology is a subset of the owning space-time's mereotopology the subset that is time invariant (in that relative space) this ensures mereological harmony Where two coordinate reference systems share the same physical basis they have the same coordinate points (worldlines) physically, they only differ by a coordinate relabelling 47 CRSs that share (or not) relative spaces 48 {5C22544A-7EE6-4342-B048-85BDC9FD1C3A} Family Plate/Global CRSs NAD83 North America NAD83(CORS96) → NAD83(NSRS2007) → NAD83(2011) ETRF Eurasian ETRF89 → ETRF93 → ETRF2000 → ETRF2014 GDA Australian GDA94 (epoch 1994.0) GDA2020 (epoch 2020.0) SIRGAS South American SIRGAS95 → SIRGAS2000 → SIRGAS2007 → SIRGAS2013 ITRF Global ITRF2014: Epochs 2005.0, 2010.0, 2015.0, 2020.0 Understandably, CRSs that share the same plate are more likely to share exactly the same physical basis. {5C22544A-7EE6-4342-B048-85BDC9FD1C3A} CRS 1 CRS 2 Difference ITRF2008 ITRF2014 different velocity fields NAD83(2011) NAD83(PA11) different plate (new Euler pole) More generally, adding in a regional deformation model with intraplate strain changes the relative space Similarly, Static CRSs (such as ED50, NAD27) cannot have the same relative space as dynamic ones Spatial object Spatial object is an object in the relative space it is composed of the relative space’s worldlines it is spatial relative to that space but not relative to other spaces The relative space’s worldlines fix the physical ‘position’ of the coordinates and of the spatial objects built from the worldlines 49 Adding the coordinate system gauge freedoms Stage Two: The second major sub-shift Gauge freedom Gauge freedom (in, for example, relativity) is the freedom to change the ‘mathematical description’ (the "gauge") of a physical system without altering the actual, measurable physics allowing for different but physically equivalent representations In the domain of coordinate systems: the timelike congruence fixes the coordinate reference systems’ physical structure – it’s physics the gauge freedoms are the rest of the components used to construct the coordinate reference system, things like: metric frame (including axes, etc.) 51 Physicalising coordinate system spatial metrics paradigm shift A coordinate gauge choice Coordinate system spatial metric gauge freedom We know that: different maps of the same territory can have different metrics this is the first level of coordinate metric gauge freedom the same map (with the same metric) can be read at different scales this is the second level of coordinate metric gauge freedom different scales (different units of measure) give different coordinate values 53 Requirement for spatial analysis 54 ISO. (2004). ISO 19125-1 Geographic information—Simple feature access—Part 1: Common architecture (No. ISO 19125-1:2004(E)). Shift to ‘spatial’ lines – distances – as objects Spatial lines are worldsheets composed of worldlines from the same relative space – so inherit time invariance. Spatial distance fixed as a set of spatial lines (curves) that are equidistant. rigid rod: stays on the same reference worldlines rigid rod: stays on the same reference worldlines line’s ‘length’ fixed at all times by the worldlines line’s component worldlines 55 Gauge choice: coordinate system spatial metric fixing what it means to be spatial worldlines relative spaces metric scale fixing distances spatial metric spatial scale fixing spatial distances Spatial metrics, and so distance is fixed within a relative space Gauge freedom to fix distances as required Multiple coordinate systems can use the same gauge choice Spatial analysis that involves distance (and area) only make sense in the same (instance of) the metric space 56 Gauge physicalising coordinate frames paradigm shift Axes schemes: one of the other gauge choices Another gauge choice: axes schemes {5C22544A-7EE6-4342-B048-85BDC9FD1C3A} Coordinate A xes System Surface Types Cartesian 3 × planes Cylindrical cylinder, half-plane and plane Spherical sphere, cone and half-plane The coordinate reference system needs an axis scheme Gauge choice offers multiple options Given the shifts so far, ontological analysis reveals the axis schemes as sets of co-oriented surfaces Cartesian Cylindrical Spherical Three common coordinate (point-labelling) axes schemes 58 Visualising the co-oriented surface objects Coordinate reference systems characterised in terms of surface types – which intersect at points intersecting surfaces uniquely identify a point Cartesian Cylindrical Spherical Each surface type is an object in the structuralist mathematical space Which has a corresponding spatial object in the coordinate reference system’s relative space Where co-oriented surfaces are constructed