A Spatial Relation Model of Three-Dimensional Electronic Navigation Charts Based on Point-Set Topology Theory

Abstract

Spatial relation models are the basis for realising three-dimensional spatial analysis. More researchers are now focusing on models that combine topological relations with distance or directional relations; however, a model that unifies all three relations has not yet been developed. In particular, it is more effective to use different spatial relations between features with different spatial characteristics in three-dimensional electronic navigation charts (3D ENC). Therefore, this paper proposes a 3D ENC feature spatial relation model (3DSRM) based on point-set topology theory, which combines 3D topological relations, distance relations and directional relations, and uses a unified model framework to describe 64 topological relations of 3D ENC features from both horizontal and vertical directions. Through the comparison and derivation of feature topological relations, it is demonstrated that the model can distinguish 3D spatial topological relations more comprehensively, realise the mutual derivation between spatial relations and spatial features, and improve the robustness of spatial relations judgement. The model can be used to judge the topological relations between features, realise 3D topological relation checking and topological creation of complex features, and improve the accuracy and interactivity of 3D ENC.

1. Introduction

As a necessary navigation tool, an ENC plays a very important role in ensuring the safe navigation of ships. In the use of a traditional two-dimensional ENC, the symbols for chart features are too abstract, dense and easy to overlap, which causes understanding difficulties for the crew, resulting in operational errors [1]. For example, the anchoring area is generally recognised by its boundary and is represented by the highlighted area shown in Figure 1. When the positional state of the ship or the scale of the chart change, it is easy to fail to correctly display the area of the ship [2]. As shown in Figure 1, the left image clearly shows that the ship is within the mooring area, while the middle image shows that the mooring area boundary cannot be displayed after changing the scale. The 3D ENC is more advantageous in these respects; by switching the viewing angle and selecting the appropriate viewpoints, the boundary information of the representation feature can be found, to more accurately judge the ship’s area, as shown in the right image in Figure 1. The 3D ENC is a 3D visualisation platform that combines ENC features with 3D marine virtual scenes. The aim is to display marine data in a realistic, real-time manner in a 3D environment, to help users identify and understand changes in water depth and environment [3], improve insight, spatial observation and spatial planning capabilities [4] and, thus, safeguard the safety of ship navigation.

Currently, 2D ENC and 3D visualisation are used separately. Two-dimensional ENC is used for chart feature management and spatial analysis, and 3D visualisation is only used as a visual aid to present the final effect. One of the reasons why the two cannot be combined is that the spatial analysis functions commonly used in geographic information systems (GIS) are difficult to use in 3D visualisations [5]. For example, in 2D space, it is relatively easy to determine the topological relation between two features, whereas in 3D space the features are displayed on the screen by projection, and when the viewing angle changes there may be a situation where the two features overlap on the screen, and it becomes complicated to determine the topological relations between the two features. In the study of 3D ENC to realise spatial data analysis functions, many specific issues are involved, such as 3D data models [6,7,8,9], 3D spatial relations [10,11], 3D topology [12,13], 3D visualization [14,15] and so on. The focus of this paper is to study the 3D spatial relation model suitable for 3D ENC, to improve the ability of spatial topology judgement, and to solve the application of 3D spatial relations in 3D ENC.

The 3D spatial relations in land scenes are more studied than in marine scenes. Zlatanova analyses possible topological relationships between multidimensional simple objects in 0, 1, 2 and 3 dimensional space [16]. Ellul analyses application areas, such as earth sciences and urban modelling which are traditionally associated with GIS, then gives a categorisation of 3D topological requirements [17]. Boguslawski proposes a terrain representation with geometrical–topological data with a high memory efficiency [18]. Stoter proposes a topological representation for 3D city models by incorporating a linear cell complex, based on the combinatorial map data structure in the CityGML data model [19]. Rahman proposed a Malaysian marine data model based on the international standard land administration domain model (LADM) to accommodate marine environmental data, but the shared boundary in land administration is not feasible to apply in the marine environment because of the marine environment’s dynamic factors [20]. However, there are significant differences between land 3D scenes and marine 3D scenes. Firstly, land 3D scenes focus on ground features, while marine 3D scenes focus on underwater, surface and above-water features; the underwater environment is especially more complex and difficult to observe. Secondly, features of the land 3D scene, such as buildings and vehicles, can be directly represented using the corresponding simple 3D model, while features of the marine 3D scene, such as channel lines, anchorages, currents and various marine weather information, need to be represented using a complex, abstracted 3D model. Thirdly, the update cycle of land 3D scenes is long, while the update cycle of marine 3D scenes is short; the Maritime Bureau, for example, will regularly release updated data of ENC.

Three-dimensional spatial relations describe the interrelationships between two objects in 3D space, and are usually divided into relative and metric relations. Relative relations are primarily topological, and reflect the logical structural relations between entities; they have greater stability than geometric data and do not change with map projections. At present, the topological relation used in ENC is the dimensionally extended 9-intersection model (DE-9IM) defined in the S-58 standard [21], which is a topological spatial relations model based on the point-set topology proposed by Egenhofer [22]. Theoretically, 2⁹ 2D topological relations can be distinguished. Actual topology relations that can be described include disjoint, touches, overlap, cover, contain, equal, covered by, inside and crosses. The goal of constructing a 3D ENC is proposed in S-101, the new generation of ENC standard [23], but the topological relation model in the 3D ENC has not been specified. Moreover, due to the lack of vertical information in the DE-9IM model, it is difficult to describe the topological relations among the 3D ENC features in detail. Some researchers have expanded on this model. For example, a point neighbourhood [24] is a model based on the point-set topology, which describes the attribution of a neighbourhood by combining the attribution relations of all points in the neighbourhood. However, the judgment process and coding of this model are more complex, which is suitable for irregular complex 3D entities. Shen used the 16-intersection model (16IM) to describe the topological relations between spatial regions with holes [25], and also proposed a model of interlinear topological relations that takes into account the metric details of the line length and endpoint number ratios [26]. He used logical reasoning and projection to determine the 3D spatial topological relations between point objects and body objects [27]. Different from traditional 3D entities, the 3D ENC features are developed from 2D geographic information. In addition to paying attention to the geographical attributes of the features, it is more important to clarify the spatial relation of the features.

Metric relations including distance relations and directional relations, usually as a refinement of topological relations, can make more detailed descriptions of some topologically ambiguous spatial relations, make up for the lack of description of topological relations in some complex situations, and accurately determine the relative position relation between two features [28]. Distance relations are mainly represented using continuous and discrete methods, such as the region connection calculus (RCC) [29] and the distance-based topological relation model (D-TRM) [30]. The RCC method mainly considers the existence of connections between spatial objects, while the D-TRM combines actual and signed distances with topological relations. The directional relations are mainly represented using four- and eight-directional conical models [31,32], topologically augmented direction relations [33,34], projection models [35] and their related extended models. The conical model represents the target object as the origin, and divides the surrounding space into four or eight directions. Goyal et al. add overlapping relations to the eight directions to form nine directional relations [36]. Dube proposes topologically augmented direction relations that assign the values of the 9-intersection model (9IM) matrix as spatial relations, rather than an empty or non-empty designation. Gu used a dual projection matrix to divide the space into 27 regions, using two 9IM matrices to represent the 3D orientation relations. Tao proposed the objects interaction matrix (OIM) for determining the underlying orientation between regional objects [37]. Yan proposed a 3DR29 directional relation model considering internal division, and a 3D topological relation model representing ocean body data from the perspective of ocean depth [38]. These methods can describe the relations between features from different perspectives, but no effective connections have been established between distance, directional and topological relations, and a unified model of spatial relations has yet to be formed.

This paper proposes a spatial relation model of 3D ENC features based on the point-set topology theory. This model takes into account the 3D spatial characteristics of 3D ENC features, combines them with 3D distance and directional relations, and divides the spatial relations into three parts, using a unified framework representation: topological relations, distance relations and directional relations. It enables the derivation of 3D spatial relations by judging the spatial relations between features and eliminating some of the spatial relations that will not occur, and is used to create and check the topology of the features.

Section 2 introduces the spatial characteristics of the 3D ENC features. Section 3 introduces the definition of the spatial relation model and sub-models of the 3D ENC features. Section 4 verifies the correctness and validity of this model. Section 5 introduces the application of the spatial relation model in 3D ENC. Finally, conclusions and prospects are given in Section 6.

2. Spatial Characteristics of 3D ENC Features

The S-101 IHO Electronic Navigational Chart Product Specification (S-101) is a specification to guide the future development direction of ENC [23]. The S-101 provides a detailed division and definition of the features in ENC, including 172 geographical features, 9 metadata features, 1 cartographic feature and 5 information types. Geographical features are the main information displayed in ENC, and are the subject of this paper, and can be divided into 18 broad categories, as shown in

Table 1. Spatial types of 3D ENC.

