Open Geospatial Consortium 
Submission Date: 20190501 
Approval Date: 20201010 
Publication Date: 20201022 
External identifier of this OGC® document: http://www.opengis.net/doc/AS/2Dtiles/1.0 
Internal reference number of this OGC® document: 19014r3 
Version: 1.0 
Category: OGC® Abstract Specification 
Editor: Carl Reed, PhD 
OGC Abstract Specification Topic 22  Core Tiling Conceptual and Logical Models for 2D Euclidean Space 
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Copyright © 2020 Open Geospatial Consortium 
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Document type: OGC® Abstract Specification 
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Document stage: Approved 
Document language: English 
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 1. Scope
 2. Conformance
 3. References
 4. Terms and Definitions
 5. Conventions
 6. Introduction
 7. Part 1: Conceptual Model for tiling any space
 8. Part 2: Logical Model for 2D Euclidean Space Tiles and Tiling: Normative
 Annex A: Conformance Class Abstract Test Suite (Normative)
 9. Annex B: Additional Informative Material: Simple Example of an XML encoding
 Annex B: Revision History
 Annex C: Bibliography
i. Abstract
This OGC Abstract Specification (AS) defines:

A conceptual model for tiling space in any dimension and;

A logical model for 2D tiled structures and by extension tiling. The logical model is based on the conceptual model.
The conceptual model specified in this Abstract Specification could be a subclass in a more comprehensive Spatial Partitioning Conceptual Model. Additional Parts may be added to this AS for other dimensions, such as 3D, or other uses cases.
ii. Keywords
The following are keywords to be used by search engines and document catalogues.
ogcdoc, OGC document, tiles, tiling, 2D, tessellation, tile set
iii. Preface
Numerous OGC standards specify some type of tiling scheme. These standards include Web Map Tiling Service (WMTS), CDB, and GeoPackage. There have also been a number of OGC Innovation Initiatives focused on tiling and tiling schemes. However, a general tiling conceptual model and related logical model for 2D Euclidean space have not been defined. This Abstract Specification defines a tiling conceptual model and a logical model for tiling 2D planar space that is based on a tiling conceptual model.
Attention is drawn to the possibility that some of the elements of this document may be the subject of patent rights. The Open Geospatial Consortium shall not be held responsible for identifying any or all such patent rights.
Recipients of this document are requested to submit, with their comments, notification of any relevant patent claims or other intellectual property rights of which they may be aware that might be infringed by any implementation of the standard set forth in this document, and to provide supporting documentation.
iv. Submitting organizations
The following organizations submitted this Document to the Open Geospatial Consortium (OGC):

CAE

Carl Reed & Associates

Esri

UK Met Office
v. Submitters/Supporters
All questions regarding this submission should be directed to the editor or the supporters:
Name  Affiliation 

Carl Reed 
Carl Reed & Associates 
CAE 

Chris Little 
UK Met Office 
Keith Ryden 
Esri 
1. Scope
1.1. What this Abstract Specification is
This OGC Abstract Specification consists of two Parts: A General Tiling Conceptual Model and, based on the Conceptual Model, a Logical Model for the regular tessellation (tiling) of 2D Euclidean Space.
The partitioning of 2D Euclidean space into physical tiles for the storage and visualization use cases is common in geospatial technology. Examples are implementations of the OGC CDB Standard, the OGC Two Dimensional Tile Matrix Set Standard, and the OGC GeoPackage Standard. However, there are also common elements and/or semantics for any approach to partitioning space in any dimension. As such, the logical model described in this document defines a set of common required elements and then follows with more specific requirements for the 2D Euclidean space case.
Part 1 of the Abstract Specification describes a general conceptual model for tiling space. The conceptual model is applicable to any dimension. The mode introduces three key classes: Tile and Tileset and TileSetMetadata that describes the properties of a TileSet. The conceptual model makes no assumptions regarding content, use cases, implementation scenarios, or how the space is to be tessellated (tiled). The conceptual model is abstract and cannot be implemented as is.
Part 2 of this Abstract Specification defines a detailed logical model for the tessellation of 2D Euclidean Space. One or more logical models are required to provide the requirements and structure necessary for implementation. Therefore, in addition to the conceptual model, this Abstract Specification also specifies a core logical model for the regular tessellation of 2D planar (Euclidean) use case.
Other Parts that specify additional logical models, such as for 3D Euclidean space, may be added in the future.
1.2. What this Abstract Specification is not
This Abstract Specification does not:

Specify the content that could be organized in a tiled structure;

Address concepts such as styling, levels of detail, attributes, and levels;

Make any suggestions regarding how content will be processed and stored in the tiled structure;

Provide guidance on formats, encodings, or any other implementation details.
Note

In this Abstract Specification, “tile” is NOT a packaged blob of data to download in a chunky streaming optimization scheme! This is a general misconception based on implementation specific requirements. Other OGC standards such as Tile Matrix Set (TMS) Standard provide requirements and details on how to organize and access tiled blobs of geographic data. 
The above concepts and implementation guidance is defined in profiles, extensions, and profiles with extensions based on this Abstract Specification. Examples of this type of guidance is the OGC Two Dimensional Tile Matrix Set Standard, the OGC I3S Community Standard, and the OGC CDB standard.
2. Conformance
This Abstract Specification currently defines two conformance classes for the 2D Euclidean Space use case. Additional conformance classes may be added as additional tiling models are identified.
2.1. Standardization Target for 2D Eucidean Space Logical Model
The standardization target for the 2D Euclidean use case are as follows.

Tile Set Scheme Requirement Class ((http://www.opengis.net/spec/2dtilemodel/1.0/req/tileset). The target is a service, a client, or an encoding needing to create/access a TileSet or to declare conformity to the tileSet conformance class.

