Topological relation refers to the relationship betweenspatial data The relationship between.That is, the adjacency, association, inclusion and connectivity between entities represented by nodes, arcs and polygons.For example, the adjacency relationship between points, the inclusion relationship between points and faces, the separation relationship between lines and faces, and the coincidence relationship between faces.
Topological relationship refers toGraphic elementThe connection and adjacency in the space between them do not consider the specific location.This topological relationship is formed by digitized point, line, and area data, and graph selection, overlapping, and merging are carried out according to user's query or application analysis requirements
Operation.Establishing topological relationships between spatial features belongs to map decoration.
The spatial connection between entities such as points, lines, and faces, such as connectivity, adjacency, inclusion, etc.Connectivity refers to the identification of the connection between line segments;It can be represented by a list of line segments assembled on each node.Adjacency usually refers to the adjacency between polygons;The containment relationship usually means that a polygon contains points or other polygons.[1]
topology
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It represents the positions of various objects as abstract positions.[2]In the network, topology vividly describes the arrangement and configuration of the network, including various nodes and the relationship between nodes.Topology does not care about the details of things or the proportion relationship between them. It only shows the relationship between the things within the scope of discussion, and shows the relationship between these things through a graph.
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Non topological properties
The distance between two points;The direction in which one point points to another;The length of the arc segment;The perimeter of an area;The area of an area.
Topology Properties
A point is at the end of an arc segment;A simple arc segment will not self intersect;A point is on the boundary of an area;A point is inside an area;A point is outside of an area;A point is inside a ring;A simple face is a continuous face.
Topology data structure
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1. Basic elements of topology
① Topological segment (arc)
This segment is not connected with other segments.
② Node
The two endpoints of a topological segment are the first node and the last node
③ Poly
Connected by several topological segments
Topology data example：
Topology data example
Illustration
2. Creation of topology relation table
Node code: ① ② ③ ④ ⑤ ⑥
Line segment code: 1 2 3 4 5 6 7 8 9
Polygon code: (1) (2) (3) (4) (5)
Illustration
Common topology
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Star structure
The star structure is based on anodeAs a central processing system, all types of network access machines are directly connected to the central node through physical links.
The advantages of star structure are simple structure, easy network construction and relatively simple control.Its disadvantages are centralized control, overload of main node, low reliability and low utilization rate of communication line.
Bus structure
The bus structure is a commonly used way, which connects all the computers connected to the network to a communication line. In order to prevent signal reflection, terminators are generally connected at both ends of the bus to match the line impedance.
Bus structureIts advantages are high channel utilization, simple structure and relatively cheap price.The disadvantage is that there can only be two at a timenetwork nodeMutual communication, limited network extension distance and limited number of nodes that the network can accommodate.stayBusThe normal operation of the entire network will be affected if there is a connection problem at one point on the.This structure is often used in local area networks.
Annular structure
The ring structure is to connect all networked computers into a closed ring with communication lines.
ring topology Is a point-to-pointAnnular structure。Each device is directly connected to the ring, or connected to the ring through an interface device and branch cable.On initial installation,ring topology The network is relatively simple.With the onlinenodeWith the increase of, the difficulty of reconfiguration also increases, which limits the maximum length of the ring and the total number of devices on the ring.It is easy to find the fault point of the cable.The range of equipment affected by the fault is large. Any error on the single loop system will affect all equipment on the network.
Tree structure
Star network topologyOne extension of is the star tree.The connection between each Hub and the end user is still star shaped, and the hierarchical connection of the Hub forms a tree.However, it should be noted that the number of Hub cascades is limited and varies with different manufacturers.
The tree structure is a hierarchical centralized control network. Compared with the star structure, its total length of communication lines is short, the cost is low, the nodes are easy to expand, and it is convenient to find paths. But except for leaf nodes and their connected lines, the failure of any node or its connected lines will affect the system.
Application: It is only applicable to low-speed signals without impedance control. For example, if there is no power supply layer, this topology can be used for power wiring.