Feature Types	Feature Name	Geometric				Spatial
Magnetic Data	Magnetic Variation, Local Magnetic Anomaly	P	C	S		L	S
Natural Features	Coastline, Slope Topline		C			L
	Land Area, Rapids, Land region, Vegetation	P	C	S		L
	Island group				N	L
	Land elevation, Waterfall	P	C			L
	River, River, Tideway		C	S		L
	Lake, Ice area			S		L
	Sloping Ground	P		S		L
Cultural Features	Built-up area, Building, Airport/Airfield, Pylon/Bridge Support	P		S		L
	Runway	P	C	S		L
	Bridge		C	S	N	L
	Span Fixed, Span Opening, Conveyor, Road, Tunnel		C	S		L
	Cable Overhead, Pipeline Overhead, Fence/wall, Railway		C			L
Landmarks	Landmark, Fortified Structure	P	C	S		L
	Silo/tank, Production/Storage Area	P		S		L
	Wind Turbine	P				L
Ports	Checkpoint, Hulks	P		S		L
	Piles, Shoreline Construction, Berth, Gate, Crane, Mooring/Warping Facility, Floating Dock, Pontoon	P	C	S		L
	Dyke, Causeway, Canal, Dam		C	S		L
	Distance Mark	P				L
	Dry Dock, Dock Area, Gridiron, Lock Basin			S		L
	Mooring Trot				N	L
Topographic Terms	Sea Area/Named Water Area	P		S			S
Tides, Currents	Tidal Stream–Flood/Ebb, Tidal Stream Panel Data	P		S			S
	Current–Non-Gravitational, Water Turbulence	P	C	S			S
Depths	Sounding, Depth–No Bottom Found	A						U
	Dredged Area, Swept Area, Depth Area, Unsurveyed Area			S				U
	Depth Contour		C					U
Nature of the Seabed	Seabed Area, Sandwaves	P	C	S				U
	Weed/Kelp	P		S				U
	Spring	P						U
Rocks, Wrecks, Foul Ground, Obstructions	Underwater/Awash Rock	P						U
	Wreck, Discoloured Water	P		S				U
	Obstruction, Foul Ground, Fishing Facility, Marine Farm/Culture	P	C	S				U
Offshore Installations	Offshore Platform	P		S			S
	Cable Submarine, Pipeline Submarine/On Land		C				S
	Cable Area, Offshore Production Area			S			S
	Submarine Pipeline Area	P		S			S
Tracks and Routes	Navigation Line, Recommended Track, Recommended Route Centreline, Deep Water Route Centreline, Traffic Separation Line, Traffic Separation Scheme Boundary, Archipelagic Sea Lane Axis, Radar Line		C				S
	Range System, Fairway System, Two-Way Route, Deep Water route, Traffic Separation Scheme, Archipelagic Sea Lane				N		S
	Fairway, Two-Way Route Part, Deep Water Route Part, Inshore Traffic Zone, Traffic Separation Scheme Lane Part, Traffic Separation Zone, Traffic Separation Scheme Crossing, Traffic Separation Scheme Roundabout, Archipelagic Sea Lane Area, Radar Range			S			S
	Recommended Traffic Lane Part, Precautionary Area	P		S			S
	Ferry Route		C	S			S
	Radar Station	P					S
Areas, limits	Anchorage Area, Seaplane Landing Area, Dumping Ground, Military Practice Area, Cargo Transhipment Area, Caution Area, Log Pond	P		S			S
	Anchor Berth, Oil Barrier, Straight Territorial Sea Baseline, Collision Regulations Limit		C				S
	Administration Area, Contiguous Zone, Continental Shelf Area, Custom Zone, Exclusive Economic Zone, Fishery Zone, Fishing Ground, Free Port Area, Harbour Area (Administrative), Territorial Sea Area, Submarine Transit Lane, Pilotage District			S			S
	Information Area	P	C	S			S
Restricted Areas	Restricted Area Navigational, Restricted Area Regulatory			S			S
Lights	Light All Around, Light Sectored, Light Fog Detector, Light Air Obstruction	P					S
Buoys, Beacons	Buoy Lateral, Buoy Cardinal, Buoy Isolated Danger, Buoy Safe Water, Buoy Special Purpose/General, Buoy Emergency Wreck Marking, Buoy Installation, Beacon Lateral, Beacon Cardinal, Beacon Isolated Danger, Beacon Safe Water, Beacon Special Purpose/General, Daymark, Light Float, Light Vessel, Retroreflector, Radar Reflector, Fog Signal	P					S
Radar, Radio	Physical AIS Aid to Navigation, Radio Station, Radar Transponder Beacon, Virtual AIS Aid to Navigation	P					S
Services	Pilot Boarding Place, Coastguard Station, Signal Station Warning, Signal Station Traffic, Rescue Station, Harbour Facility, Small Craft Facility	P		S			S
	Vessel Traffic Service Area			S			S

. The geometric characteristics of the features can be divided into the point, curve, surface and point-set features; the physical characteristics of the features can also be divided into environmental features, substance features and virtual features [39]. To be able to concisely describe the 3D spatial characteristics of features and improve the accuracy of the spatial relationships between features, this paper divides geographical features into three parts, according to the spatial characteristics of each feature: land features, sea surface features and underwater features. For example, in two dimensions, the topological relation between the recommended route and the nature of the seabed are intersecting, but in three dimensions they are on the surface and the seabed, respectively, and are not in the same space, so using only the intersecting topological relation to represent is not accurate enough.

1.

Land features are those features that exist within the land area, both above and below land, such as buildings, harbours, dykes, bridges, tunnels, etc.;

2.

Sea surface features are those features that are above the sea surface within the marine domain, such as buoys, lighthouses, anchorages, channel lines, etc.;

3.

Underwater features are features that are below the surface of the sea within the marine domain, such as depth, submarine geology, submerged obstruction, wrecks, submarine pipelines, etc.

It is also worth noting that there are also some features of the marine environment, such as magnetic data, which are usually distributed globally, and therefore belong to both sea surface and land features. Table 1 summarises the geometric and spatial characteristics of each ENC feature, with the abbreviations for the geometric characteristics being: point (P), curve (C), surface (S), point-set (A) and none (N); with the abbreviations for the spatial characteristics being: land features (L), sea surface features (S) and underwater features (U).

3. The Spatial Relation Model of 3D ENC Features

This section proposes a 3DSRM, a spatial relation model for the 3D ENC features based on the point-set topology theory, which consists of three parts to represent the relative and metric relationships between two 3D ENC features: a 3D topological relation model (3DSRM-T), a 3D distance relation model (3DSRM-D) and a 3D directional relation model (3DSRM-A).

3.1. 3D Topological Relation Model

The 3D topological relation model is based on the 9IM, which describes topological relations by defining the boundary, interior and exterior of features, and lacks information at the spatial level when extended to 3D. Therefore, this paper redefines the boundary and interior of features, and divides the exterior of features into either the same layer exterior or different layer exteriors, to achieve a more detailed description of spatial relations, as shown in Figure 2. In the 3D coordinates, the direction along the x and z axes is called the horizontal direction, and the direction along the y axis is called the vertical direction. The specific definition is as follows, assuming that A and B are two features in 3D space, $\mathrm{A},\mathrm{B}\subset {\mathbb{R}}^3$.

Definition 1.

The boundary of feature A is composed of several area faces surrounding the feature, represented as $\partial A$.

Definition 2.

The interior of feature A is the interior of a polyhedron consisting of ∂A, represented as $A^{°}$. The interior of feature A does not intersect the boundary, i.e., $A^{°}\cap \partial A=\varnothing$.

Definition 3.

The same layer exterior of feature A is the exterior of a polyhedron consisting of $\partial A$, and on the same vertical region as A, represented as $A^{-}$. The same layer exterior of feature A does not intersect the boundary or the interior, i.e., $A^{-}\cap \partial A=\varnothing , A^{-}\cap A^{°}=\varnothing$.

Definition 4.

The different layer exterior of feature $A$ is the exterior of a polyhedron consisting of $\partial A$, and not on the same vertical region as $A$, represented as $\nabla A$. The different layer exterior of feature A does not intersect the boundary, interior or same layer exterior, i.e., $\nabla A\cap \partial A=\varnothing$, $\nabla A\cap A^{°}=\varnothing$ and $\nabla A\cap A^{-}=\varnothing$. A 3D space can consist of these four components, i.e., $\partial A\cup A^{°}\cup A^{-}\cup \nabla A={\mathbb{R}}^3$.

Definition 5.

The combination of the boundary, the interior and the same layer exterior of feature A are represented as $A^{+}$, defining the entire space over that vertical region, i.e., $A^{+}=\partial A\cup A^{°}\cup A^{-}$. Per definition 4, it is in turn possible to obtain $A^{+}\cup \nabla A={\mathbb{R}}^3$.

From the above five definitions, it follows that the topological relation between features A and B is the set of relations between ${\mathrm{A}}^{+}$ and ${\mathrm{B}}^{+}$, ${\mathrm{A}}^{+}$ and $\nabla \mathrm{B}$, $\nabla \mathrm{A}$ and ${\mathrm{B}}^{+}$, and $\nabla \mathrm{A}$ and $\nabla \mathrm{B}$, denoted by ${\mathrm{R}}_{3\mathrm{D}\mathrm{T}\mathrm{R}\mathrm{M}\text{-}\mathrm{T}}$, in matrix form as

$${\mathrm{R}}_{3\mathrm{D}\mathrm{S}\mathrm{R}\mathrm{M}\text{-}\mathrm{T}}\left(\mathrm{A},\mathrm{B}\right)=\left[\begin{array}{cc}{\mathrm{A}}^{+}\cap {\mathrm{B}}^{+}& {\mathrm{A}}^{+}\cap \nabla \mathrm{B}\\ \nabla \mathrm{A}\cap {\mathrm{B}}^{+}& \nabla \mathrm{A}\cap \nabla \mathrm{B}\end{array}\right]$$

where the topological relation between ${\mathrm{A}}^{+}$ and ${\mathrm{B}}^{+}$ is the same as that represented by 9IM, which is represented by the matrix as

$${\mathrm{R}}_{9\mathrm{I}\mathrm{M}}\left(\mathrm{A},\mathrm{B}\right)=\left[\begin{array}{ccc}{\mathrm{A}}^{°}\cap {\mathrm{B}}^{°}& {\mathrm{A}}^{°}\cap \partial \mathrm{B}& {\mathrm{A}}^{°}\cap {\mathrm{B}}^{-}\\ \partial \mathrm{A}\cap {\mathrm{B}}^{°}& \partial \mathrm{A}\cap \partial \mathrm{B}& \partial \mathrm{A}\cap {\mathrm{B}}^{-}\\ {\mathrm{A}}^{-}\cap {\mathrm{B}}^{°}& {\mathrm{A}}^{-}\cap \partial \mathrm{B}& {\mathrm{A}}^{-}\cap {\mathrm{B}}^{-}\end{array}\right]$$

Although 9IM can theoretically distinguish 2⁹ 2D topological relations, in reality, many topological relations do not exist. For example, the value of ${\mathrm{A}}^{-}\cap {\mathrm{B}}^{-}$ for all topological relations must be 1. This is because no one feature will cover the whole space, and there must be an external space that intersects the external space of another feature. By the same token, it follows that $\nabla \mathrm{A}\cap \nabla \mathrm{B}$ and ${\mathrm{A}}^{-}\cap {\mathrm{B}}^{-}$ have the same value of 1. Thus, the topological relation between features A and B is represented by the matrix.