Tile Requirement Class (http://www.opengis.net/spec/2dtilemodel/1.0/req/tile). The target is a service, a client, or an encoding needing to create/access a tileSet or to declare conformity to the Tile conformance class.
Conformance with this standard shall be checked using all the relevant tests specified in Annex A (normative) of this document. The framework, concepts, and methodology for testing, and the criteria to be achieved to claim conformance are specified in the OGC Compliance Testing Policies and Procedures and the OGC Compliance Testing web site.
All requirementsclasses and conformanceclasses described in this document are owned by the standard(s) identified.
3. References
The following normative documents contain provisions that, through reference in this text, constitute provisions of this document. For dated references, subsequent amendments to, or revisions of, any of these publications do not apply. For undated references, the latest edition of the normative document referred to applies.

CEN: ENV 1613:1995 Medical informatics  Messages for exchange of laboratory information, 1995

ISO/TC 211: ISO 19106:2004 Geographic information — Profiles, 2004 https://www.iso.org/standard/26011.html

ISO/OGC: ISO 19111 and OGC 18005r4 OGC Abstract Specification Topic 2: Referencing by coordinates, 2019 http://docs.opengeospatial.org/as/18005r4/18005r4.html

ISO/TC 211: ISO 191151:2014 Geographic information — Metadata — Part 1: Fundamentals, 2014 https://www.iso.org/standard/53798.html

ISO/TC 211: ISO/WD 191231 Geographic information — Schema for coverage geometry and functions — Part 1: Fundamentals, 2005 https://www.iso.org/standard/70743.html

OGC: OGC 07057r7, OGC Web Map Tile Service Implementation Standard, 2010 http://portal.opengeospatial.org/files/?artifact_id=35326

OGC: OGC 15113r5, Volume 1: CDB Core Standard: Model and Physical Data Store Structure, 2018 https://www.opengeospatial.org/standards/cdb

OGC: OGC 15104r5, Topic 21: Discrete Global Grid Systems Abstract Specification, 2017 http://docs.opengeospatial.org/as/15104r5/15104r5.html

OGC: OGC 12128r15, GeoPackage Encoding Standard  with Corrigendum, 2018 https://www.opengeospatial.org/standards/geopackage

OGC: OGC 05020r27 Technical Committee Policies and Procedures, 2019 http://docs.opengeospatial.org/pol/05020r27/05020r27.html

OGC: OGC 17083r2, Two Dimensional Tile Matrix Set Standard, 2019 https://www.opengeospatial.org/standards/tms

OGC: OGC 06121r9, Web Services Common Encoding Standard, 2010 https://www.opengeospatial.org/standards/common
4. Terms and Definitions
This document uses the terms defined in Subclause 5.3 of [OGC 06121r8], which is based on the ISO/IEC Directives, Part 2, Rules for the structure and drafting of International Standards. In particular, the word “shall” (not “must”) is the verb form used to indicate a requirement to be strictly followed to conform to this standard.
For the purposes of this document, the following additional terms and definitions apply.
4.1
conceptual model
description of common concepts and their relationships, particularly in order to facilitate exchange of information between parties within a specific domain. A conceptual model is explicitly chosen to be independent of design or implementation concerns.
(SOURCE: CEN ENV 1613:1995)
4.2
coordinate reference system
coordinate system that is related to the real world by a datum
(SOURCE: ISO 19111)
4.3
coordinate system
set of mathematical rules for specifying how coordinates are to be assigned to points
(SOURCE: ISO 19111)
4.4
equilateral triangle
triangle in which all three sides are equal in length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular: That is, all three internal angles are also equal in size and are each 60°.
4.5
extension
set of one or more conformance clauses that defines new options for an existing standard. Extensions are generally designed to add additional capabilities that are not part of the core standard. An example is OGC GeoPackage Extension for Tiled Gridded Coverage Data.
(SOURCE: OGC Technical Committee policies and Procedures)
4.6
hexagon
sixsided polygon or 6gon. The total of the internal angles of any simple (nonselfintersecting) hexagon is 720°.
Note 1 to entry: A regular hexagon is defined as a hexagon that is both equilateral and equiangular. It is bicentric, meaning that it is both cyclic (has a circumscribed circle) and tangential (has an inscribed circle). All internal angles are 120 degrees.
4.7
periodic tiling
tiling that repeats itself at regular intervals. If a region of the tiling can be outlined with a parallelogram and then tile the rest of the plane by translating that parallelogram (rotations and reflections are not allowed), then the resulting tiling is periodic. Examples of periodic tilings include regular tessellations, tilings that use only congruent regular polygons, such as regular hexagons.
4.8
profile
proper subset of an existing standard including restrictions on or deletions of conformance clauses related to the subsetting. An example of a profile is the GML Simple Feature Profile.
(SOURCE: ISO 19106:2004, Type 1 Profile}
4.9
profile with extension(s)
set of one or more conformance clauses from a base standard that includes at least one new conformance clause (extension).
EXAMPLE: OpenGIS® Web Map Services  Profile for EO Products.
(SOURCE: OGC Technical Committee policies and Procedures)
4.10
space partitioning
process of dividing a geometric space (usually a Euclidean space) into two or more disjoint subsets (see also partition of a set). Space partitioning divides a space into nonoverlapping regions. Any point in the space can then be identified to lie in exactly one of the regions.
(SOURCE: Wikipedia, August 2020)
4.11
square
A regular quadrilateral with four equal sides and four 90 degree angles.
(SOURCE: Wikipedia, August 2020)
4.12
tessellation
partitioning of a space into a set of conterminous subspaces having the same dimension as the space being partitioned
(SOURCE: ISO 19123)
Note 1 to entry: In mathematics, tessellations can be generalized to higher dimensions and a variety of geometries. Tessellations may be regular, semiregular, or irregular. Regular tessellations are made up entirely of congruent regular polygons all meeting vertex to vertex, and the arrangement of the polygons around every vertex is the same. There are only three regular tessellations. These consist of a network of equilateral triangles, squares or hexagons. Irregular tessellations consist of figures that are not composed of regular polygons but still interlock without gaps or overlaps. The pattern of the common brick laying bond is also not a regular tessellation, as some vertices are along the sides of the polygons. Tessellations with more than one kind of polygon are also not regular.
4.13
tile
geometric shape with known properties that may or may not be the result of a tiling (tessellation) process. A tile consists of a single connected "piece" (topological disc) without "holes" or "lines".
Note 1 to entry: In the above definition, the term hole
means that any given tile in a tileset cannot have subtiles or exclusion areas. The use of the term `hole' in the above definition should not be confused with exclusion areas in a polygon geometry (aka donuts or islands).
Note 2 to entry: In the unstructured case (GeneralTileSet), tiles may or may not overlap and have gaps between tiles.
4.14
tile scheme
scheme that defines the unique properties of each individual tile in a tile set. These properties include a unique identifier for each tile, the tile origin, and the extent of the tile.
4.15
tiling
in mathematics, a tiling (tessellation) is a collection of subsets of the space being tiled, i.e. tiles that cover the space without gaps or overlaps.
4.16
tile set
set of tiles  a collection of subsets of the space being partitioned.
Note 1 to entry: A tile set may be 1.) structured with known, common properties that meets the definition of a tessellation or 2.) unstructured in that there is not a common set of properties and/or the result of a tessellation.
4.17
tile set metadata
additional metadata beyond the common properties defining the tile set. Such metadata could be an abstract, the owner, the author, or other common metadata.
Note 1 to entry: This may be essential information about a tile set needed to support users in their efforts to discover, select and modify tile sets.
4.18
tile set scheme
scheme that defines how space is partitioned into individual tiles. The scheme defines the spatial reference system, the geometric properties of tiles in a tile set, which space a uniquely identified tile occupies, and reversely which unique identifier corresponds to a space satisfying the geometric properties to be a tile.
Note 1 to entry: While in the general sense, a tiling scheme is not restricted to a coordinate reference system or a tile matrix set and allows for other spatial reference systems such as DGGS and other organizations including irregular ones, the logical model in this abstract specification focuses on the 2D case.
5. Conventions
This Abstract Specification defines conceptual and logical models that specify mandatory and recommended requirements for tiles, tilesets, and tiling of a 2D Euclidean plane. This Abstract Specification does not mandate any particular coding or implementation pattern (e.g., XML, JSON)
5.1. Abbreviations

