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 (external events) 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,Gerhard Albers, Leonidas J. Guibas, Joseph S. B. Mitchell, and Thomas Roos. Voronoi diagrams of moving points. Int. J. Comput. Geometry Appl., 8(3):365{380, 1998. in which the ratio of the total number of events to the number of external events is .
Other triangulations Kaplan et al. developed a randomized triangulation scheme that experiences an expected number of external events, where is the maximum number of times each triple of points can become collinear, , and is the maximum length of a Davenport-Schinzel sequence of order s + 2 on n symbols.
Pseudo-triangulations There is a kinetic data structure (due to Agarwal et al.) which maintains a pseudo-triangulation in events total.Pankaj K. Agarwal, Julien Basch, Leonidas J. Guibas, John Hershberger, and Li Zhang. Deformable free-space tilings for kinetic collision detection. I. J. Robotic Res., 21(3):179{198, 2002. All events are external and require time to process.
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