$${\mathrm{R}}_{3\mathrm{D}\mathrm{S}\mathrm{R}\mathrm{M}\text{-}\mathrm{T}}\left(\mathrm{A},\mathrm{B}\right)=\left[\begin{array}{cc}\begin{array}{cc}{\mathrm{A}}^{°}\cap {\mathrm{B}}^{°}& {\mathrm{A}}^{°}\cap \partial \mathrm{B}\end{array}& \begin{array}{cc}{\mathrm{A}}^{°}\cap {\mathrm{B}}^{-}& {\mathrm{A}}^{+}\cap {\mathrm{B}}^{+}\end{array}\\ \begin{array}{cc}\partial \mathrm{A}\cap {\mathrm{B}}^{°}& \partial \mathrm{A}\cap \partial \mathrm{B}\end{array}& \begin{array}{cc}\partial \mathrm{A}\cap {\mathrm{B}}^{-}& {\mathrm{A}}^{+}\cap \nabla \mathrm{B}\end{array}\\ \begin{array}{cc}{\mathrm{A}}^{-}\cap {\mathrm{B}}^{°}& {\mathrm{A}}^{-}\cap \partial \mathrm{B}\end{array}& \begin{array}{cc}{\mathrm{A}}^{-}\cap {\mathrm{B}}^{-}& \nabla \mathrm{A}\cap {\mathrm{B}}^{+}\end{array}\end{array}\right]$$

3.2. 3D Distance Relation Model

Topological relations describe the relative relations between two features, but this does not uniquely determine their position. Combining the two metric relations of distance and direction provides a complementary description of the spatial relation between features, and a more accurate description of the position of the features relative to each other. Using accurate distance relations makes it easier for a crew to understand the spatial relations between ships and ENC features, providing richer information support for ship navigation, route planning or maritime activities. For example, a large container ship sailing in the Suez Canal is usually represented as a point feature in large-scale charts, while in some small-scale charts it is represented as a surface feature, and in 3D ENC it is represented as a solid. To unify the description of the 3D distance relation between ship and ENC features, this paper proposes a 3D distance relation model 3DSRM-D, which divides the 3D distance relation into two parts: horizontal distance and vertical distance, and the relevant definitions are as follows:

Definition 6.

Let V and W be the boundary points of the planes of features A and B in space, respectively. The distance between V and W is positive when W is outside of A, zero when W is at the boundary of A, and negative when W is inside of A [30]. As shown in the left panel of Figure 3, V is the boundary point of feature A, and W₁–W₃ are the boundary points of feature B. According to the above definition, the distance between V and W₁ is positive, the distance between V and W₂ is zero, and the distance between V and W₃ is negative. The horizontal distances are expressed using the matrix as

$$R\left(A,B\right)=\left[\begin{array}{cc}{Min}_D\left(\partial A,\partial B\right)& {Min}_D\left(\partial B,\partial A\right)\\ {Max}_D\left(\partial A,\partial B\right)& {Max}_D\left(\partial B,\partial A\right)\end{array}\right]$$

where ${\mathrm{M}\mathrm{i}\mathrm{n}}_{\mathrm{D}}\left(\partial \mathrm{A},\partial \mathrm{B}\right)$ represents the minimum plane distance from $\partial \mathrm{A}$ to $\partial \mathrm{B}$, ${\mathrm{M}\mathrm{a}\mathrm{x}}_{\mathrm{D}}\left(\partial \mathrm{A},\partial \mathrm{B}\right)$ represents the maximum plane distance from $\partial \mathrm{A}$ to $\partial \mathrm{B}$, ${\mathrm{M}\mathrm{i}\mathrm{n}}_{\mathrm{D}}\left(\partial \mathrm{B},\partial \mathrm{A}\right)$ represents the minimum plane distance from $\partial \mathrm{B}$ to $\partial \mathrm{A}$, and ${\mathrm{M}\mathrm{a}\mathrm{x}}_{\mathrm{D}}\left(\partial \mathrm{B},\partial \mathrm{A}\right)$ represents the maximum plane distance from $\partial \mathrm{B}$ to $\partial \mathrm{A}$.

Definition 7.

Let P and Q be the vertical boundary points of features A and B in space, respectively. The distance between P and Q is positive when Q is outside of A, zero when Q is at the boundary of A, and negative when Q is inside of A. As shown in the right panel of Figure 3, P is the boundary point of feature A, and Q₁–Q₃ are the boundary points of feature B. According to the above definition, the distance between P and Q₁ is positive, the distance between P and Q₂ is zero, and the distance between P and Q₃ is negative. The vertical distances are expressed using the matrix as

$$R\left(A,B\right)=\left[\begin{array}{cc}{Min}_H\left(\partial A,\partial B\right)& {Min}_H\left(\partial B,\partial A\right)\\ {Max}_H\left(\partial A,\partial B\right)& {Max}_H\left(\partial B,\partial A\right)\end{array}\right]$$

where ${\mathrm{M}\mathrm{i}\mathrm{n}}_{\mathrm{H}}\left(\partial \mathrm{A},\partial \mathrm{B}\right)$ represents the minimum vertical distance from $\partial \mathrm{A}$ to $\partial \mathrm{B}$, ${\mathrm{M}\mathrm{a}\mathrm{x}}_{\mathrm{H}}\left(\partial \mathrm{A},\partial \mathrm{B}\right)$ represents the minimum vertical distance from $\partial \mathrm{A}$ to $\partial \mathrm{B}$, ${\mathrm{M}\mathrm{i}\mathrm{n}}_{\mathrm{H}}\left(\partial \mathrm{B},\partial \mathrm{A}\right)$ represents the minimum vertical distance from $\partial \mathrm{B}$ to $\partial \mathrm{A}$ and ${\mathrm{M}\mathrm{a}\mathrm{x}}_{\mathrm{H}}\left(\partial \mathrm{B},\partial \mathrm{A}\right)$ represents the maximum vertical distance from $\partial \mathrm{B}$ to $\partial \mathrm{A}$.

In summary, the 3D distance relation between features A and B is represented by the matrix:

$${\mathrm{R}}_{3\mathrm{D}\mathrm{S}\mathrm{R}\mathrm{M}\text{-}\mathrm{D}}\left(\mathrm{A},\mathrm{B}\right)=\left[\begin{array}{c}\begin{array}{cc}{\mathrm{M}\mathrm{i}\mathrm{n}}_{\mathrm{D}}\left(\partial \mathrm{A},\partial \mathrm{B}\right)& {\mathrm{M}\mathrm{i}\mathrm{n}}_{\mathrm{D}}\left(\partial \mathrm{B},\partial \mathrm{A}\right)\\ {\mathrm{M}\mathrm{a}\mathrm{x}}_{\mathrm{D}}\left(\partial \mathrm{A},\partial \mathrm{B}\right)& {\mathrm{M}\mathrm{a}\mathrm{x}}_{\mathrm{D}}\left(\partial \mathrm{B},\partial \mathrm{A}\right)\end{array}\\ \begin{array}{cc}{\mathrm{M}\mathrm{i}\mathrm{n}}_{\mathrm{H}}\left(\partial \mathrm{A},\partial \mathrm{B}\right)& {\mathrm{M}\mathrm{i}\mathrm{n}}_{\mathrm{H}}\left(\partial \mathrm{B},\partial \mathrm{A}\right)\\ {\mathrm{M}\mathrm{a}\mathrm{x}}_{\mathrm{H}}\left(\partial \mathrm{A},\partial \mathrm{B}\right)& {\mathrm{M}\mathrm{a}\mathrm{x}}_{\mathrm{H}}\left(\partial \mathrm{B},\partial \mathrm{A}\right)\end{array}\end{array}\right]$$

3.3. 3D Directional Relation Model

Directional relations, which provide information on the metrics between features in terms of direction, are often used more frequently in ENC than distance relations, and can avoid some of the misunderstandings that are expressed using distance relations. For example, in a ship navigating in a narrow channel, where both sides of the ship are close to an area of dangerous depth, using the distance relation alone would produce an early warning. However, if the ship’s bow is in the direction of the extended channel, it is safe to navigate without yawing, and considering the directional relation can avoid some false warnings under normal navigation. To describe the directional relation accurately in the 3D ENC, this paper uses azimuth and pitch angle to represent the position relation between two features in horizontal and vertical directions, respectively, which is expressed by the matrix ${\mathrm{R}}_{3\mathrm{D}\mathrm{S}\mathrm{R}\mathrm{M}\text{-}\mathrm{A}}\left(\mathrm{A},\mathrm{B}\right)$, as defined below.

$${\mathrm{R}}_{3\mathrm{D}\mathrm{S}\mathrm{R}\mathrm{M}\text{-}\mathrm{A}}\left(\mathrm{A},\mathrm{B}\right)=\left[\begin{array}{cc}{\mathrm{R}}_{\mathrm{\measuredangle }}\left(\mathrm{A},\mathrm{B}\right)& {\mathrm{R}}_{\mathrm{\sphericalangle }}\left(\mathrm{A},\mathrm{B}\right)\end{array}\right]$$

Definition 8.

The azimuth from point feature A to the surface feature B is the horizontal angle between the line pointing north from point A and the line connecting it to area B in a clockwise direction, represented as $\measuredangle \left(A,\partial B\right)$. The minimum azimuth from A to $\partial B$ is represented as ${Min}_{\measuredangle }\left(A,\partial B\right)$, the maximum azimuth from A to $\partial B$ is represented as ${Max}_{\measuredangle }\left(A,\partial B\right)$, and the azimuth angle from A to $C_B$ is represented as $\measuredangle \left(A,C_B\right)$. The azimuth angle takes values in the range (−180°, 180°], i.e., starting from due north, positive is clockwise, negative is counterclockwise, 0° is due north, 180° is due south, 90° is due east and −90° is due west. This is shown on the left in Figure 4.

Three inferences can be derived from Definition 8.

Inference 1.

When the maximum azimuth angle from A to $\partial B$ is equal to 180°, and the minimum azimuth angle tends infinitely to −180°, the point feature A lies inside feature B.

Inference 2.

When the maximum azimuth from A to $\partial B$ is less than 180°, or the minimum azimuth is greater than −180°, the point feature A lies outside of feature B.

Inference 3.

When the difference in azimuth from A to any two adjacent points on B is equal to 180°, the point features A lies on the boundary of feature B.

