What are the most promising directions for polygon triangulation algorithms?

4 Optimal polygon triangulation

A challenging direction for polygon triangulation algorithms is to find the optimal polygon triangulation, which is the triangulation that minimizes a certain objective function, such as the total length, area, or perimeter of the triangles. Finding the optimal polygon triangulation is known to be NP-hard, which means that there is no efficient algorithm that can solve it in polynomial time. However, there are some approximation algorithms that can find near-optimal solutions in reasonable time, such as dynamic programming, greedy algorithms, or randomized algorithms. The advantage of these algorithms is that they can achieve better results than the previous methods, especially for polygons with many vertices or special properties. The disadvantage is that they can be very slow, complex, or impractical for real-world applications.

5 Polygon triangulation with constraints

A practical direction for polygon triangulation algorithms is to consider polygons with constraints, such as holes, boundaries, regions, attributes, or quality measures. These constraints can reflect the physical, logical, or aesthetic requirements of the application domain, such as engineering, mapping, or art. Polygon triangulation with constraints is a generalization of the constrained Delaunay triangulation, and it can be solved by using various techniques, such as inserting additional vertices, splitting or merging edges, or modifying the objective function. The advantage of these algorithms is that they can accommodate diverse and complex scenarios and produce customized and satisfactory solutions. The disadvantage is that they can be difficult to design, implement, and evaluate, and they may not have a clear or unique definition of optimality.

6 Polygon triangulation with holes

A special case of polygon triangulation with constraints is polygon triangulation with holes, which is the problem of triangulating a polygon that has one or more interior polygons that are not part of the domain. Polygon triangulation with holes is important for many applications that involve irregular or non-convex domains, such as image processing, computer vision, or computational biology. Polygon triangulation with holes can be solved by using various methods, such as transforming the polygon into a simple polygon, using a constrained Delaunay triangulation, or using a hierarchical decomposition. The advantage of these methods is that they can handle polygons with arbitrary shapes and sizes of holes, and they can preserve the topology and geometry of the domain. The disadvantage is that they can introduce additional complexity and overhead, and they may not guarantee the optimal or minimum number of triangles.

7 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?