Kinetic triangulation
A kinetic triangulation data structure is a kinetic data structure that maintains a triangulation of a set of moving points. Maintaining a kinetic triangulation is important for applications that involve motion planning, such as video games, virtual reality, dynamic simulations and robotics.
Choosing a triangulation scheme
The efficiency of a kinetic data structure is defined based on the ratio of the number of internal events to external events, thus good runtime bounds can sometimes be obtained by choosing to use a triangulation scheme that generates a small number of external events.For simple affine motion of the points, the number of discrete changes to the convex hull is estimated by, thus the number of changes to any triangulation is also lower bounded by. Finding any triangulation scheme that has a near-quadratic bound on the number of discrete changes is an important open problem.
Delaunay triangulation
The Delaunay triangulation seems like a natural candidate, but a tight worst-case analysis of the number of discrete changes that will occur to the Delaunay triangulation was considered an open problem until 2015; it has now been bounded to be between and.There is a kinetic data structure that efficiently maintains the Delaunay triangulation of a set of moving points, in which the ratio of the total number of events to the number of external events is.