The horizontal direction relation between two surface features can be further analysed based on the horizontal direction relation from the point feature to the surface feature, and the topological relation can be deduced. The relation between surface feature A and surface feature B can be seen as the set of relations between each boundary point on feature A and feature B. This can be expressed as a matrix.

$${\mathrm{R}}_{\mathrm{\measuredangle }}\left(\mathrm{A},\mathrm{B}\right)=\left[\begin{array}{cc}\mathrm{B}\mathrm{P}\left(\mathrm{A},\mathrm{B}\right)& \mathrm{B}\mathrm{P}\left(\mathrm{B},\mathrm{A}\right)\\ \mathrm{R}\mathrm{P}\left(\mathrm{A},\mathrm{B}\right)& \mathrm{R}\mathrm{P}\left(\mathrm{B},\mathrm{A}\right)\end{array}\right]$$

where $\mathrm{B}\mathrm{P}\left(\mathrm{A},\mathrm{B}\right)$ indicates whether the point of feature A is on the boundary of B, $\mathrm{B}\mathrm{P}\left(\mathrm{B},\mathrm{A}\right)$ indicates whether the point of feature B is on the boundary of A, with a value of 1 if yes, 0 if no. $\mathrm{R}\mathrm{P}\left(\mathrm{A},\mathrm{B}\right)$ indicates whether the rest of the points of feature A, except those on the boundary of B, are inside or outside of B, $\mathrm{R}\mathrm{P}\left(\mathrm{B},\mathrm{A}\right)$ indicates whether the rest of the points of feature B, except those on the boundary of A, are inside or outside of A, with a value of 1 if they are all external, −1 if they are all internal, and 0 otherwise.

Definition 9.

The pitch angle from feature A to feature B is the vertical angle from the horizontal line of A to the line connecting the area of feature B. The minimum pitch angle from A to B is represented as ${Min}_{\sphericalangle }\left(A,B\right)$ and the maximum pitch angle from A to B is represented as ${Max}_{\sphericalangle }\left(A,B\right)$. The pitch angle takes the values in the range (−90°, 90°), starting from the horizontal direction: counterclockwise is positive, clockwise is negative and horizontal is 0° (as shown in Figure 4, right).

Three inferences can be derived from Definition 9.

Inference 1.

When the maximum pitch angle from A to B is greater than 0°, and the minimum pitch angle is less than 0°, point feature A is located in the interior of feature B.

Inference 2.

When both the maximum and minimum pitch angles from A to B are greater than 0°, or both the maximum and minimum pitch angles are less than 0°, point feature A is located outside of feature B.

Inference 3.

When the maximum or minimum pitch angle from A to B is equal to 0°, the point feature A lies on the boundary of feature B.

The vertical direction relation between the two surface features can be further analysed from the vertical direction relation from the point feature to the surface feature and the topological relation can be deduced. This can be expressed as a matrix.

$${\mathrm{R}}_{\mathrm{\sphericalangle }}\left(\mathrm{A},\mathrm{B}\right)=\left[\begin{array}{cc}{\mathrm{M}\mathrm{i}\mathrm{n}}_{\mathrm{\sphericalangle }}\left({\mathrm{A}}_{\mathrm{T}},\mathrm{B}\right)& {\mathrm{M}\mathrm{i}\mathrm{n}}_{\mathrm{\sphericalangle }}\left({\mathrm{A}}_{\mathrm{L}},\mathrm{B}\right)\\ {\mathrm{M}\mathrm{a}\mathrm{x}}_{\mathrm{\sphericalangle }}\left({\mathrm{A}}_{\mathrm{T}},\mathrm{B}\right)& {\mathrm{M}\mathrm{a}\mathrm{x}}_{\mathrm{\sphericalangle }}\left({\mathrm{A}}_{\mathrm{L}},\mathrm{B}\right)\end{array}\right]$$

where ${\mathrm{M}\mathrm{i}\mathrm{n}}_{\mathrm{\sphericalangle }}\left({\mathrm{A}}_{\mathrm{T}},\mathrm{B}\right)$ represents the minimum pitch angle from the highest point of feature A in the vertical direction to feature B, ${\mathrm{M}\mathrm{a}\mathrm{x}}_{\mathrm{\sphericalangle }}\left({\mathrm{A}}_{\mathrm{T}},\mathrm{B}\right)$ represents the maximum pitch angle from the highest point of feature A in the vertical direction to feature B, ${\mathrm{M}\mathrm{i}\mathrm{n}}_{\mathrm{\sphericalangle }}\left({\mathrm{A}}_{\mathrm{L}},\mathrm{B}\right)$ represents the minimum pitch angle from the lowest point of feature A in the vertical direction to feature B, and ${\mathrm{M}\mathrm{a}\mathrm{x}}_{\mathrm{\sphericalangle }}\left({\mathrm{A}}_{\mathrm{L}},\mathrm{B}\right)$ represents the maximum pitch angle from the lowest point of feature A in the vertical direction to feature B.

In summary, the 3D directional relation between features A and B is represented by the matrix as:

$${\mathrm{R}}_{3\mathrm{D}\mathrm{S}\mathrm{R}\mathrm{M}\text{-}\mathrm{A}}\left(\mathrm{A},\mathrm{B}\right)=\left[\begin{array}{c}\begin{array}{cc}\mathrm{B}\mathrm{P}\left(\mathrm{A},\mathrm{B}\right)& \mathrm{B}\mathrm{P}\left(\mathrm{B},\mathrm{A}\right)\\ \mathrm{R}\mathrm{P}\left(\mathrm{A},\mathrm{B}\right)& \mathrm{R}\mathrm{P}\left(\mathrm{B},\mathrm{A}\right)\end{array}\\ \begin{array}{cc}{\mathrm{M}\mathrm{i}\mathrm{n}}_{\mathrm{\sphericalangle }}\left({\mathrm{A}}_{\mathrm{T}},\mathrm{B}\right)& {\mathrm{M}\mathrm{i}\mathrm{n}}_{\mathrm{\sphericalangle }}\left({\mathrm{A}}_{\mathrm{L}},\mathrm{B}\right)\\ {\mathrm{M}\mathrm{a}\mathrm{x}}_{\mathrm{\sphericalangle }}\left({\mathrm{A}}_{\mathrm{T}},\mathrm{B}\right)& {\mathrm{M}\mathrm{a}\mathrm{x}}_{\mathrm{\sphericalangle }}\left({\mathrm{A}}_{\mathrm{L}},\mathrm{B}\right)\end{array}\end{array}\right]$$

4. Verification of Spatial Relation Model of 3D ENC Features

4.1. Topological Relation Comparison

Eight 2D topological relations between two features can be described using either the 9IM model or the D-TRM model, including disjoint, touches, overlap, cover, contain, equal, covered by and inside, with the corresponding matrices and 2D diagram for each topological relation shown in

Table 2. Both 9IM and D-TRM can describe eight topological relations.

Spatial Relation	9IM	D-TRM	2D Diagram
Disjoint	$\left[\begin{array}{ccc}0& 0& 1\\ 0& 0& 1\\ 1& 1& 1\end{array}\right]$	$\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$
Touches	$\left[\begin{array}{ccc}0& 0& 1\\ 0& 1& 1\\ 1& 1& 1\end{array}\right]$	$\left[\begin{array}{cc}0& 1\\ 0& 1\end{array}\right]$
Overlap	$\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right]$	$\left[\begin{array}{cc}-1& 1\\ -1& 1\end{array}\right]$
Cover	$\left[\begin{array}{ccc}1& 1& 1\\ 0& 1& 1\\ 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]$
Contain	$\left[\begin{array}{ccc}1& 1& 1\\ 0& 0& 1\\ 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cc}-1& -1\\ 1& 1\end{array}\right]$
Equal	$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$
CoveredBy	$\left[\begin{array}{ccc}1& 0& 0\\ 1& 1& 0\\ 1& 1& 1\end{array}\right]$	$\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$
Inside	$\left[\begin{array}{ccc}1& 0& 0\\ 1& 0& 0\\ 1& 1& 1\end{array}\right]$	$\left[\begin{array}{cc}1& 1\\ -1& -1\end{array}\right]$

.

However, these eight topological relations are not sufficient for describing the features of 3D space. For any two features in 3D space, one topological relation can be determined when they are projected on the XOZ plane, while different topological relations may occur when they are projected on the XOY or ZOY planes.

Table 3. Topological relations that cannot be described by 9IM but can be described by 3DSRM-T.

Spatial Relation	9IM	3DTRM-T	3D Diagram
Disjoint × Disjoint	$\left[\begin{array}{ccc}0& 0& 1\\ 0& 0& 1\\ 1& 1& 1\end{array}\right]$	$\left[\begin{array}{cc}\begin{array}{cc}0& 0\\ 0& 0\\ 1& 1\end{array}& \begin{array}{cc}1& 0\\ 1& 1\\ 1& 1\end{array}\end{array}\right]$
Disjoint × Equal		$\left[\begin{array}{cc}\begin{array}{cc}0& 0\\ 0& 0\\ 1& 1\end{array}& \begin{array}{cc}1& 1\\ 1& 0\\ 1& 0\end{array}\end{array}\right]$
Disjoint × Overlap		$\left[\begin{array}{cc}\begin{array}{cc}0& 0\\ 0& 0\\ 1& 1\end{array}& \begin{array}{cc}1& 1\\ 1& 1\\ 1& 1\end{array}\end{array}\right]$
Disjoint × Contain		$\left[\begin{array}{cc}\begin{array}{cc}0& 0\\ 0& 0\\ 1& 1\end{array}& \begin{array}{cc}1& 1\\ 1& 1\\ 1& 0\end{array}\end{array}\right]$
Disjoint × Inside		$\left[\begin{array}{cc}\begin{array}{cc}0& 0\\ 0& 0\\ 1& 1\end{array}& \begin{array}{cc}1& 1\\ 1& 0\\ 1& 1\end{array}\end{array}\right]$

lists, as an example, the 3D topological relations that cannot be described by 9IM, but can be described by 3DSRM-T using the 2D planes of disjoint topology.

Features A and B are two features in 3D space that are disjointed when projected onto the XOZ plane, whereas topological relations such as disjoint, equal, overlap, contain and inside are possible when projected onto the XOY plane. In this case, the matrix form is the same if described using the 9IM model, whereas, with the 3DSRM-T model, it can be described using five different matrices. Therefore, the following inferences can be drawn.

Inference 1.

If the value of $A^{+}\cap B^{+}$ is equal to 0, then it means that feature A and feature B are in different layers in space, indicating that the topological relation between these two features in the XOY plane is disjoint.

Inference 2.

If the value of $A^{+}\cap B^{+}$ equals 1, if the value of $A^{+}\cap \nabla B$ equals 0, and if the value of ∇A∩B+ equals 0, then it means that feature A and feature B are in the same layer in space, indicating that the topological relation between these two features in the XOY plane is equal.

