Computational Geometry Questions Long
The Delaunay triangulation is a fundamental concept in computational geometry that provides a way to partition a set of points into a triangulated mesh such that no point lies inside the circumcircle of any triangle. This triangulation has several desirable properties, such as maximizing the minimum angle of all triangles and minimizing the overall edge length.
The algorithm for constructing the Delaunay triangulation can be summarized as follows:
1. Input: A set of points P in the plane.
2. Create a super-triangle that encloses all the points in P. This super-triangle should be large enough to ensure that no point lies outside its circumcircle. Typically, an equilateral triangle with vertices at infinity is used.
3. Insert the super-triangle into the triangulation.
4. For each point p in P, do the following:
a. Locate the triangle T in the current triangulation that contains p.
b. Remove T from the triangulation.
c. Create three new triangles by connecting p to the vertices of T.
d. Check the Delaunay condition for each new triangle. If any triangle violates the condition, flip its diagonal edge.
e. Add the three new triangles to the triangulation.
5. Remove all triangles that share an edge with the super-triangle.
6. Output: The resulting triangulation is the Delaunay triangulation of the input points.
The key step in this algorithm is the flipping of diagonal edges. This operation ensures that the Delaunay condition is satisfied, which guarantees that no point lies inside the circumcircle of any triangle. The flipping process continues until all triangles in the triangulation satisfy the Delaunay condition.
The time complexity of this algorithm is O(n log n), where n is the number of input points. This is because the algorithm involves a series of point location operations, which can be efficiently performed using techniques such as binary search trees or the Bowyer-Watson algorithm.
In conclusion, the algorithm for constructing the Delaunay triangulation involves creating a super-triangle, inserting it into the triangulation, and iteratively adding points while maintaining the Delaunay condition through edge flipping. The resulting triangulation provides a well-defined mesh with desirable geometric properties.