1. Triangulation

        Triangulation: Suppose V is a finite set of points on the two-dimensional real number field, the edge e is a closed line segment composed of points in the point set as endpoints, and E is the set of e. Then a triangulation T=(V, E) of the point set V is a plane graph G that satisfies the conditions:
        1. Except for the endpoints, the edges in the plane graph do not contain any points in the point set.
        2. No intersecting edges.
        3. All faces in a plane graph are triangular faces, and the collection of all triangular faces is the convex hull of the scatter set V.

2. Delaunay triangulation

1. Delaunay side

Suppose that for an edge e in E (two endpoints are a, b), there is a circle passing through two points a and b, and the circle does not contain any other points in the point set V, and there are at most three points on the circle, then e called Delaunay side.

ps: Such a circle without other points is called an empty circle.

2, Delaunay triangulation

If a triangulation T of a point set V contains only Delaunay edges, then T is called a Delaunay triangulation.

3. Condition of point set

A necessary and sufficient condition for a point set to have a Delaunay triangulation is that there is no empty circle with four points.

4. Characteristics of Delaunay Triangulation

(1) If there is no point set with four points on an empty circle, then there is a unique Delaunay triangulation.

(2) Maximize the minimum angle feature: Among the triangulations that may be formed by the scatter set, the minimum angle of the triangle formed by the Delaunay triangulation is the largest.

(3) Regional: When adding, deleting, or moving a vertex, only the adjacent triangles are affected.

3. Thiessen Polygon

1. Tyson Polygon

The Thiessen polygon (division) is based on Delaunay triangulation. Based on the result of this division, the vertical bisectors of all sides are drawn to form a Thiessen polygon.

2. Characteristics of Thiessen Polygons

(1) There is one and only one point (sample) in each polygon

(2) Any position within a polygon is the closest to the sample point of the polygon, and farther from the sample points of other polygons.

Plane division