Inference 3.

If the value of $A^{+}\cap B^{+}$ is equal to 1, if the value of $A^{+}\cap \nabla B$ is equal to 1, and if the value of $\nabla A\cap B^{+}$ is equal to 1, then it means that feature A and feature B have parts that are in the same layer in space and parts that are in different layers, indicating that the topological relation between these two features in the XOY plane is overlap.

Inference 4.

If the value of $A^{+}\cap B^{+}$ is equal to 1, if the value of $A^{+}\cap \nabla B$ is equal to 1, and if the value of $\nabla A\cap B^{+}$ is equal to 0, then it means that feature B is in the same layer of feature A in space, while feature A has parts in different layers of feature B, indicating that the topological relation between these two features in the XOY plane is contain.

Inference 5.

If the value of $A^{+}\cap B^{+}$ is equal to 1, if the value of $A^{+}\cap \nabla B$ is equal to 0, and if the value of $\nabla A\cap B^{+}$ is equal to 1, then it means that feature A is all in the same layer of feature B in space, while feature B has parts that are in different layers of feature A, indicating that the topological relation between these two features in the XOY plane is inside.

Similarly, the other seven topological relations projected on the XOZ plane can also be subdivided, so that the 3DSRM-T model can distinguish between 40 topological relations. However, the topological relations distinguished using the 3DSRM-T model alone are not yet complete, and need to be distinguished by the distance and directional relation models.

Using the 3DSRM-D and 3DSRM-A models, it is equally possible to describe eight 2D topological relations between two features and to represent them in 3D space, with the corresponding matrix and 3D diagram for each topological relation shown in

Table 4. Eight topological relations described using 3DSRM-D and 3DSRM-A.

Spatial Relation	3DSRM-D	3DSRM-A	3D Diagram
Disjoint × Equal	$\left[\begin{array}{c}\begin{array}{cc}1& 1\\ 1& 1\end{array}\\ \begin{array}{cc}-1& -1\\ 0& 0\end{array}\end{array}\right]$	$\left[\begin{array}{c}\begin{array}{cc}0& 0\\ 1& 1\end{array}\\ \begin{array}{cc}-1& 0\\ 0& 1\end{array}\end{array}\right]$
Touches × Equal	$\left[\begin{array}{c}\begin{array}{cc}0& 1\\ 0& 1\end{array}\\ \begin{array}{cc}-1& -1\\ 0& 0\end{array}\end{array}\right]$	$\left[\begin{array}{c}\begin{array}{cc}1& 1\\ 1& 1\end{array}\\ \begin{array}{cc}-1& 0\\ 0& 1\end{array}\end{array}\right]$
Overlap × Equal	$\left[\begin{array}{c}\begin{array}{cc}-1& 1\\ -1& 1\end{array}\\ \begin{array}{cc}-1& -1\\ 0& 0\end{array}\end{array}\right]$	$\left[\begin{array}{c}\begin{array}{cc}0& 0\\ -1& -1\end{array}\\ \begin{array}{cc}-1& 0\\ 0& 1\end{array}\end{array}\right]$
Cover × Equal	$\left[\begin{array}{c}\begin{array}{cc}-1& 0\\ 0& -1\end{array}\\ \begin{array}{cc}-1& -1\\ 0& 0\end{array}\end{array}\right]$	$\left[\begin{array}{c}\begin{array}{cc}1& 1\\ 1& -1\end{array}\\ \begin{array}{cc}-1& 0\\ 0& 1\end{array}\end{array}\right]$
Contain × Equal	$\left[\begin{array}{c}\begin{array}{cc}-1& -1\\ 1& 1\end{array}\\ \begin{array}{cc}-1& -1\\ 0& 0\end{array}\end{array}\right]$	$\left[\begin{array}{c}\begin{array}{cc}0& 0\\ 1& -1\end{array}\\ \begin{array}{cc}-1& 0\\ 0& 1\end{array}\end{array}\right]$
Equal × Equal	$\left[\begin{array}{c}\begin{array}{cc}0& 0\\ 0& 0\end{array}\\ \begin{array}{cc}-1& -1\\ 0& 0\end{array}\end{array}\right]$	$\left[\begin{array}{c}\begin{array}{cc}1& 1\\ 0& 0\end{array}\\ \begin{array}{cc}-1& 0\\ 0& 1\end{array}\end{array}\right]$
CoveredBy × Equal	$\left[\begin{array}{c}\begin{array}{cc}0& 1\\ -1& 0\end{array}\\ \begin{array}{cc}-1& -1\\ 0& 0\end{array}\end{array}\right]$	$\left[\begin{array}{c}\begin{array}{cc}1& 1\\ -1& 1\end{array}\\ \begin{array}{cc}-1& 0\\ 0& 1\end{array}\end{array}\right]$
Inside × Equal	$\left[\begin{array}{c}\begin{array}{cc}1& 1\\ -1& -1\end{array}\\ \begin{array}{cc}-1& -1\\ 0& 0\end{array}\end{array}\right]$	$\left[\begin{array}{c}\begin{array}{cc}0& 0\\ -1& 1\end{array}\\ \begin{array}{cc}-1& 0\\ 0& 1\end{array}\end{array}\right]$

.

In the 3DSRM-D and 3DSRM-A models, the four values in the upper half of the matrix represent 2D planar topological relations, while the four values in the lower half are used to distinguish 3D topological relations.

Table 5. Topological relations that cannot be described by D-TRM but can be described by 3DSRM-D and 3DSRM-A.

Spatial Relationship	D-TRM	3DSRM-D	3DSRM-A	3D Diagram
Disjoint × Disjoint	$\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$	$\left[\begin{array}{c}\begin{array}{cc}1& 1\\ 1& 1\end{array}\\ \begin{array}{cc}1& 1\\ 1& 1\end{array}\end{array}\right]$	$\left[\begin{array}{c}\begin{array}{cc}0& 0\\ 1& 1\end{array}\\ \begin{array}{cc}1& 1\\ 1& 1\end{array}\end{array}\right]$ or $\left[\begin{array}{c}\begin{array}{cc}0& 0\\ 1& 1\end{array}\\ \begin{array}{cc}-1& -1\\ -1& -1\end{array}\end{array}\right]$
Disjoint × Touches		$\left[\begin{array}{c}\begin{array}{cc}1& 1\\ 1& 1\end{array}\\ \begin{array}{cc}0& 0\\ 1& 1\end{array}\end{array}\right]$	$\left[\begin{array}{c}\begin{array}{cc}0& 0\\ 1& 1\end{array}\\ \begin{array}{cc}0& 1\\ 1& 1\end{array}\end{array}\right]$ or $\left[\begin{array}{c}\begin{array}{cc}0& 0\\ 1& 1\end{array}\\ \begin{array}{cc}-1& -1\\ -1& 0\end{array}\end{array}\right]$
Disjoint × Overlap		$\left[\begin{array}{c}\begin{array}{cc}1& 1\\ 1& 1\end{array}\\ \begin{array}{cc}-1& -1\\ 1& 1\end{array}\end{array}\right]$	$\left[\begin{array}{c}\begin{array}{cc}0& 0\\ 1& 1\end{array}\\ \begin{array}{cc}-1& 1\\ 1& 1\end{array}\end{array}\right]$ or $\left[\begin{array}{c}\begin{array}{cc}0& 0\\ 1& 1\end{array}\\ \begin{array}{cc}-1& -1\\ -1& 1\end{array}\end{array}\right]$
Disjoint × Cover		$\left[\begin{array}{c}\begin{array}{cc}1& 1\\ 1& 1\end{array}\\ \begin{array}{cc}-1& -1\\ 1& 0\end{array}\end{array}\right]$	$\left[\begin{array}{c}\begin{array}{cc}0& 0\\ 1& 1\end{array}\\ \begin{array}{cc}-1& 1\\ 0& 1\end{array}\end{array}\right]$
Disjoint × Contain		$\left[\begin{array}{c}\begin{array}{cc}1& 1\\ 1& 1\end{array}\\ \begin{array}{cc}-1& -1\\ 1& -1\end{array}\end{array}\right]$	$\left[\begin{array}{c}\begin{array}{cc}0& 0\\ 1& 1\end{array}\\ \begin{array}{cc}-1& 1\\ -1& 1\end{array}\end{array}\right]$
Disjoint × Equal		$\left[\begin{array}{c}\begin{array}{cc}1& 1\\ 1& 1\end{array}\\ \begin{array}{cc}-1& -1\\ 0& 0\end{array}\end{array}\right]$	$\left[\begin{array}{c}\begin{array}{cc}0& 0\\ 1& 1\end{array}\\ \begin{array}{cc}-1& 0\\ 0& 1\end{array}\end{array}\right]$
Disjoint × CoveredBy		$\left[\begin{array}{c}\begin{array}{cc}1& 1\\ 1& 1\end{array}\\ \begin{array}{cc}-1& -1\\ 0& -1\end{array}\end{array}\right]$	$\left[\begin{array}{c}\begin{array}{cc}0& 0\\ 1& 1\end{array}\\ \begin{array}{cc}-1& 0\\ 1& 1\end{array}\end{array}\right]$
Disjoint × Inside		$\left[\begin{array}{c}\begin{array}{cc}1& 1\\ 1& 1\end{array}\\ \begin{array}{cc}-1& -1\\ -1& 1\end{array}\end{array}\right]$	$\left[\begin{array}{c}\begin{array}{cc}0& 0\\ 1& 1\end{array}\\ \begin{array}{cc}-1& -1\\ 1& 1\end{array}\end{array}\right]$

shows, as an example, the 3D topological relations that cannot be described by D-TRM, but can be described by 3DSRM-D and 3DSRM-A using the 2D planar disjoint topological relations.

1.

3DSRM-D model inferences

Inference 1.

If all four values in the lower half are greater than 0, then any point on feature A lies outside of feature B in the vertical direction, and vice versa. This means that the relation between the two features in the vertical direction is disjoint.

Inference 2.

If ${Min}_H\left(\partial A,\partial B\right)$ is equal to 0, ${Max}_H\left(\partial A,\partial B\right)$ is greater than 0, ${Min}_H\left(\partial B,\partial A\right)$ is equal to 0, and ${Max}_H\left(\partial B,\partial A\right)$ is greater than 0, then it means that any point on feature A lies outside or on the boundary of feature B in the vertical direction, and vice versa. This means that the relation between the two features in the vertical direction is touches.