from worldlines (from the same reference space) 59 Summary Geospatial coordinate reference systems data infrastructure Taking an ontological look at the infrastructure foundations revealed implicit structure and opportunities for improvement “making explicit what had been tacit, and precise what had been vague; …, smoothing kinks, lopping off vestigial growths, clearing ontological slums” (Quine) lead to a substantial paradigm shift a new way of looking at the domain, with different objects reflecting where our understanding has changed More broadly, Spatial structures are relative to a spatial framework Pattern common to spatial breakdown structures 61 Paradigm shift: (partial) object view 62 Features Worldlines ABC Relative Space ABC Relative Space Worldlines DEF Relative Space DEF Relative Space Worldlines UoW worldline Space-time Geometry (Mathematical) Objects ℝ 3 ℝ 2 Space and time UoW point UoW point at t 1 abc <x, y, z> def <e, n> UoW point at t 2 BEFORE AFTER multiple objects one object, multiple roles space & time Paradigm shift: topological and metric granularity view 63 space-time 3D relative space spatial metric (scale) 2D relative space (reference ellipsoid surface) ( ℝ 2 ) 2D Coordinate Reference System ( ℝ 3 ) 3D Coordinate Reference System ( ℝ 2 ) 2D Coordinate Reference System time (mereotopology) invariant spatial objects distance invariant spatial objects distance invariant spatial objects ( ℝ 3 ) 3D Coordinate Reference System spatial metric (scale) time/distance invariant geometric ℝ 3 objects space-time original (operational) paradigm time/distance invariant geometric ℝ 2 objects new spatial paradigm BEFORE AFTER datum realization map projection clear picture of what a spatial object is and where it appears clear picture of invariant metric spatial properties Opportunities to radically improve archaic critical data infrastructure Infrastructure is often not maintained or modernised well enough, leading to a deteriorating condition that both increases risks and reduces opportunities for exploitation and improvement this is as true for data as physical infrastructure As you have seen, geospatial coordinate reference systems data structures are an example multiple other examples of critical data infrastructure that needs file system management systems data structures network management systems data structures cyber management system data structures In cases like these, ontologically driven data paradigm shifts are a cost-effective way of modernising the data infrastructure foundations 64 Some related prior papers Partridge, C. (2011). An Information Model for Geospatial and Temporal References. https://www.academia.edu/39988229 Partridge, C. (2013). Geospatial and Temporal Reference – A Case Study Illustrating (Radical) Refactoring. ONTOBRAS-2013 6th Ontology Research Seminar in Brazil. https://www.academia.edu/27433806/ Partridge, C., Mitchell, A., Loneragan, M., Atkinson, H., de Cesare, S., & Khan, M. (2019). Coordinate Systems: Level Ascending Ontological Options. 2019 ACM/IEEE 22nd International Conference on Model Driven Engineering Languages and Systems Companion (MODELS-C), 78–87. https://www.academia.edu/40354620 Partridge, C., Mitchell, A., Loneragan, M., Atkinson, H., de Cesare, S., & Khan, M. A. (2020). The Fantastic Combinations and Permutations of Coordinate Systems’ Characterising Options The Game of Constructional Ontology. https://www.academia.edu/118060077/ 65 Questions 66 67 questions 68 Features Worldlines Geometry Objects Space-time unifying paradigm shift: conceptual data model view 69 Features Worldlines UoW worldline space-time universe Earth Geometry Objects ℝ 3 Geometry Objects ℝ 2 Geometry Objects abc <x, y, z> def <e, n> Ellipsoids abc reference ellipsoid ℝ 3 Coordinates ℝ 2 Coordinates ABC Realisations ABC_DEF Projections simplified Features Worldlines Physicalising coordinate mathematics: conceptual data model view 70 ABC Worldlines Features Worldlines UoW worldline space-time universe Earth ABC Ellipsoids abc reference ellipsoid ℝ 3 Objects ABC ℝ 3 Objects ℝ 2 Objects DEF ℝ 2 Objects DEF Worldlines ABC ℝ 3 Coordinates DEF ℝ 2 Coordinates abc <x, y, z> def <e, n> simplified Features Physicalising coordinate system bases: conceptual data model view 71 ABC Relative Space Worldlines Features Worldlines UoW worldline ℝ 3 Objects ABC Relative Space ℝ 2 Objects DEF Relative Space DEF Relative Space Worldlines ABC ℝ 3 Coordinates DEF ℝ 2 Coordinates abc <x, y, z> def <e, n> Relative Spaces ABC ℝ 3 Objects Relative Space Worldlines DEF ℝ 2 Objects ℝ 3 Structure ℝ 2 Structure ABC Coordinate System DEF Coordinate System ABC Coordinate System ℝ 3 Structure DEF Coordinate System ℝ 2 Structure Features Powerset