Delaunay Triangulation
Definition of Delaunay Triangulation
A collection of connected but non-overlapping triangles whose circumcircle does not contain any other points of the region.
To sum up, any triangle in the generated triangulation network cannot contain the vertices of other triangles.
Important Criteria for Delaunay Triangulation
Among them, the empty circumscribed circle criterion and the maximum opening angle criterion are the two most commonly used criteria
Lawson's Local Optimization Algorithm (LOP)
For the process of generating a triangulation, inserting new points into the already generated triangulation will cause the new triangulation to no longer conform to the triangulation criterion. Based on the maximum and minimum angle rule, Lawson also gave a local optimization algorithm (LOP), exchanging the diagonals of the convex quadrilateral, keeping the short diagonal, and maximizing the minimum angle of all triangles in the triangulation.
Therefore, in the process of network construction, or inserting a new point set into the original triangulation, it is not necessary to traverse the entire triangulation every time to find triangles that do not meet the triangulation and optimize it, because the points other than the newly inserted TINs are all triangulations. Therefore, LOP only needs to detect the new triangle and its adjacent triangles to make them conform to the triangle profile, which greatly saves time consumption.
Establishment of Triangular Irregular Network (TIN)
Triangulation construction method:
Split Merge Algorithm
Use divide and conquer strategy to simplify complex problems:
First divide the data points into point subsets that are easy to triangulate (such as 3 or 4 points per subset), then triangulate each subset separately, and optimize it into a D_triangulation network by LOP; then triangulate each subset The nets are merged to form the final D_triangular net.
The segmentation and merging triangulation algorithm is shown in the figure:
STEP1
Sort the data set in ascending order with the abscissa as the main and the ordinate as the auxiliary.
STEP2
If the number of points in the data set is greater than the threshold, continue to divide the data set into two subsets with approximately equal number of points, and do the following work for each subset:
① Obtain the convex hull of each subset;
② Use the convex hull as the data boundary to perform triangulation, and use LOP to optimize it into a D triangulation network;
③ Find the bottom line and top line connecting the two convex hulls of the left and right subsets;
④ Merge two TINs from the bottom line to the top line.
STEP3
If the number of points in the data set is not greater than the threshold, the triangulation result will be output directly.
The data point set adopts the recursive segmentation and quick sorting method; the generation of the subset convex hull can adopt the Graham algorithm (see below); the subset triangulation can adopt any method, such as the minimum subset of 3 or 4 points can be directly triangulated For subdivision and subnetwork merging, it is necessary to first find the bottom line and top line of the left and right subset convex hulls (see the algorithm below), and then gradually merge the triangulations to obtain the final D triangulation.
