Three Ground Object Elements in Euclidean Space and Euclidean Space

Many geographic phenomena models are built on the basis of embedding in a coordinate space. In this coordinate space, the distance and direction between points can be measured according to common formulas. This space model with coordinates is called Euclidean space, which converts the spatial characteristics into Tuples characteristics of real numbers, and the two-dimensional model is called Euclidean plane. In Euclidean space, the most commonly used reference system is Cartesian Coordinates, which consists of a fixed and special point as the origin, and a pair of mutually perpendicular lines passing through the origin as the coordinate axis. In addition, in some cases, other coordinate systems, such as polar coordinates, are often used.

By embedding geographic elements into Euclidean space, three kinds of objects of ground objects are formed, namely point objects, line objects and polygonal objects.

Basic concepts of factor model

Element-based spatial models emphasize individual phenomena, which are studied independently or in a way that correlate with other phenomena. Any phenomenon, regardless of size, can be identified as an object, assuming that it can be conceptually separated from its neighborhood phenomena. Elements can be composed of different objects, and they can have special relations with other separated objects. In an application related to the owner record of land and property, a factor-based perspective is adopted, because each land block and each building must be different and must be uniquely identified and can be measured individually. A factor-based view is appropriate for organized boundary phenomena, although not limited. Therefore, it is also suitable for man-made phenomena, such as buildings, roads, facilities and management areas. Some natural phenomena, such as lakes, rivers, islands and forests, are often represented in factor-based models because they can be seen as discrete phenomena for certain purposes, but it should be remembered that the boundaries of such phenomena are seldom fixed over time, so their actual location definitions are seldom accurate at any time.

The element-based spatial information model decomposes the information space into objects or entities. An entity must meet three conditions：

. Can be identified;

. Important (related to the problem);

. It can be described (characterized).

The characteristics that related entities can be described by static attributes (such as city names), dynamic behavior characteristics and structural characteristics. Unlike the field-based model, the factor-based model regards the information space as a collection of many objects (cities, towns, villages, districts), which have their own attributes (such as population density, centroid and boundary). The entities in the element-based model can define attributes in many dimensions, including spatial dimension, temporal dimension, graphic dimension and text/digital dimension.

Spatial objects are called “space” because they exist in “space”, which is called “embedded space”. The definition of spatial object depends on the structure of embedded space. The commonly used types of embedded space are: (1) Euclidean space, which allows the measurement of distance and orientation between objects, and objects in Euclidean space can be represented by a set of coordinate groups; (2) metric space, which allows the measurement of distance (but not necessarily directional) between objects; (3) topological space, which allows the description of topological relationships among objects.（ There are not necessarily distances and directions; (4) Set-oriented space, which only uses general set-based relationships, such as inclusion, merge and intersection.

1) Types of Spatial Objects on Euclidean Plane

It shows a possible hierarchy of object inheritance on a continuous two-dimensional Euclidean plane(Figure 3-8).

The object with the highest abstraction level is the “space object” class, which derives from zero-dimensional point and extended objects which can derive from one-dimensional and two-dimensional object classes (the above figure). Two subclasses of one-dimensional objects: arc and ring (Loop), if they do not intersect, they are called Simple Arc and Simple Loop. In the two-dimensional space object class, the connected object is called the area object, and the simple area object without “hole” is called the area unit object.

2) Spatial Objects on Discrete Euclidean Planes

The plane of Euclidean space can not be calculated because of its continuity. It must be discretized before it is suitable for calculation. All discrete forms of continuous type exist in Figure 3-8. Figure 3-9 shows the inheritance hierarchy of partially discrete one-dimensional objects.

Object behavior is defined by some operations. These operations are used for one or more objects (operation objects) and produce a new object (result). Spatial operations acting on spatial objects can be divided into two categories: static and dynamic. Static operations do not lead to essential changes in operational objects, while dynamic operations change (or even generate or delete) one or more operational objects.

Although the system’s object-oriented method and element-based spatial data model are similar in concept, there are still obvious differences between them. The implementation of factor-based models does not necessarily require the use of object-oriented methods; on the other hand, object-oriented methods can be used as a framework for describing both the spatial model of the field and the spatial model based on the elements. For element-based models, object-oriented description is obviously appropriate. For field-based models, object-oriented method can also be used to build.