2D 2dimensional

3D 3dimensional

CDB Abbreviated reference for the OGC CDB Standard.

CRS Coordinate Reference System

DGGS Discrete Global Grid System

EPSG European Petroleum Survey Group

MVT MapBox Vector Tiles

OWS OGC Web Services

PNG Portable Network Graphics

TMS Tile Matrix Set

WMTS Web Map Tiling Service

XML eXtensible Markup Language
6. Introduction
For many geospatial technology applications and use cases, there is the need to partition the space of interest. Examples are numerous. Within the OGC standards baseline, CDB defines how to partition the globe into a set of 2D tiles for 24 levels of detail. The Web Map Tile Service (WMTS) standard defines the rules and requirements for a tile matrix set as a way to index space based on a set of regular grids defining a domain (tile matrix) for a limited list of scales in a restricted set of Coordinate Reference Systems (CRS). The new  and more general  Tile Matrix Set standard specifies the concept of a tile matrix set and tile matrix set limits and its implementation in 2D space. Finally, the OGC I3S Community Standard defines how to partition 3D space using a hierarchical, nodebased spatial index structure in which each node’s payload may contain features with associated geometry, textures and attributes.
In geometry, space partitioning is the process of dividing a space (usually a Euclidean space) into two or more disjoint subsets. Space partitioning divides a space into nonoverlapping regions. Any point in the space can then be identified to lie in exactly one of the regions.
Within that context, the requirements for partitioning space based on dimension can be considered. This OGC Abstract Specification presents a conceptual and logical model for partitioning (tiling) 2D Euclidean space. Tiling of 2D Euclidean space is the most commonly known approach to partitioning space. However, there are common concepts for any approach to partitioning space in any dimension. This document defines the common concepts that apply to any dimension and then follows with mandatory and recommended requirements for the tiling of 2d Euclidean space use case.
While developing the conceptual model and the associated logical model, the tiling sections/extensions to the OGC CDB, GeoPackage, and WMTS standards along with the commercial MapBox Vector Tiles (MVT) specification were reviewed. The OGC DGGS Abstract Specification and various Engineering Reports from OGC Test Bed 13, Test Bed 14 and the Vector Tiles Pilot were also considered. Finally, the new Tile Matrix Set Standard was reviewed. Much of the content and requirements specified in this abstract specification are derived by abstracting what is contained in the existing and candidate standards and specifications.
The goal is to define a simple conceptual model that can support any and all requirements for tiled data stores and applications including extensions for visualization, portrayal, analytics, filtering, levels of detail, and so forth. Various application use cases and/or workflows, such as styling or dealing with topology, are not part of the core logical model. Instead, such applications may be thought of as profiles and/or extensions of the model that restrict core requirements or define additional requirements. For example, the majority of the CDB standard that deals with tiles defines additional requirements, such as level of detail, to optimize the data store for that domain of interest.
6.1. Key concept  The assumption of a 2D planar space
The tiling logical model defined in this document assumes that the tiling target is a two dimensional plane in Euclidean space. However, the content to be stored in a tiled structure is typically earth (or planetary body) referenced. Therefore, the transformation from a spheroidal coordinate system into a planar system is required. The following paragraph, extracted from the Two Dimensional Tile Matrix Set Standard (TMS), describes the solution for coordinate reference systems other than geographic (Latitude, Longitude).
As stated in OGC 08015r6 Abstract Specification Topic 2: Spatial referencing by coordinates, a coordinate system is a set of mathematical rules for specifying how coordinates are to be assigned to points in space. A Coordinate Reference System (CRS) is a coordinate system that is related to the real world by a reference datum. An example of mathematical rules is the application of a sphere or an ellipsoid centered in the datum and the use of a projection to transform the sphere or the ellipsoid into a planar representation of the world.
6.2. Two key use cases
There are two distinct but related use cases for implementing and using a tiled structure: storage and visualization. In both cases, the driving force is the desire to enhance performance. Performance could be in terms of access latency, caching, and/or streaming of relevant content to a client.
6.2.1. Storage Use Case
In this use case, geospatial content is subdivided into small units called tiles. The intent is to significantly enhance such operations as search, update, and presentation of the source content in its "native" form. In this use case, the original geospatial data (raster, grid, vector, point clouds, etc.) is maintained. There is no special processing to create forms of the data specifically for high speed caching and rendering on the client. Such processing is typically required for the Visualization use case and is described in the next clause.
A very typical storage use case would be a city defining the tile structure for its spatial data store. They could begin by defining the geographic extent of the entire area of interest and could include a fringe area for future growth. Typically in the United States, cities and counties work in State Plane coordinates, such as Colorado North zone (NAD 83) for the City of Ft. Collins. The parameters are available in the EPSG database (EPSG 2876). The units of measure are feet. They pick a tile set origin that is the same for all tile sets in the data store. Typically, GIS practitioners think of origins being in the lower left corner so that option is selected for both the tile set and for each tile in a tile set. They then choose a regular square tessellation. They wish to create three tile sets with unique tileset IDs of Parcels, Streams, and Roads. Each tile set will have a different tile size. Parcels will have a tile width and height of 1000 feet. Streams will have a tile width and height of 5000 feet. Roads will have a tile width and height of 2000 feet.
Figure1. Example tile set for above parcel map use case