Inference 3.

If ${Min}_H\left(\partial A,\partial B\right)$ is less than 0, ${Max}_H\left(\partial A,\partial B\right)$ is greater than 0, ${Min}_H\left(\partial B,\partial A\right)$ is less than 0 and ${Max}_H\left(\partial B,\partial A\right)$ is greater than 0, then this indicates that there are points on feature A that lie outside and inside feature B in the vertical direction, and vice versa. This means that the relation between the two features in the vertical direction is overlap.

Inference 4.

If ${Min}_H\left(\partial A,\partial B\right)$ is less than 0, ${Max}_H\left(\partial A,\partial B\right)$ is greater than 0, ${Min}_H\left(\partial B,\partial A\right)$ is less than 0 and ${Max}_H\left(\partial B,\partial A\right)$ is greater than 0, then it means that any point on feature B lies inside or on the boundary of feature A in the vertical direction. This means that the relation between the two features in the vertical direction is cover.

Inference 5.

If ${Min}_H\left(\partial A,\partial B\right)$ is equal to 0, ${Max}_H\left(\partial A,\partial B\right)$ is greater than 0, ${Min}_H\left(\partial B,\partial A\right)$ is equal to 0 and ${Max}_H\left(\partial B,\partial A\right)$ is less than 0, then it means that any point on feature B lies inside feature A in the vertical direction. This means that the relation between the two features in the vertical direction is contain.

Inference 6.

If ${Min}_H\left(\partial A,\partial B\right)$ is less than 0 and ${Max}_H\left(\partial A,\partial B\right)$ is equal to 0, and ${Min}_H\left(\partial B,\partial A\right)$ is less than 0 and ${Max}_H\left(\partial B,\partial A\right)$ is equal to 0, then it means that any point on feature A lies inside or on the boundary of feature B in the vertical direction, and vice versa. This means that the relation between the two features in the vertical direction is equal.

Inference 7.

If ${Min}_H\left(\partial A,\partial B\right)$ is less than 0, ${Max}_H\left(\partial A,\partial B\right)$ is equal to 0, ${Min}_H\left(\partial B,\partial A\right)$ is less than 0 and ${Max}_H\left(\partial B,\partial A\right)$ is less than 0, then it means that any point on feature A lies inside or on the boundary of feature B in the vertical direction. This means that the relation between the two features in the vertical direction is covered by.

Inference 8.

If ${Min}_H\left(\partial A,\partial B\right)$ is less than 0, ${Max}_H\left(\partial A,\partial B\right)$ is less than 0, ${Min}_H\left(\partial B,\partial A\right)$ is less than 0 and ${Max}_H\left(\partial B,\partial A\right)$ is greater than 0, then it means that any point on feature A lies inside feature B in the vertical direction. This means that the relation between the two features in the vertical direction is inside.

2.

3DSRM-A model inferences

Inference 1.

If all four values in the lower half are greater than 0 or less than 0, i.e., both the maximum and minimum pitch angles are greater than 0° or less than 0°, then it means that any point on feature A lies outside of feature B in the vertical direction, and vice versa. This means that the two features are disjointed in the vertical direction.

Inference 2.

If ${Min}_{\sphericalangle }\left(A_T,B\right)$ is equal to 0, ${Max}_{\sphericalangle }\left(A_T,B\right)$ is greater than 0, ${Min}_{\sphericalangle }\left(A_L,B\right)$ is greater than 0, ${Max}_{\sphericalangle }\left(A_L,B\right)$ is greater than 0, or if ${Min}_{\sphericalangle }\left(A_T,B\right)$ is less than 0, ${Max}_{\sphericalangle }\left(A_T,B\right)$ is less than 0, ${Min}_{\sphericalangle }\left(A_L,B\right)$ is less than 0 and ${Max}_{\sphericalangle }\left(A_L,B\right)$ is equal to 0, then any point on feature A lies on feature B in the vertical direction the exterior or boundary of the feature. This means that the relation between the two features in the vertical direction is touches.

Inference 3.

If ${Min}_{\sphericalangle }\left(A_T,B\right)$ is less than 0, ${Max}_{\sphericalangle }\left(A_T,B\right)$ is greater than 0, ${Min}_{\sphericalangle }\left(A_L,B\right)$ is greater than 0, ${Max}_{\sphericalangle }\left(A_L,B\right)$ is greater than 0, or if ${Min}_{\sphericalangle }\left(A_T,B\right)$ is less than 0, ${Max}_{\sphericalangle }\left(A_T,B\right)$ is less than 0, ${Min}_{\sphericalangle }\left(A_L,B\right)$ is less than 0 and ${Max}_{\sphericalangle }\left(A_L,B\right)$ is greater than 0, then there are points on feature A that lie on feature B in the vertical direction of the external and internal points. This means that the relation between the two features in the vertical direction is overlap.

Inference 4.

If ${Min}_{\sphericalangle }\left(A_T,B\right)$ is less than 0, ${Max}_{\sphericalangle }\left(A_T,B\right)$ is equal to 0, ${Min}_{\sphericalangle }\left(A_L,B\right)$ is greater than 0, and ${Max}_{\sphericalangle }\left(A_L,B\right)$ is greater than 0, then it means that any point on feature B lies inside or at the boundary of feature A in the vertical direction. This means that the relation between the two features in the vertical direction is cover.

Inference 5.

If ${Min}_{\sphericalangle }\left(A_T,B\right)$ is less than 0, ${Max}_{\sphericalangle }\left(A_T,B\right)$ is less than 0, ${Min}_{\sphericalangle }\left(A_L,B\right)$ is greater than 0 and ${Max}_{\sphericalangle }\left(A_L,B\right)$ is greater than 0, then it means that any point on feature B lies in the interior of feature A in the perpendicular direction. This means that the relation between the two features in the vertical direction is contain.

Inference 6.

If ${Min}_{\sphericalangle }\left(A_T,B\right)$ is less than 0, ${Max}_{\sphericalangle }\left(A_T,B\right)$ is equal to 0, ${Min}_{\sphericalangle }\left(A_L,B\right)$ is equal to 0 and ${Max}_{\sphericalangle }\left(A_L,B\right)$ is greater than 0, then it means that any point on feature A lies inside or on the boundary of feature B in the vertical direction, and vice versa. This means that the relation between the two features in the vertical direction is equal.

Inference 7.

If ${Min}_{\sphericalangle }\left(A_T,B\right)$ is less than 0, ${Max}_{\sphericalangle }\left(A_T,B\right)$ is greater than 0, ${Min}_{\sphericalangle }\left(A_L,B\right)$ is equal to 0 and ${Max}_{\sphericalangle }\left(A_L,B\right)$ is greater than 0, then it means that any point on feature A lies inside or on the boundary of feature B in the vertical direction. This means that the relation between the two features in the vertical direction is covered by.

Inference 8.

If ${Min}_{\sphericalangle }\left(A_T,B\right)$ is less than 0, ${Max}_{\sphericalangle }\left(A_T,B\right)$ is greater than 0, ${Min}_{\sphericalangle }\left(A_L,B\right)$ is less than 0 and ${Max}_{\sphericalangle }\left(A_L,B\right)$ is greater than 0, then it means that any point on feature A lies in the interior of feature B in the perpendicular direction. This means that the relation between the two features in the vertical direction is inside.

4.2. Derivation of the Spatial Relation of 3D ENC Features

The distance relation model or the directional relation model alone can derive the corresponding 3D topological relation, but when some information is missing from the model, it will affect the judgement of the topological relation. Combining the 3DSRM-D and 3DSRM-A models can increase the way to derive the spatial relation and improve the robustness of the spatial relation judgement.

Since both models consist of two-part relations, horizontal and vertical, they can be derived distributively. The 3D distance relationship matrix and the 3D orientation relationship matrix are divided into four sub-matrices, as follows.

$${\mathrm{R}}_{3\mathrm{D}\mathrm{S}\mathrm{R}\mathrm{M}\text{-}\mathrm{D}}\left(\mathrm{A},\mathrm{B}\right)=\left\{\left[\begin{array}{c}{\mathrm{M}\mathrm{i}\mathrm{n}}_{\mathrm{D}}\left(\partial \mathrm{A},\partial \mathrm{B}\right)\\ {\mathrm{M}\mathrm{a}\mathrm{x}}_{\mathrm{D}}\left(\partial \mathrm{A},\partial \mathrm{B}\right)\end{array}\right],\left[\begin{array}{c}{\mathrm{M}\mathrm{i}\mathrm{n}}_{\mathrm{D}}\left(\partial \mathrm{B},\partial \mathrm{A}\right)\\ {\mathrm{M}\mathrm{a}\mathrm{x}}_{\mathrm{D}}\left(\partial \mathrm{B},\partial \mathrm{A}\right)\end{array}\right],\left[\begin{array}{c}{\mathrm{M}\mathrm{i}\mathrm{n}}_{\mathrm{H}}\left(\partial \mathrm{A},\partial \mathrm{B}\right)\\ {\mathrm{M}\mathrm{a}\mathrm{x}}_{\mathrm{H}}\left(\partial \mathrm{A},\partial \mathrm{B}\right)\end{array}\right],\left[\begin{array}{c}{\mathrm{M}\mathrm{i}\mathrm{n}}_{\mathrm{H}}\left(\partial \mathrm{B},\partial \mathrm{A}\right)\\ {\mathrm{M}\mathrm{a}\mathrm{x}}_{\mathrm{H}}\left(\partial \mathrm{B},\partial \mathrm{A}\right)\end{array}\right]\right\}$$

$${\mathrm{R}}_{3\mathrm{D}\mathrm{S}\mathrm{R}\mathrm{M}\text{-}\mathrm{A}}\left(\mathrm{A},\mathrm{B}\right)=\left\{\left[\begin{array}{c}\mathrm{B}\mathrm{P}\left(\mathrm{A},\mathrm{B}\right)\\ \mathrm{R}\mathrm{P}\left(\mathrm{A},\mathrm{B}\right)\end{array}\right],\left[\begin{array}{c}\mathrm{B}\mathrm{P}\left(\mathrm{B},\mathrm{A}\right)\\ \mathrm{R}\mathrm{P}\left(\mathrm{B},\mathrm{A}\right)\end{array}\right],\left[\begin{array}{c}{\mathrm{M}\mathrm{i}\mathrm{n}}_{\mathrm{\sphericalangle }}\left({\mathrm{A}}_{\mathrm{T}},\mathrm{B}\right)\\ {\mathrm{M}\mathrm{a}\mathrm{x}}_{\mathrm{\sphericalangle }}\left({\mathrm{A}}_{\mathrm{T}},\mathrm{B}\right)\end{array}\right],\left[\begin{array}{c}{\mathrm{M}\mathrm{i}\mathrm{n}}_{\mathrm{\sphericalangle }}\left({\mathrm{A}}_{\mathrm{L}},\mathrm{B}\right)\\ {\mathrm{M}\mathrm{a}\mathrm{x}}_{\mathrm{\sphericalangle }}\left({\mathrm{A}}_{\mathrm{L}},\mathrm{B}\right)\end{array}\right]\right\}$$

Any combination of the first two matrices of each group can derive all the topological relations of the two features in the horizontal direction, and any combination of the last two matrices can derive most of the topological relations of the two features in the vertical direction, as shown in

Table 6. Results of the derivation of spatial relations.

