The spatial objects specified in 2.3.1, 2.3.2, 2.3.3, and 2.3.5.1 represent the simple objects required for digital spatial processing which can be *used to construct well-defined aggregates or user-defined composite objects* that represent a more complex realization of the real world. The following* zero, one, and two dimension *definitions are valid in planar and non-planar, Euclidean geometry, as well as simple curved surfaces such as the sphere or ellipsoid. Each object type is associated with one or more two-character object representation codes. Use of the codes is explained in 5.6, Vector Modules, 5.5, Composite Module and, 5.7, Raster Modules.

Composite objects (object representation code FF) are constructed from the simple objects by aggregation. Specifically, a composite object consists of one or more other objects, either simple or composite.

A zero-dimensional object that specifies geometric location. One coordinate pair or triplet specifies the location (see Figure 1).

The three point definitions that follow are special implementations of the general case.

A point used for identifying the location of point features (or areal features collapsed to a point), such as towers, buoys, buildings, places, etc.

A reference point used for displaying map and chart text (e.g., feature names) to assist in feature identification.

A representative point within an area usually carrying attribute information about that area.

A zero-dimensional object that is a topological junction of two or more links or chains, or an end point of a link or chain (see Figure 2).

A line is a generic term for a one-dimensional object.

A direct line between two points. (see Figure 3)

A connected nonbranching sequence of line segments specified as the ordered sequence of points between those line segments. Note: A string may intersect itself or other strings (see Figure 4).

A locus of points that forms a curve that is defined by a mathematical expression (see Figure 5).

A topological connection between two nodes. A link may be directed by ordering its nodes (see Figure 6).

A directed nonbranching sequence of nonintersecting line segments and (or) arcs bounded by nodes, not necessarily distinct, at each end.

The following three objects are special cases of chain. They share all characteristics of the general case as defined above.

A chain that explicitly references left and right polygons and start and end nodes (see Figure 7). It is a component of a two-dimensional manifold (2.3.4.5.2.)

A chain that explicitly references left and right polygons and not start and end nodes (see Figure 8). It is a component of a two-dimensional manifold (2.3.4.5.2).

A chain that explicitly references start and end nodes and not left and right polygons (see Figure 9). It is a component of a network (2.3.4.5.3).

A sequence of nonintersecting chains or strings and (or) arcs, with closure. A ring represents a closed boundary, but not the interior area inside the closed boundary.

A ring created from strings and (or) arcs (see Figure 10).

A ring created from complete and (or) area chains (see Figure 11).

An area is a generic term for a bounded, continuous, two-dimensional object that may or may not include its boundary.

An area not including its boundary (see Figure 12).

An area consisting of an interior area, one outer G-ring and zero or more nonintersecting, nonnested inner G-rings. No ring, inner or outer, must be collinear with or intersect any other ring of the same G-polygon (see Figure 13).

An area that is an atomic two-dimensional component of one and only one two-dimensional manifold. The boundary of a GT-polygon may be defined by GT-rings created from its bounding chains. A GT-polygon may also be associated with its chains (either the bounding set, or the complete set) by direct reference to these chains. The complete set of chains associated with a GT-polygon may also be found by examining the polygon references on the chains.

Defines the part of the universe that is outside the perimeter of the area covered by other GT-polygons ("covered area") and completes the two-dimensional manifold. This polygon completes the adjacency relationships of the perimeter links. The boundary of the universe polygon is represented by one or more inner rings and no outer ring. Attribution of the universe polygon may not exist, or may be substantially different from the attribution of the covered area (see Figure 15).

Defines a part of the two-dimensional manifold that is bounded by other GT-polygons, but otherwise has the same characteristics as the universe polygon. The geometry and topology of a void polygon are those of a GT-polygon. Attribution of a void polygon may not exist, or may be substantially different from the attribution of the covered area (see Figure 15).

A two-dimensional *(geospatial)* picture element that is the smallest nondivisible element of a digital image (2.3.4.1). (see Figure 16).

A two-dimensional *(geospatial)* object that represents the smallest nondivisible element of a grid (2.3.4.2). (see Figure 17).

Certain *two-dimensional *aggregate spatial objects must be defined to provide context for many of the simple objects defined above. These aggregate objects are necessary for the definition of (a) raster objects (grid, image, layer, and raster) and (b) topology objects (three types of graphs, with or without geometry).

A two-dimensional *(geospatial)* array of regularly spaced picture elements (pixels) constituting a picture. (see Figure 18).

*A two-dimensional (geospatial) set of grid cells forming a regular tesselation of a surface.* (see Figure 19).

*Each row and column of the grid may have an independent thickness or width (see Figure 20).*

An areally distributed set of spatial data representing entity instances within one theme, or having one common attribute or attribute value in an association of spatial objects. In the context of raster data, a layer is specifically a two,* three or N-dimensional array of attribute values associated with all or part of a grid, image, voxel space or any other type of raster data.(see Figure 21)*

One or more *related* o*verlapping layers for the same grid, digital image, voxel space or any other type of raster data. The corresponding cells between layers are registered to the same raster object scan reference system. The layers overlap but need not be of the same spatial extent.*

A set of topologically interrelated zero-dimensional (node), one-dimensional (link or chain), and sometimes two-dimensional (GT-polygon) objects that conform to a set of defined constraint rules. Numerous rule sets can be used to distinguish different types of graphs. Three such types, planar graph, network, and two-dimensional manifold, are used in this standard. All three share the following rules: each link or chain is bounded by an ordered pair of nodes, not necessarily distinct; a node may bound one or more links or chains; and links or chains may only intersect at nodes.

Planar graphs and networks are two specialized types of graphs, and a two-dimensional manifold is an even more specific type of planar graph.

The node and link or chain objects of the graph occur or can be represented as though they occur upon a planar surface. Not more than one node may exist at any given point on the surface. Links or chains may only intersect at nodes (see Figure 23).

A planar graph and its associated two dimensional objects. Each chain bounds two and only two, not necessarily distinct, GT-polygons. The GT-polygons are mutually exclusive and completely exhaust the surface (see Figure 24).

A graph without two dimensional objects. If projected onto a two-dimensional surface, a network can have either more than one node at a point and (or) intersecting links or chains without corresponding nodes (see Figure 25).

*A volume is a generic term for a bounded, continuous, three-dimensional object that may or may not include its bounding surfaces.*

*A three-dimensional (geospatial) object that represents the smallest nondivisible unit of a voxel space (volume). (The voxel is the three-dimensional conceptual equivalent of the two-dimensional pixel or grid cell.) (see Figure 26).*

*A three-dimensional (geospatial) array of voxels in which the volumetric dataset (the object) resides. The volume represents some measurable properties or independent variables of a real object or phenomenon. (see Figure 27).*

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