6.2.2. Visualization Use Case
The Visualization use case is focused on providing high speed geospatial content rendering and visualization capabilities on a client. In this use case, the source geospatial data is heavily processed, restructured, and reformatted to be optimized for visualization on one or more clients. For example, source vector data that is topologically structured and semantically rich may be processed into a tile matrix of PNG images and all geometry, topology, and semantics are lost.
Figure2. Example tile matrix for visualization
For example, the OGC Tile Matrix Set standard specifies rules for defining a tile matrix. From that standard:
For the case of a two dimensional space, given the top left point of the tile matrix in CRS coordinates (tileMatrixMinX, tileMatrixMaxY), the width and height of the tile matrix in tile units (matrixWidth, matrixHeight), the width and height of a tile (tileWidth, tileHeight) in original grid cells (often referred to as pixels), the coefficient to convert the coordinate reference system (CRS) units into meters (metersPerUnit) and the scale (1:scaleDenominator), the bottom right corner of the bounding box of a tile matrix (tileMatrixMaxX, tileMatrixMinY) can be calculated as follows: . . .
The TMS model is entirely consistent with the conceptual and logical models defined in this abstract specification – although the property names are different. The OGC TMS can be thought of as a profile with extensions of this abstract specification.
6.3. Characteristics of a Conceptual Model
The terms and definitions clause in this Abstract Specification provides a short definition for "conceptual model". This clause provides additional information on the OGC use of "conceptual model".
A conceptual model organizes the vocabulary needed to communicate consistently and thoroughly about the knowhow of a problem domain. The aim of a conceptual model is to express the meaning of terms and concepts used by domain experts to discuss the problem, and to find the correct relationships between different concepts.
A conceptual model:

is a representation of a system, made of the composition of concepts which are used to help people know, understand, or simulate a subject the model represents. A documented conceptual model represents 'concepts' (entities), the relationships between them, and a vocabulary;

is explicitly defined to be independent of design or implementation concerns;

organizes the vocabulary needed to communicate consistently and thoroughly about the knowhow of a problem domain;

starts with a glossary of terms and definitions. There is a very high premium on highquality, designindependent definitions, free of data or implementation biases; the model also emphasizes rich vocabulary; and

is always about identifying the correct choice of terms to use in communications, including statements of rules and requirements, especially where high precision and subtle distinctions need to be made. The core concepts of a geospatial problem domain are typically quite stable over time.
6.4. Logical Model
A logical data model or logical schema is a data model of a specific problem domain expressed independently of a particular database management product or storage technology (physical data model) but in terms of data structures such as relational tables and columns, objectoriented classes, or XML tags. This is as opposed to a conceptual data model, which describes the semantics of an organization without reference to technology.
Logical data models represent the abstract structure of a domain of information. They are often diagrammatic in nature and are most typically used in business processes that seek to capture things of importance to an organization and how they relate to one another. Once validated and approved, the logical data model can become the basis of a physical data model and form the design of a database.
Logical data models should be based on the structures identified in a preceding conceptual data model, since this describes the semantics of the information context, which the logical model should also reflect. Even so, since the logical data model anticipates implementation on a specific computing system, the content of the logical data model is adjusted to achieve certain efficiencies.
6.4.1. Conceptual Model vs. Logical (Data) Model
A conceptual model differs from a logical model in important ways cite:[Gupta2012]. The goal of a conceptual model is to support the expression of naturallanguage statements, and supply their semantics — not unify, codify (and sometimes simplify) data. Therefore the vocabulary included in a conceptual model is far richer, as suits knowledgeintensive problem domains. In short, conceptual models are conceptcentric; logical models are thingentityorclasscentric.
Logical models can usually be rather easily derived from conceptual models; the reverse is much harder (or impossible). Like logical models, conceptual models are often rendered graphically, but free of such distractions to stakeholders such as cardinalities.
7. Part 1: Conceptual Model for tiling any space
Part 1
of this Abstract Specification describes a general tiling conceptual model. The conceptual model is applicable to any dimension. The conceptual model makes no assumptions regarding content, use cases, implementation scenarios, or how the space is to be tessellated (tiled). The conceptual model is abstract and cannot be implemented as is.
Part 2
of this Abstract Specification defines a detailed logical model for the periodic tessellation of 2D Euclidean Space.
Additional Parts to this Abstract Specification could define one or more logical models for other use cases or dimensions, such as 3D Euclidean space.
7.1. General Tile/Tiling Conceptual Model
Figure 3 (below) portrays the core conceptual model for a space paritioning based on tiles. This conceptual model can be applied to spatial data independent of any space/time dimension.
Figure3. Conceptual for space partitioning based on tiles.
Figure 3 captures the concepts for defining and implementing a tiled structure in any space. While these terms are normatively defined in the Terms and Definitions clause, the following provides additional detail for the concepts depicted in the figure.
*GeneralTileSet
: Incorporates the concepts that tiles or collections of tiles may be structured or unstructured, may or may not have known properties, and may or may not be the result of a tessellation. Hence, at a fundamental level a general tile set is simply a collection of tiles.
Note