No	Submatrices		Disjoint	Touches	Overlap	Cover	Contain	Equal	CoveredBy	Inside
1	$\left[\begin{array}{c}\mathrm{B}\mathrm{P}\left(\mathrm{A},\mathrm{B}\right)\\ \mathrm{R}\mathrm{P}\left(\mathrm{A},\mathrm{B}\right)\end{array}\right]$	$\left[\begin{array}{c}{\mathrm{M}\mathrm{i}\mathrm{n}}_{\mathrm{D}}\left(\partial \mathrm{A},\partial \mathrm{B}\right)\\ {\mathrm{M}\mathrm{a}\mathrm{x}}_{\mathrm{D}}\left(\partial \mathrm{A},\partial \mathrm{B}\right)\end{array}\right]$	$\left[\begin{array}{cc}0& 1\\ 1& 1\end{array}\right]$	$\left[\begin{array}{cc}1& 0\\ 1& 0\end{array}\right]$	$\left[\begin{array}{cc}0& -1\\ -1& -1\end{array}\right]$	$\left[\begin{array}{cc}1& -1\\ 1& 0\end{array}\right]$	$\left[\begin{array}{cc}0& -1\\ 1& 1\end{array}\right]$	$\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]$	$\left[\begin{array}{cc}1& 0\\ -1& -1\end{array}\right]$	$\left[\begin{array}{cc}0& 1\\ -1& -1\end{array}\right]$
2	$\left[\begin{array}{c}\mathrm{B}\mathrm{P}\left(\mathrm{B},\mathrm{A}\right)\\ \mathrm{R}\mathrm{P}\left(\mathrm{B},\mathrm{A}\right)\end{array}\right]$	$\left[\begin{array}{c}{\mathrm{M}\mathrm{i}\mathrm{n}}_{\mathrm{D}}\left(\partial \mathrm{A},\partial \mathrm{B}\right)\\ {\mathrm{M}\mathrm{a}\mathrm{x}}_{\mathrm{D}}\left(\partial \mathrm{A},\partial \mathrm{B}\right)\end{array}\right]$	$\left[\begin{array}{cc}0& 1\\ 1& 1\end{array}\right]$	$\left[\begin{array}{cc}1& 0\\ 1& 0\end{array}\right]$	$\left[\begin{array}{cc}0& -1\\ -1& -1\end{array}\right]$	$\left[\begin{array}{cc}1& -1\\ -1& 0\end{array}\right]$	$\left[\begin{array}{cc}0& -1\\ -1& 1\end{array}\right]$	$\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]$	$\left[\begin{array}{cc}1& 0\\ 1& -1\end{array}\right]$	$\left[\begin{array}{cc}0& 1\\ 1& -1\end{array}\right]$
3	$\left[\begin{array}{c}\mathrm{B}\mathrm{P}\left(\mathrm{A},\mathrm{B}\right)\\ \mathrm{R}\mathrm{P}\left(\mathrm{A},\mathrm{B}\right)\end{array}\right]$	$\left[\begin{array}{c}{\mathrm{M}\mathrm{i}\mathrm{n}}_{\mathrm{D}}\left(\partial \mathrm{B},\partial \mathrm{A}\right)\\ {\mathrm{M}\mathrm{a}\mathrm{x}}_{\mathrm{D}}\left(\partial \mathrm{B},\partial \mathrm{A}\right)\end{array}\right]$	$\left[\begin{array}{cc}0& 1\\ 1& 1\end{array}\right]$	$\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$	$\left[\begin{array}{cc}0& 1\\ -1& 1\end{array}\right]$	$\left[\begin{array}{cc}1& 0\\ 1& -1\end{array}\right]$	$\left[\begin{array}{cc}0& -1\\ 1& 1\end{array}\right]$	$\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]$	$\left[\begin{array}{cc}1& 1\\ -1& 0\end{array}\right]$	$\left[\begin{array}{cc}0& 1\\ -1& -1\end{array}\right]$
4	$\left[\begin{array}{c}\mathrm{B}\mathrm{P}\left(\mathrm{B},\mathrm{A}\right)\\ \mathrm{R}\mathrm{P}\left(\mathrm{B},\mathrm{A}\right)\end{array}\right]$	$\left[\begin{array}{c}{\mathrm{M}\mathrm{i}\mathrm{n}}_{\mathrm{D}}\left(\partial \mathrm{B},\partial \mathrm{A}\right)\\ {\mathrm{M}\mathrm{a}\mathrm{x}}_{\mathrm{D}}\left(\partial \mathrm{B},\partial \mathrm{A}\right)\end{array}\right]$	$\left[\begin{array}{cc}0& 1\\ 1& 1\end{array}\right]$	$\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$	$\left[\begin{array}{cc}0& 1\\ -1& 1\end{array}\right]$	$\left[\begin{array}{cc}1& 0\\ -1& -1\end{array}\right]$	$\left[\begin{array}{cc}0& -1\\ -1& 1\end{array}\right]$	$\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]$	$\left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]$	$\left[\begin{array}{cc}0& 1\\ 1& -1\end{array}\right]$
5	$\left[\begin{array}{c}{\mathrm{M}\mathrm{i}\mathrm{n}}_{\mathrm{\sphericalangle }}\left({\mathrm{A}}_{\mathrm{T}},\mathrm{B}\right)\\ {\mathrm{M}\mathrm{a}\mathrm{x}}_{\mathrm{\sphericalangle }}\left({\mathrm{A}}_{\mathrm{T}},\mathrm{B}\right)\end{array}\right]$	$\left[\begin{array}{c}{\mathrm{M}\mathrm{i}\mathrm{n}}_{\mathrm{H}}\left(\partial \mathrm{A},\partial \mathrm{B}\right)\\ {\mathrm{M}\mathrm{a}\mathrm{x}}_{\mathrm{H}}\left(\partial \mathrm{A},\partial \mathrm{B}\right)\end{array}\right]$	$\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$$\left[\begin{array}{cc}-1& 1\\ -1& 1\end{array}\right]$	$\left[\begin{array}{cc}0& 0\\ 1& 1\end{array}\right]$$\left[\begin{array}{cc}-1& 0\\ -1& 1\end{array}\right]$	$\left[\begin{array}{cc}-1& -1\\ 1& 1\end{array}\right]$$\left[\begin{array}{cc}-1& -1\\ -1& 1\end{array}\right]$	$\left[\begin{array}{cc}-1& -1\\ 0& 1\end{array}\right]$	$\left[\begin{array}{cc}-1& -1\\ -1& 1\end{array}\right]$	$\left[\begin{array}{cc}-1& -1\\ 0& 0\end{array}\right]$	$\left[\begin{array}{cc}-1& -1\\ 1& 0\end{array}\right]$	$\left[\begin{array}{cc}-1& -1\\ 1& -1\end{array}\right]$
6	$\left[\begin{array}{c}{\mathrm{M}\mathrm{i}\mathrm{n}}_{\mathrm{\sphericalangle }}\left({\mathrm{A}}_{\mathrm{L}},\mathrm{B}\right)\\ {\mathrm{M}\mathrm{a}\mathrm{x}}_{\mathrm{\sphericalangle }}\left({\mathrm{A}}_{\mathrm{L}},\mathrm{B}\right)\end{array}\right]$	$\left[\begin{array}{c}{\mathrm{M}\mathrm{i}\mathrm{n}}_{\mathrm{H}}\left(\partial \mathrm{A},\partial \mathrm{B}\right)\\ {\mathrm{M}\mathrm{a}\mathrm{x}}_{\mathrm{H}}\left(\partial \mathrm{A},\partial \mathrm{B}\right)\end{array}\right]$	$\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$$\left[\begin{array}{cc}-1& 1\\ -1& 1\end{array}\right]$	$\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]$$\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]$	$\left[\begin{array}{cc}1& -1\\ 1& 1\end{array}\right]$$\left[\begin{array}{cc}-1& -1\\ 1& 1\end{array}\right]$	$\left[\begin{array}{cc}1& -1\\ 1& 1\end{array}\right]$	$\left[\begin{array}{cc}1& -1\\ 1& 1\end{array}\right]$	$\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$	$\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$	$\left[\begin{array}{cc}-1& -1\\ 1& -1\end{array}\right]$
7	$\left[\begin{array}{c}{\mathrm{M}\mathrm{i}\mathrm{n}}_{\mathrm{\sphericalangle }}\left({\mathrm{A}}_{\mathrm{T}},\mathrm{B}\right)\\ {\mathrm{M}\mathrm{a}\mathrm{x}}_{\mathrm{\sphericalangle }}\left({\mathrm{A}}_{\mathrm{T}},\mathrm{B}\right)\end{array}\right]$	$\left[\begin{array}{c}{\mathrm{M}\mathrm{i}\mathrm{n}}_{\mathrm{H}}\left(\partial \mathrm{B},\partial \mathrm{A}\right)\\ {\mathrm{M}\mathrm{a}\mathrm{x}}_{\mathrm{H}}\left(\partial \mathrm{B},\partial \mathrm{A}\right)\end{array}\right]$	$\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$$\left[\begin{array}{cc}-1& 1\\ -1& 1\end{array}\right]$	$\left[\begin{array}{cc}0& 0\\ 1& 1\end{array}\right]$$\left[\begin{array}{cc}-1& 0\\ -1& 1\end{array}\right]$	$\left[\begin{array}{cc}-1& -1\\ 1& 1\end{array}\right]$$\left[\begin{array}{cc}-1& -1\\ -1& 1\end{array}\right]$	$\left[\begin{array}{cc}-1& -1\\ 0& 0\end{array}\right]$	$\left[\begin{array}{cc}-1& -1\\ -1& -1\end{array}\right]$	$\left[\begin{array}{cc}-1& -1\\ 0& 0\end{array}\right]$	$\left[\begin{array}{cc}-1& -1\\ 1& -1\end{array}\right]$	$\left[\begin{array}{cc}-1& -1\\ 1& 1\end{array}\right]$
8	$\left[\begin{array}{c}{\mathrm{M}\mathrm{i}\mathrm{n}}_{\mathrm{\sphericalangle }}\left({\mathrm{A}}_{\mathrm{L}},\mathrm{B}\right)\\ {\mathrm{M}\mathrm{a}\mathrm{x}}_{\mathrm{\sphericalangle }}\left({\mathrm{A}}_{\mathrm{L}},\mathrm{B}\right)\end{array}\right]$	$\left[\begin{array}{c}{\mathrm{M}\mathrm{i}\mathrm{n}}_{\mathrm{H}}\left(\partial \mathrm{B},\partial \mathrm{A}\right)\\ {\mathrm{M}\mathrm{a}\mathrm{x}}_{\mathrm{H}}\left(\partial \mathrm{B},\partial \mathrm{A}\right)\end{array}\right]$	$\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$$\left[\begin{array}{cc}-1& 1\\ -1& 1\end{array}\right]$	$\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]$$\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]$	$\left[\begin{array}{cc}1& -1\\ 1& 1\end{array}\right]$$\left[\begin{array}{cc}-1& -1\\ 1& 1\end{array}\right]$	$\left[\begin{array}{cc}1& -1\\ 1& 0\end{array}\right]$	$\left[\begin{array}{cc}1& -1\\ 1& -1\end{array}\right]$	$\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$	$\left[\begin{array}{cc}0& -1\\ 1& -1\end{array}\right]$	$\left[\begin{array}{cc}-1& -1\\ 1& 1\end{array}\right]$

