Tyson Polygon (Voronoi graph) generation algorithm
first. Purpose of the document
This paper describes the algorithm used to generate the Tyson polygon in the Geomodel module.
second. Overview
In GIS and geo-analysis, it is a common tool to solve the adjacency problem by using Tyson Polygon for fast interpolation and analyzing the influence area of geographic entity.
Dutch climate biologist A. H. Thiessen proposes a method of calculating the average rainfall based on the rainfall of a discrete-distributed meteorological station, which is to connect all adjacent meteorological stations into triangles, making the perpendicular bisector of each of these triangles, so that several perpendicular bisector around each station are surrounded by a polygon. Use the rainfall intensity of a unique weather station contained within this polygon to represent the intensity of rainfall in this polygon area, and call this polygon a Tyson polygon. 1, where the dotted line polygon is the Tyson Polygon. Tyson Polygon each vertex is the circumscribed circle center of each triangle. Tyson Polygons are also known as Voronoi graphs, or Dirichlet graphs.
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The characteristics of the Tyson Polygon are: |
Each Tyson polygon contains only one discrete point data
The distance from the point within the Tyson polygon to the corresponding discrete point is nearest
The distance from the point on the Tyson Polygon edge to the discrete point on either side of it is equal
Tyson polygons can be used for qualitative analysis, statistical analyses, proximity analysis, and so on. For example, the properties of a discrete point can be used to describe the properties of the Tyson Polygon region, and the data of the Tyson Polygon region could be computed by using discrete points, and it can be concluded that a discrete point adjacent to any other discrete points is directly based on the Tyson Polygon, and that the Jottessen polygon is n-sided, then adjacent to n discrete points. When a data point falls into a Tyson polygon, it is closest to the corresponding discrete point, eliminating the need to calculate distances.
In the construction of Tyson Polygon, it is first to make a triangular network of discrete points. This triangular network is called the Delaunay triangle Network.
Third, the construction of Delaulay Triangle
The construction of Delaunay Triangle Network is also called the construction of irregular triangular network, which is the construction of triangular network by discrete data points, 2, that is to determine which three data points constitute a triangle, also known as Automatic connection triangle network. That is, for n discrete points on a plane, whose plane coordinates are (xi,yi), I=1,2,...,n, the closest three points are the best triangles, so that each discrete point becomes the vertex of the triangle.
The result of automatically joining a triangular network is the marking of three vertices of all triangles, such as: 1,2,8;2,8,3;3,8,7;
In order to obtain the best triangle, in the triangular network, we should make the triangle's three internal angles as sharp angles as possible, that is, conform to the Delaunay Triangle production criteria:
1. Any circumscribed circle within a Delaunay triangle cannot contain any other discrete points.
2. The adjacent two Delaunay triangles form a convex quadrilateral, and after swapping the diagonal of the convex quadrilateral, the smallest of the six inner angles no longer increases. This property is the maximum criterion of the minimum angle.
The general algorithm of constructing Delaunay triangles in n dimensional Euclidean space is presented in Tsai (1993)---convex hull interpolation algorithm.
(i), convex package generation
1, find out the point set to meet min (x-y), Min (x+y), Max (x-y), Max (X+y) Four points, and a counter-clockwise to form a point of the list. These 4 points are the closest point in a discrete point to the 4 corners of the bounding rectangle that contains the discrete points. These 4 points form a polygon as the initial convex hull.
2, for each bump on the point I, set its subsequent point is J, calculate the vector segment ij to the right of all points to the distance from the IJ, to find the largest point K.
3. Insert k between I and J and assign K to J.
4, repeat 2, 3 steps until the point set is not on the right side of the line IJ point.
5, the J is assigned to I,j to take its follow-up point, repeat 2, 3, 4 steps.
6, when there is no discrete point on the right side of any adjacent two-point connection in convex hull, the end-point set convex hull is obtained.
Upon completion of this step, a polygon (convex hull) containing all the discrete points is formed, as shown in 3.
(ii), ring-cut boundary method convex package triangulation
Each time the convex hull list is searched for a triangle consisting of two adjacent convex edges, no other points on the convex hull are included in the inner and boundary of the triangle. This point is removed and a new convex chain list is obtained. Repeat this process until there are only three discrete points left in the convex chain list. The last three discrete points in the convex chain list are formed into a triangle to end the convex hull triangulation process.
After this step, the points in the convex hull are formed into several Delaunay triangles, as shown in 4.
(iii), discrete point interpolation
After triangulation of the convex hull, the remaining discrete points on the convex hull can be divided by point interpolation method. The basic process is:
1. Select a discrete point that has not yet formed a triangle
2. In the triangle that has been generated, find the triangle of the discrete point (the discrete point is inside the triangle or on the edge of the triangle)
3, if the discrete point inside the triangle, then the triangle and the edge of the triangle is deleted, and then the three vertices and the discrete points are connected to form three new triangles. If the discrete point is on the edge of the triangle, the edge of the record point is e, according to the topological relationship, find the left and right neighboring triangle t1,t2 of the edge, add four new edges and four new triangular NT, delete the T1,t2 and the Edge e.
For newly generated triangles, it is necessary to circumscribed circle the edges of each of them in a heartbeat. For the edge e of the newly generated triangle, find the two triangles adjacent to the edge, and determine whether the diagonal vertices of that side are located inside the circumscribed circle of the other triangle. If it is, then the Edge e is deleted, and then the two diagonal is joined together to form two new triangles. For the edges of the new triangle, an empty circumscribed circle detection is also required, so continue until all newly generated triangles are detected by an empty circumscribed circle.
4, repeat 1, 2, 3, until all non-convex shell discrete points are inserted so far. Upon completion of this step, the construction of the Delaunay triangular network is completed, as shown in 5.
Fourth, the creation of Tyson Polygon steps
The key to the establishment of the Tyson Polygon algorithm is to construct the Delaunay triangular network by reasonably connecting the discrete data points to the triangular network. The steps to build the Tyson Polygon are:
1, the discrete point automatically constructs the triangular network, namely constructs the Delaunay triangle net. The number of discrete points and triangles formed, and the three discrete points in which each triangle is recorded.
2. Find the number of all triangles adjacent to each discrete point and record them. This simply finds all triangles with one of the same vertices in the constructed triangular network.
3. The triangles adjacent to each discrete point are sorted in a clockwise or counterclockwise direction so that the next connection is generated by the Tyson Polygon. The sort method can be shown in 6. Set the discrete point to O. Find a triangle with O as its vertex, set to A; take triangle a other than O vertex, set to a, the other vertex can also be found, that is f; then the next triangle must be in the side of, that is, the triangle F; the other vertex of the triangle F is e, the next triangle is on the OE side; so repeated, Until you return to the OA side.
4, calculate the circumscribed circle center of each triangle, and record it.
5, according to each discrete point of the adjacent triangle, connecting the circumscribed Circle center of these adjacent triangles, that is, Tyson Polygon. For the triangle network edge of the Tyson Polygon, can be used as perpendicular bisector and the map, together with the map to form Tyson Polygon. Source: https://www.douban.com/note/475322367/
Tyson Polygon-the strongest brain program knowledge