The logical model for the 2D Euclidean space use case defined in this Abstract Specification restricts this concept to require a tessellation with known properties. An additional Part to this Abstract Specification would need to be defined to support a general tile set use case. 
Tile
: The fundamental unit for partitioning space. A Tile
is a geometric shape with known properties that may or may not be the result of the tiling (Tessellation
) of the space defined as by the TileSetScheme
. A Tile
consists of a single connected "piece" without "holes" or "lines" (topological disc). In two dimensional space, a Tile
could be regular (e.g. square) or irregular (e.g. Thiessen polygon). For tiles in 3D Euclidean space, see the note below. Specific properties of a Tile
, such as tile width and height, are specified in a logical model.
TileSet
: A set of tiles with common properties that meets the definition of the TileSetScheme
and may or may not have been tiled based on the tessellation rules. The common properties of a TileSet
are specified in the TileSetScheme
.
Note

The concept TileSet is bounded by two invariant constraints: The coverage and homogeneity constraints. The homogeneity constraint specifies that all tiles in a given TileSet are related to and are the result of the same tessellation. The coverage constraint states that tiles within a TileSet do not overlap and completely cover the identified space. 
TileSetScheme
: A scheme that specifies a set of common properties that defines the TileSet
. The scheme could define the spatial reference system, the geometric properties of a tile, which space a uniquely identified tile occupies, and conversely which unique identifier corresponds to a space satisfying the geometric properties to be a tile. The properties required for a tiling approach that can be implemented are specified in a logical model.
Tessellation
: In addition to the definition provided in the Terms and Definition, the Tessellation
concept includes the rules for tessellating space into a TileSet
. The specific allowed/required Tessellation
approaches are specified in the logical model. For the purposes of this Abstract Specification, tiling
is consider a synonym.
TileSetMetadata
: In addition to the common properties that define a tile set, additional metadata may be provided. Such metadata could be an abstract, the owner, the author, or other common metadata. The logical model specifies which metadata elements are recommended.
Note

Higher dimensions: This conceptual model is applicable to higher dimensions. For geospatial content, the ability to tessellate 3D Euclidean space is highly important. As such the rules for the 2D Euclidean space logical model defined in this Abstract Specification are applicable to the 3D use case. Specifically, in geometry, a honeycomb is a space filling or close packing of polyhedral or higherdimensional cells, so that there are no gaps. A 3dimensional uniform honeycomb is a honeycomb in 3space composed of uniform polyhedral cells, and having all vertices the same (i.e., the group of [isometries of 3space that preserve the tiling] is transitive on vertices). This is an example of the more general mathematical tiling or tessellation in any number of dimensions. 
Note

On tessellations: Honeycombs, tiled GIS data, bathroom floors and designs by artist M.C. Escher have something in common: They are composed of repeating patterns of the same shape without any overlaps or gaps. This type of pattern is called tiling, or tessellation. Tessellate originates from the Greek tesseres, which means "four." The first tilings were made from square tiles. The concept tessellation is particularly rich in mathematics, with ties to geometry, topology and group theory.

8. Part 2: Logical Model for 2D Euclidean Space Tiles and Tiling: Normative
Part 2 of this Abstract Specification defines the mandatory and recommended requirements of the core model for the tessellation (tiling) of 2D Euclidean space. The specified mandatory elements SHALL be implemented regardless and independent of the implementation platform, programming languages, encodings, or any other implementation specific requirements.
The following figure, based on the Conceptual Model diagram, shows the properties by class (concept) and the relationships between the classes.
8.1. Restrictions to definitions of key concepts for the tiling of 2D plane
Concept definitions provided in the Terms and Definitions clause and in the Conceptual Model are further restricted as follows for the 2D Euclidean use case. The key restrictions for the 2D use case are highlighted.
tile
: A geometric shape with known properties that is the result of the tiling (tessellation) of a 2D plane. A tile consists of a single connected "piece" without "holes" or "lines" (topological disc).
tile set
: A set of tiles with common properties that meets the definition of tile and are tiled based on the tessellation rules. In short, a collection of subsets of the plane. However, in the 2D Logical Model tiles in a tile set are nonoverlapping.
tessellation
: partitioning of a space into a set of conterminous subspaces having the same dimension as the space being partitioned. For the 2D Euclidean space model, the tiles and associated tile set are generated by a regular (see Note below) tessellation of a 2D plane.
tiling
: When tiling in 2D planar space, the tiles are periodic. More specifically, a tiling consists of an arrangement of shapes covering the plane without overlap or gaps. The simplest tilings are periodic. In mathematical language, a pattern that repeats in a regular way is called periodic. Periodic tiles consist of a metatiling of duplicated regions under translation. Three common periodic tilings used in GIS and clarified in the logical model for 2D tiling of Euclidean space are ones consisting of triangles, squares, or hexagons. Note: Therefore rectangular tiling is NOT supported by the logical model.
Note