. The individual topological relations that cannot be determined are {Equal, CoveredBy} for combination 6 and {Cover, Equal} for combination 7. Some of the topological relations that may be confusing are {Overlap, Contain} for combination 5, {Overlap, Cover, Contain} for combination 6, and {Overlap, Inside} for combination 7 and combination 8.

The spatial relations and spatial characteristics of features can also be derived from each other. The topological relations that may exist between land, sea surface and underwater features in the horizontal direction are disjoint and touches, and all eight topological relations may exist in the vertical direction. All eight topological relations between the sea surface and underwater features may exist in the horizontal direction, and the topological relation that may exist in the vertical direction is disjoint.

Table 7. Derivation of spatial relations and spatial characteristics.

	L	S	U
L		horizontal {Disjoint, Touches}	horizontal {Disjoint, Touches}
		vertical {All}	vertical {All}
S	horizontal {Disjoint, Touches}		Horizontal {All}
	vertical {All}		vertical {Disjoint}
U	horizontal {Disjoint, Touches}	Horizontal {All}
	vertical {All}	vertical {Disjoint}

shows the derivation of spatial relations and spatial properties.

5. Application of 3D ENC Features Spatial Relation Model

5.1. 3D Topological Relation Checking

Judging whether the topological relation between the 3D ENC features is correct is one of the prerequisites for displaying and using the chart. The focus of the 3D topological relation check is to determine the 3D topological relation between body and point, curve and surface features, as well as between features with different spatial characteristics, which consists of two main parts: one is to check for errors in the features themselves, and the other is to check for errors between features.

In 3D ENC, some point features, such as marker points, are still represented as point features with textual information, while features such as buoys and lights, for example, are in the form of 3D substantial models. Where the model is correctly displayed as a whole, representing a particular feature of the chart, errors occur when the model is split into several entities, due to the absence of key points. For example, a buoy will not overlap with other substantial features when displayed correctly, as shown in Figure 5 left, and if two buoy features appear to be equal or overlapping horizontally, then the model is in error, as shown in Figure 5 right.

For checking errors between features, there are also checks for topological relations between features of the same type and between features of different types. Features of the same type use the same model, e.g., the topological relation between two buoys in the vertical direction is equal, as shown in Figure 6a. If the topological relation in the vertical direction is covered by, a feature error may occur, as shown in Figure 6b, or an error may occur where the features are not of the same size, as shown in Figure 6c. If the topological relations in the vertical direction overlap, errors in the position of the features may occur, as shown in Figure 6d. For different types of features, when the spatial characteristics are the same (for example, the recommended route and buoy are both sea features), the recommended route is represented by a surface and the buoy by a body, and there are various possible topological relations in the horizontal direction, but the topological relation in the vertical direction must be touched, as shown in Figure 6e, if there are other spatial relations, such as overlap, then an error in the feature position is indicated, as shown in Figure 6f. When the spatial characteristics are not the same (for example, a wreck is an underwater feature and a buoy is a sea feature), the topological relations in the horizontal direction may be equal or internal, and the topological relation in the vertical direction must be touched, as shown in Figure 6g; if other spatial relations appear then the feature is in error, as shown in Figure 6h. In addition, complex features consisting of multiple features may also have topology errors. For example, a traffic separation scheme aggregation consists of multiple features, such as lateral buoys, isolated danger buoys, and two-way routes. The lateral buoys are used to indicate the left and right sides of the channel, and the topological relations between the lateral buoys and the channel in the horizontal direction are inside, as shown in Figure 6i, while an error is indicated if there is a topological relation such as overlap or disjoint, as shown in Figure 6j.

5.2. Topology Creation of Complex Features

The S-100 universal hydrological data model (S-100) defines a complex feature of ENC, which is a collection of features of the same type, or several features of different types [40]. The creation and analysis of a single type of feature are relatively simple, with a single topological relation, while complex features have more overlapping points, lines, surfaces and complex topological relations, and bring an increase in computation. The S-57 universal hydrological data model (S-57) does not define complex features, and the ENCs displayed are drawn separately with a single feature as a layer, making it difficult to represent the topological relations of different sub-features in complex features [41]. In the 3D ENC under the S-100 standard, creating topology rules for complex features ensures that they do not change due to geospatial changes, improves the authenticity of the features displayed while ensuring data quality, and reduces the computation of 3D scene creation by drawing uniformly. The method for creating complex feature topologies is shown in Figure 7.

1.

Create the topology rules

First, enter the complex features and set the topology rules for the features, including both the basic topology rules and the special topology rules. The basic topology rules are rules made according to the spatial characteristics of the features, such as that the topology relation between land features and sea surface features in the horizontal direction must be disjoint or touches, and the topology relation between sea surface features and underwater features in the vertical direction must be disjoint, and the same types of features cannot overlap. Special topology rules are based on the S-100 standard, such as depth contours cannot intersect each other, wreck points must be inside the hazardous area, etc.

2.

Topology verification

Calculate the spatial relation matrix between each sub-feature in the complex features, and verify according to the set topology rules; if there is a case of wrong topology relation, mark to show the topology error, and then carry out topology modification. If there is no topology relation error, then topology editing can be carried out.

3.

Topology modification

Topology modification is the process of readjusting the features from both the features themselves and the topology rules. For example, the fairway auxiliary feature is an association feature of the fairway and related features auxiliary to the fairway, which is composed of various features, such as beacon cardinal, beacon isolated danger, recommended track, restricted area navigational, etc., and contains both land and sea surface features. When two beacon cardinals of this complex feature overlap in the horizontal direction, topological modifications are required. Firstly, the features themselves can be adjusted by adding or deleting features, adjusting their coordinates, adjusting their accuracy, adjusting their type and adjusting their properties. Secondly, the topology rules can be adjusted by adding and deleting topology rules and adjusting topology types.

4.

Topology editing

Once the complex feature topology has been validated, topology editing is carried out to describe and record the topological relations that have been created, including defining topology names, topology tolerances, topology levels and topology categories. The topology name is named using the spatial relation, e.g., Disjoint×Equal, and the topology tolerance represents the minimum distance between two coordinates, e.g., 0.001m. The topology level represents the levels of detail (LOD) in the 3D scene, with different coordinate accuracy levels: the highest accuracy level is 1, and the lowest is 10. The topology category records the set topology rules.

5.

Topology query

After the topology of complex features is created, topology queries can be performed. This includes feature query, rule query, topology query, etc.

6. Conclusions

The spatial relation model of 3D ENC features can improve the accuracy and interactivity of 3D ENC, be used for the real-time drawing of 3D ENC features, enhance the convenience of using 3D ENC, reduce the misunderstanding of chart information, and provide a guarantee for the safety of ship navigation. The main contribution of this paper is to propose a 3D ENC spatial relation model based on the point-set topology theory, which combines 3D topological relations, 3D distance relations and 3D directional relations in a unified framework, and realises the mutual transformation and derivation of spatial relations. The model is a combination of qualitative and quantitative research on 3D spatial relations, which is more in line with the need for accuracy in determining the spatial relations of the features of 3D ENC. Compared with 9IM and D-TRM, it increases the information for considering topological relations from the vertical direction, has a higher detail description ability, and can effectively distinguish eight topological relations of disjoint, touches, overlap, cover, contain, equal, covered by and inside in horizontal and vertical directions. The model provides a basis for checking and creating the topological relations of 3D ENC features, and can judge the 3D topological relations between features themselves and features with different spatial characteristics, realise the creation of topology of complex features, help the combination of 3D virtual scenes and 2D ENC, and improve the spatial analysis and judgment ability.

The spatial relation model of 3D ENC features proposed in this paper is a preliminary study of 3D relations in ocean space, aiming to establish a unified model of 3D topological relations, 3D distance relations and 3D directional relations, and to solve the problems in the future quality verification of 3D ENC. The current research is mainly based on the geometric and spatial characteristics of the features, and has not yet discussed the issue of spatial relations after considering the properties of the features. For example, ocean water is a 3D space, and can be considered as a whole without considering the attributes of the water, but ocean water is a space with attributes such as temperature, salinity and density, and cannot be considered as a whole after considering the attributes of the water. As the different densities of the water have an important influence on underwater targets, such as submarines and underwater vehicles, it is necessary to analyse the spatial relations between underwater targets and the different attributes of the water. The analysis of the 3D spatial relation between the attributes of ocean water and the underwater target is one for future research directions.