This paper discusses two algorithms for constructing a Delaunay triangulation over a planar set of N points. The Delaunay triangulation is a specific type of triangulation where the circumcircle of any triangle in the triangulation contains no other points of the set in its interior. The paper reviews the geometric properties of Delaunay triangulations and their applications, particularly in computational geometry and digital terrain modeling. The first algorithm uses a divide-and-conquer approach and runs in O(N log N) time, which is asymptotically optimal. The second algorithm is iterative and has a worst-case time complexity of O(N²), but its average-case performance is comparable to the first algorithm. The Delaunay triangulation is closely related to the Voronoi tessellation. Voronoi polygons are regions around each point in the set, containing all points closer to that point than to any other. These polygons form a tessellation of the plane, with each polygon associated with a point in the set. The paper defines the Delaunay triangulation and its properties, and presents two algorithms for its construction. It also discusses applications of the triangulation, including its use in modeling terrain surfaces. The Delaunay triangulation is noted as an excellent choice for this application due to its efficiency and visual quality.This paper discusses two algorithms for constructing a Delaunay triangulation over a planar set of N points. The Delaunay triangulation is a specific type of triangulation where the circumcircle of any triangle in the triangulation contains no other points of the set in its interior. The paper reviews the geometric properties of Delaunay triangulations and their applications, particularly in computational geometry and digital terrain modeling. The first algorithm uses a divide-and-conquer approach and runs in O(N log N) time, which is asymptotically optimal. The second algorithm is iterative and has a worst-case time complexity of O(N²), but its average-case performance is comparable to the first algorithm. The Delaunay triangulation is closely related to the Voronoi tessellation. Voronoi polygons are regions around each point in the set, containing all points closer to that point than to any other. These polygons form a tessellation of the plane, with each polygon associated with a point in the set. The paper defines the Delaunay triangulation and its properties, and presents two algorithms for its construction. It also discusses applications of the triangulation, including its use in modeling terrain surfaces. The Delaunay triangulation is noted as an excellent choice for this application due to its efficiency and visual quality.