The 2D Euclidean space logical model is restricted to regular tessellations of the plane: equilateral triangles, regular hexagons and squares. This 2D logical model DOES NOT address nonperiodic (aka aperiodic) tiling. This includes tiling based on rectangles. An Abstract Specification for rectangular tiling would be an additional Part to this Abstract Specification. 
8.3. Requirement Class Tile Set Scheme
The clause defines the common properties that describe a tile set. This set of common properties is the same for all tiles contained within a tile set. The requirement for each property is contained in the requirements class tileSetScheme
Element name: tileSetScheme
Requirements Class 

http://www.opengis.net/spec/2dtilemodel/1.0/req/tilesetscheme 

Target type 
Service, Client, Encoding 
Dependency 

Requirement 2 
http://www.opengis.net/spec/2dtilemodel/1.0/req/core/tileset/id 
Requirement 3 
http://www.opengis.net/spec/2dtilemodel/1.0/req/core/tileset/crs 
Requirement 4 
http://www.opengis.net/spec/2dtilemodel/1.0/req/core/tileset/uom 
Requirement 5 
http://www.opengis.net/spec/2dtilemodel/1.0/req/core/tileset/extent 
Requirement 6 
http://www.opengis.net/spec/2dtilemodel/1.0/req/core/tileset/origin 
Requirement 7 
http://www.opengis.net/spec/2dtilemodel/1.0/req/core/tileset/tessellation 
Each of the properties that defines a tile set is now described in more detail.
8.3.1. Requirement 2  Tile Set Identifier (ID)
Requirement 2 
/req/core/tileset/id 
Each tile set in a tiled data store shall have a unique identifier. This identifier could be a number or an alphanumeric string. For example, the ID could be CodeType, as an adaptation of MD_Identifier class ISO 19115. An example of an alphanumeric identifier might be "Parcels" or "Street Network".
Element name: tileSetIdentifier
8.3.2. Requirement 3  Tile Set Coordinate Reference System (CRS)
Requirement 3 
/req/core/tileset/crs A tile scheme for a tile set SHALL have a coordinate reference system (CRS) that is consistent for all tiles in a given tile set. 
A tile set scheme shall specify a coordinate reference system (CRS) that is the same for all tiles in a given tile set. The CRS for a given tile set could be based on an engineering datum or an earth centric datum. The CRS could be restricted to meet very specific requirements, say for visualization of prerendered tiles as defined in the WMTS or TMS standards. Further, at the core conceptual level, there is no requirement that all CRSs be the same for all tile sets in a tiled data store.
Having the same CRS for all tile sets in a data store is a requirement that would be specified in a profile. For example, CDB specifies WGS 84 (EPSG4326) for all tiles.
Element name: tileSetCRS
8.3.3. Requirement 4  Tile Set Units of Measure (UoM)
Requirement 4 
/req/core/tileset/uom A tile scheme for a tile set shall have a units of measure that is consistent for all tiles in a given tile set. 
A tile set scheme shall specify a unit of measure (UoM) for a tile set, such as meters or feet, that is consistent for any given tile set. The UoM may be specified in the CRS definition or may be different from the CRS definition. At the core conceptual level, there is no requirement that all UoMs be the same for all tile sets in a tiled data store. Having a consistent UoM across all tile sets is a requirement that would be specified in a profile. (Example: “tileSetUoM=ft”)
Element name: tileSetUoM
8.3.4. Requirement 5  Tile Set Extent
Requirement 5 
/req/core/tileset/extent Each tile set SHALL have an extent in the TileSet coordinate reference system (CRS). The extent SHALL be expressed following the rules as specified in Clause 10.2.1 Basic bounding box parameters (OWS Common). 
In any tiling scheme, there are two extents to be considered. The extent of an individual tile and the extent of the tile set. This requirement is for defining the extent of the tile set.
Note: If there is one tile in a tile set, the extents are identical. In this tiling model there is no requirement, such as in DGGS, for a tile or tile set to cover the entire globe. A tile or tile set could have an extent that covers a building site, a city, a county, a country, and so forth. Any restrictions on the extent of a tile or tile set would be specified in one or more requirements in a profile or a profile with extensions of this abstract specification.
Element name: tileSetExtent
8.3.5. Requirement 6  Tile Set Origin
Requirement 6 
/req/core/tileset/origin The tile set origin SHALL specify the spatial origin reference point for the entire tile set. Values SHALL be one of lower_left, upper_left, lower_right, upper_right, or center. The tile set origin also includes the coordinate of the tile set origin. The coordinate is expressed in the CRS of the tile set. 
The tile set origin defines where the spatial origin reference point is for the entire tile set. Typically, for a GIS implemented in the northern hemisphere, the tile set origin would be specified in the coordinates of CRS using the lower left hand corner. This Abstract Specification does not specify where the origin of a tile set is located. There is also not the requirement that all tile set origins in a tiled data store be the same although using consistent tile set origins helps ensure interoperability through a workflow process such as searchaccessvisualize. These rules would be specified in a profile or be specific to an implementation. For example, in WMTS the tile set origin is the upper left corner. In any case, there is a tile set origin.
Element name: tileSetOrigin
Example: tileSetOrigin=lower_left
8.4. Tessellation class
Related to the Tile Set Scheme is the actual tessellation process used to generate the tile set. The following is the tessellation requirements class.
Requirement 7 
/req/core/tileset/tessellation Any tiling process used to create a tile set SHALL be a regular tessellation of a 2D plane
using a geometric shape of type square (rectilinear), hexagon, or equilateral triangle. The
resulting tile set SHALL consist of one or more tiles, with no overlaps and no gaps.
The tiling scheme SHALL be consistent for all tiles within a tile set and shall adhere to
requirements 2 through 6. The possible values for this requirement are: 
A regular tessellation 2D planar surface can be performed using one of three geometric shapes: squares, hexagons, and equilateral triangles. Squares (a special case of rectangle in which each side is the same length) represent the majority of current and past approaches for tessellating a 2D plane in geospatial systems. However, hexagons or triangles have been used in a variety of systems.
Typically there are also rules applied to specify the size and or extent for each tesselation type. For example, in a DGGS implementation, equal area hexagons are often used. In WMTS and TMS, each tile in a tile set are the same width and height.
Element name: tessellationType
8.4.1. Tessellation Type square
If the tessellationType
is square
, then the valid following properties SHALL be specified:
property name  description 

tileWidth 
Width of an individual tile. The default unit of measure is as specified by tileSetUoM, 
tileHeight 
Height of an individual tile. The default unit of measure is as specified by tileSetUoM. 
equalArea 
If all tiles are to be equal area this is the area in the unit of measure specified in tileSetUoM. If 