Fields and objects can coexist at many levels. For spatial data modeling, field-based methods and element-based methods are not mutually exclusive. Some applications can naturally apply field modeling to establish field models, such as the climate change in a region mentioned in the previous example. However, even in this case, field models are not suitable for all situations. For example, if the points collecting rainfall data are scattered and irregularly distributed in space, and these points have their own characteristics, then an object may be more suitable for describing the changes of regional climate attributes, containing two attributes, location and average rainfall. In a word, the field-based model and the factor-based model have their own advantages. They should be properly integrated to model. In the high-level modeling of GIS application model, data structure design and GIS application, the integration of these two models will be encountered. Figure 3-10 depicts the comparison between the element model and the field model.

Vector Data Model

Vector method (Fig. 3-11) emphasizes the existence of discrete phenomena. The boundary is determined by boundary lines (points, lines and surfaces), so it can be seen as element-based. However, in some vector-based GIS, the convenience of surface representation brings the possibility of simulating two-dimensional field. The most common example is surface elevation. Grid technology is often described as location-based because it focuses on the content of the location of spatial grid pixels. The raster data model seems to be similar to the field view, but the stored spatial information model is not a description of a continuous variable. A set of grid-pixel values, which can certainly be regarded as sampling a field model can also be sampled as an object-based model.

Vector data model regards phenomena as a set of primitive entities and constitutes spatial entities. In the two-dimensional model, the prototype entity is point or line or surface. In the three-dimensional model, the prototype also includes surface and body. The scale of observation or the degree of generalization determine the type of prototype. In a small scale representation, phenomena such as towns can be represented by individual points, while roads and rivers are represented by lines. When the scale of performance increases, the scale of phenomena must be taken into account. On a medium scale, a town can be represented by a specific prototype, such as a line, to record its boundaries. On a larger scale, towns will be represented as a complex collection of specific prototypes, including building boundaries, roads, parks and other natural and management phenomena.

The expression of vector model originates from the prototype space entity itself, which is usually defined by coordinates. The position of a point can be described by a single set of coordinates in two or three dimensions. A line is usually represented by an orderly set of two or more coordinate pairs. The path of a line between specific coordinates can be a linear function or a higher mathematical function, and the line can be determined by the set of intermediate points. A surface is usually defined by a boundary, which consists of one or more lines forming a closed ring. If there is a hole in the region, then multiple rings can be used to describe it.

According to the type of application, there are some special requirements for describing three-dimensional model with vector data. The application of topographic models requires either simple, single-valued surfaces (a single-valued surface refers to a single, determined elevation value for any location), which can only represent the elevation of the surface. Or they can be combined with the topographic features of the topographic surface. In the landscape structure, it is necessary to combine the three-dimensional representation of the topographic surface and features, such as the buildings and vegetation [4]_. For cartographic purposes, contours are not a convenient representation for analytical purposes. If surfaces are sampled as contours (perhaps digitized from a map), they are usually converted into the most general GIS-based terrain representation, such as regular grids and irregular triangles. The regular grid or matrix of point values can be directly derived from an original sampling scheme of rules. Usually, it is an interpolation of irregular distribution values, which can include digital contours and discrete points. The characteristic of TIN is that they retain the original irregular sampling data values. It is a triangulated plane used to represent a triangle and is associated with these original values. The surface of a TIN element is regarded as a Plannar by default, but surface functions can also be used to interpolate between vertices.

If TIN is used to represent a single value surface (whether terrain data or other), it provides a digital representation of a two-dimensional field when combined with an interpolation function. Similarly, if a grid of sampling points is accompanied by an interpolation function between sampling points, it can also be used to implement a field model.

If volume objects are stored in vector-based GIS, they are usually defined by one or more closed surfaces, while surfaces can be defined by polygons surrounded by three-dimensional lines. Lines and their constituent points, or sets of vertices, define such surfaces as a polygonal mesh structure. Each surface of the net is considered to be flat or curved; in both cases, a mathematical function is needed to indicate the position of the surface between specific coordinates. If a smooth surface is needed, the digital surface function can be constructed by the vertices of the polygonal network, then the computer graphics display of such a surface can be realized by decomposing the digital surface into very small planes. Examples of mathematical surface functions are B-spline functions. These types of functions control the relationship between the known control points and the fitted surface, including the measurement of the surface and the approximation between the control points and the fitted surface.