Reticular structure
The reticular structure can be divided into fully connected reticular structure and incompletely connected reticular structure.In a fully connected mesh, eachnodeIt has link connection with other nodes in the network.In incomplete connection network, twonodeThere is not necessarily a direct link connection between them. Their communication depends on other nodes.The advantages of this network arenodebetweenrouteMany, collision and blocking can be greatly reduced, and localfaultIt will not affect the normal operation of the whole network and has high reliability;Network expansion andhostAccess to the network is flexible and simple.However, this kind of network relationship is complex and it is not easy to build a network,Network controlThe mechanism is complex.WANIn general, incomplete connection network structure is used.
Hybrid topology
Two or more topologies are used at the same time.
Advantages: The basic topology of the network can learn from each other.Disadvantages: network configuration is difficult.
Cellular topology
The cellular topology isWireless LANStructure commonly used in.It usesWireless transmission media(microwave, satellite, infrared, etc.) point-to-point and multi-point transmission. It is a wireless network, suitable for urban networks, campus networksEnterprise network。
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Folding of the problem of the seven bridges in Gothenburg
In mathematics, about the problem of the seven bridges in GothenburgEuler's theorem of polyhedronThe four color problem is an important problem in the history of topology.The problem of seven bridges in konisburg.In the 18th century, seven bridges were built on the river, connecting the two islands in the middle of the river and the river bank.People often take a walk on this bridge in their spare time. One day, someone asked whether they could walk on each bridge only once and finally return to the original position.This seemingly simple and interesting question attracts everyone. Many people are trying all kinds of walking methods, but no one has done it.It seems that it is not easy to get a clear and ideal answer.
In 1736, someone took this question to Euler, a mathematician at that time. After some thinking, Euler soon gave the answer in a unique way.Euler first simplified the problem. He regarded the two islands and the two banks of the river as four points, and the seven bridges as the connection between the four points.Then the question will be simplified to whether the figure can be drawn with one stroke.After further analysis, Euler came to the conclusion that it was impossible to walk every bridge once and finally return to the original position.And the conditions that all figures can be drawn in one stroke should have are given.This is the "precursor" of topology.
Euler Theorem Folding of Polyhedron
In the history of topology, there is also a famous and important theorem about polyhedron, which is also related to Euler.The content of this theorem is that if the number of vertices, edges and faces of a convex polyhedron is v, e and f, then they always have the following relationship: f+v-e=2.
According to Euler's theorem of polyhedron, we can get such an interesting fact: there are only five kinds of regular polyhedron.
The famous "four color problem" is also related to the development of topology.The four color problem, also known as the four color conjecture, is one of the three modern mathematical problems in the world.The four color conjecture came from Britain.In 1852, when Francis Guthrie, who graduated from London University, came to a scientific research unit to do map coloring work, he found an interesting phenomenon: "It seems that every map can be colored with four colors, so that countries with common borders are painted with different colors."
In 1872, Kelly, the most famous mathematician in Britain at that time, formally raised this question to the London Mathematical Society, so the four color conjecture became a concern of the world's mathematical community.Many first-class mathematicians in the world have participated in the battle of four color conjecture.Between 1878 and 1880, famous lawyers and mathematicians Kemp and Taylor submitted papers respectively to prove the four-color conjecture and announced that they had proved the four-color theorem.But later, the mathematician Herwood pointed out that Kemp's proof was wrong with his own accurate calculation.Soon, Taylor's proof was also denied.Therefore, people began to realize that this seemingly easy topic is actually a problem comparable to Fermat's conjecture.
Since the beginning of the 20th century, scientists have been basically following Kemp's ideas in proving the four-color conjecture.After the advent of electronic computers, the process of proving the four-color conjecture has been greatly accelerated due to the rapid increase of computing speed and the emergence of human-computer dialogue.In 1976, American mathematicians Apel and Haken made 10 billion judgments in 1200 hours on two different computers at the University of Illinois, and finally completed the proof of the four color theorem.However, many mathematicians are not satisfied with the achievements made by computers. They believe that there should be a simple and straightforward method of written proof.
The above examples are all about some problems related to geometry, but these problems are different from traditional geometry, but they are some new geometric concepts.These are the forerunners of "topology".