Units of Measure for equal area tiles (e.g. square miles, hectares, etc) 
8.4.2. Tessellation Type triangle
If the specified tesslationType
is triangle
, then the length of the side of the equilaterial triangle SHALL be specified.
property name  description 

length 
Length of the triangle side in the units of measure specified in TileSetUOM. All three sides of the triangle have the same length. 
equalArea 
If all tiles are to be equal area this is the area in the unit of measure specified in tileSetUoM. If 
unitofArea 
Unit of Measure for equal area tiles (e.g. square miles, hectares, etc). 
8.4.3. Tessellation Type Hexagon
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which three hexagons meet at each vertex. For the purposes of this logical model, a tesellationType hexagon
SHALL be a regular hexagon. A regular hexagon has vertices equally spaced around a circle and with all sides the same length. Further, the interior angle at each vertex is 120 degrees.
property name  description 

length 
Length of the hexagonal side in the units of measure specified in TileSetUOM. All six sides of the hexagon have the same length. 
equalArea 
If all hexagonal tiles are to be equal area, this property is the area in the unit of measure specified in TileSetUOM. If 
unitofArea 
Unit of Measure for equal area tiles (e.g. square miles, hectares, etc). 
8.5. Requirements Class: Tile
This clause specifies the requirements that define the properties of each individual tile in a tile set. Combined with the Tile Set Scheme common properties, there is enough information for any application  server or client  to create, access, process, analyze, and visualize any geospatial content provided in the tiled structure.
The Tile
requirements class is defined as:
Requirements Class 

Target type 
Token 
Dependency 
Tile Set Scheme 
Requirement 9 
http://www.opengis.net/spec/2dtilemodel/1.0/req/core/tile/address 
Requirement 10 
http://www.opengis.net/spec/2dtilemodel/1.0/req/core/tile/origin 
Requirement 11 
http://www.opengis.net/spec/2dtilemodel/1.0/req/core/tile/reference 
Requirement 12 
http://www.opengis.net/spec/2dtilemodel/1.0/req/core/tile/extent 
8.5.1. Requirement 9  Tile Address (ID)
Requirement 9 
/req/core/tile/address A tile set SHALL use a spatial referencing method to assign a unique spatial reference (or index) to each tile across the entire tile set. 
A tile set shall use a spatial referencing method to assign a unique spatial reference (or index) to each tile across the entire tile set. These indices may be any alphanumeric string including controlled vocabularies such as the USGS quad index, IP address, tile row/column, or other indexing schemes. The default is sequential numbering beginning at the tile located at the tileSet origin. A value of default signifies row/column addressing.
Element name: tileAddress
8.5.2. Requirement 10  Tile Origin
Requirement 10 
/req/core/tile/origin The tile origin SHALL be defined. The same tile origin shall apply to all tiles in a given tile set. Tile origins may be one of “lower left”,“upper left”, "lower_right", "upper_right", and "centroid". 
The tile origin defines where the spatial origin reference point is for each individual tile in a tile set. In a GIS being used in the northern hemisphere, this typically would be the lower left hand corner. All tile origin locations (such as lower left) would be the same for all tiles in a tile set. Neither the conceptual model nor the logical model specifies where the origin of a tile is located. These rules would be specified in a profile or be specific to an implementation. In any case, there is always a tile set origin.
Element name: tileOrigin
8.5.3. Requirement 11  Tile Reference
Requirement 11 
/req/core/tile/reference 
Each tile must be referenced at its centroid. The centroid is the only location that will provide a systematic and consistent spatial reference point for all tiles in a tile set regardless of their shape. NOTE: The reference information does not need to be physically stored in the tile set store. The reference can be calculated from the tile extents (see requirement 12).
Element name: tileReference
8.5.4. Requirement 12  Tile Extent
*Requirement 12 
/req/core/tile/extent 
Please note that the tile extent does not need to be stored as part of the tile metadata. However, at a minimum the tile extent must be able to be calculated. Tile extents are typically required for building a spatial index such as an RTree. See following Notes.
Element name: tileExtent
8.6. Recommended Tile Set Metadata Class
Providing more complete metadata for a tile set ensures enhanced provenance and quality of service for any content provided in a tiled structure. Metadata can provide essential information about a tile set needed to support users in their efforts to discover, select and modify tile sets.
The following suggested metadata elements are extracted from the OGC Two Dimensional Tile Matrix Set Standard. Additional metadata elements may be added to this table. If additional metadata elements are required, the OGC CDB Standard provides an enumeration of the most common metadata standards used in the geospatial industry (Volume 1 Clause 5.1.6). The CDB Standard also provides a more detailed list of key metadata elements.
Element  Description  Reference  

title 
Title of the tile set, normally used for display to a human 
LanguageString data structure, see Figure 15 in OWS Common [OGC 06121r9] 
Zero or more (optional) Include when available and useful. Include one for each language represented. 
abstract 
Brief narrative description of the tile set, normally available for display to a human 
LanguageString data structure, see Figure 15 in OWS Common [OGC 06121r9] 
Zero or more (optional) Include when available and useful Include one for each language represented 
keywords 
Unordered list of one or more commonly used or formalized word(s) or phrase(s) used to describe this tile set. 
MD_Keywords class in ISO 19115 
Zero or more (optional) One for each keyword authority used 
8.7. Tile Indexes, Tile Pyramids, and regular tessellations
8.7.1. Tile indexes
A tile index provides an efficient (fast) mechanism for finding any tile or set of tiles in a tiled data store when using a spatial filter, including a point and query operation. Typical spatial indexing are RTrees and QuadTrees. The logical model specifies no requirements for how a spatial index is constructed or the technology used. However, the tile extents, tile addresses, and tile references can be used to populate a spatial index, such as an RTree.
8.7.2. Tile Pyramids
A tile pyramid structure would be specified as a set of rules (requirements) applied using the core logical model. A tile pyramid is usually based on using a square tile tessellation for some area of interest and a common tile set extent for all tile sets that define the pyramid. The base of the pyramid could be one tile whose extent is identical to that of the tile set. The next tile set (level in the pyramid) could divide the tile set extent into 4 tiles. The next tile set could be based on dividing the tile set extent into 16 tiles and so forth. The OGC Tile Matrix Set candidate standard covers this topic extensively.
8.7.3. Regular Tessellations
Note: A tiling is said to be regular if the symmetry group of the tiling acts transitively on the flags of the tiling, where a flag is a triple consisting of a mutually incident vertex, edge and tile of the tiling. This means that, for every pair of flags, there is a symmetry operation mapping the first flag to the second. This is equivalent to the tiling being an edgetoedge tiling by congruent regular polygons. There must be six equilateral triangles, four squares or three regular hexagons at a vertex, yielding the three regular tessellations. Grünbaum, Branko; Shephard, Geoffrey C. (1977). "Tilings by regular polygons". Math. Mag. 50 (5): 227–247. doi:10.2307/2689529
Annex A: Conformance Class Abstract Test Suite (Normative)
A.2. Conformance Class Tile Set Schema
Based on the Requirement Class Tile Set Schema
requirements/rctilesetschema.adoc
A.2.1. Requirement 2
Test id: 
/conf/confclassa/requirements/REQ2_coretilesetid.adoc 

Requirement: 
/req/core/tileset/id 
Test purpose: 
To verify that a tile set has a unique identifier. 
Test method: 
Either automated test or visual inspection. 
A.2.2. Requirement 3
Test id: 
/conf/confclassa/requirements/REQ3_coretilesetcrs.adoc 

Requirement: 
/req/core/tileset/crs 
Test purpose: 
To verify that a tile set has a unique CRS identifier. 
Test method: 
Either automated test or visual inspection. 
A.2.3. Requirement 4
Test id: 
/conf/confclassa/requirements/REQ4_coretilesetuom.adoc 

Requirement: 
/req/core/tileset/uom 
Test purpose: 
To verify that a tile set has a unique UoM identifier. 
Test method: 
Either automated test or visual inspection. 
A.2.4. Requirement 5
Test id: 
/conf/confclassa/requirements/REQ5_coretilesetextent.adoc 

Requirement: 
/req/core/tileset/extent 
Test purpose: 
To verify that a tile set has an extent specified in the tile set 
Test method: 
Either automated test or visual inspection. 
A.3. Conformance Class tessellationType
Test id: 
/conf/confclassa/requirements/REQ7_coretessellation.adoc 

Requirement: 
/req/core/tileset/tesselation 
Test purpose: 
To verify that a tesselation type of 'square', 'hexagon', or 'triangle' is specified. 
Test method: 
Either automated test or visual inspection. 
A.4. Conformance Class Tile
Requirement Class Tile
requirements/rctile.adoc
A.4.1. Requirement 9
Test id: 
/conf/confclassa/requirements/REQ9_coretileaddress.adoc 

Requirement: 
/req/core/tile/address 
Test purpose: 
To verify that a spatial referencing method to assign a unique 
Test method: 
Either automated test or visual inspection. 
A.4.2. Requirement 10
Test id: 
/conf/confclassa/requirements/REQ10_coretileorigin.adoc 

Requirement: 
/req/core/tile/origin 
Test purpose: 
To verify that a spatial origin reference point is for each 
Test method: 
Either automated test or visual inspection. 
9. Annex B: Additional Informative Material: Simple Example of an XML encoding
Below is a modification of an XML example from the WMTS standard but revised to meet the above example requirements. This could just have easily been written in JSON, Javascript, and so on.
<ows:Identifier>Fort Collins Tiled Data Store</ows:Identifier>
<tileSet>
<tileSetIdentifier>Parcels<tileSetIdentifier>
<tileSetCRS>EPSG2876</tiledSetCRS>
<tileSetUOM>FT</tileSetUOM>
<tileSetOrigin>lower_left</tileSetOrigin>
<! Extent defined. lower left point of tile set bounding box >
<ows:BoundingBox>
<LowerCorner>3080000,1400000</LowerCorner>
<UpperCorner>3600000,1500000</UpperCorner>
<ows:BoundingBox>
< tessellationType>Square</tessellationType>
<! width and height of each tile in specified units >
<tileWidth>1000</tileWidth>
<tileHeight>1000</tileHeight>
<tile>
<tileOrigin>lower_left</tileOrigin>
<tileAddress>default</tileAddress>
<tileReference>default><tileReference>
<tileExtent>default</tileExtent>
<tile>
</tileSet>
Annex B: Revision History
Date  Release  Editor  Primary clauses modified  Description 

2019April28 
1.0 
C. Reed Editor 
all 
initial version 
2019June3 
1.0 
C. Reed 
Various 
Grammatical edits, add references 
2020October2 
1.0 
C. Reed 
Various 
address comments from TC vote 
Annex C: Bibliography
[1] Web: Gupta, Sanjay. Difference between Conceptual, Logical and Physical Data Models. http://uksanjay.blogspot.com/2012/06/differencebetweenconceptuallogical.html
[1] OGC: OGC Testbed 12 Annex B: Architecture. (2015).