Kinetic convex hull

A kinetic convex hull data structure is a kinetic data structure that maintains the convex hull of a set of continuously moving points.

The best known data structure for the 2-dimensional kinetic convex hull problem is by Basch, Guibas, and Hershberger. This data structure is responsive, efficient, compact and local.

The dual of a convex hull of a set of points is the upper and lower envelopes of the dual set of lines. Therefore, maintaining the upper and lower envelopes of a set of moving lines is equivalent to maintaining the convex hull of a set of moving points. Computing upper and lower envelopes are equivalent problems, so computing the upper envelope of a set of lines is equivalent to computing the convex hull of a set of moving points. The upper envelope of a set of static lines can be computed using a divide and conquer algorithm which partitions the lines into two sets of equal size, calls itself recursively on the two sets to find their upper envelopes, and then merges the two resulting upper envelopes. The merge step is performed using a vertical line sweep. Call the first set of points blue and the second set of points red.

The standard line sweep algorithm for merging upper envelopes sweeps though all of vertices of the red and blue upper envelopes, from left to right. The most recently encountered red and blue points are maintained as the line sweeps. When a point is encountered, the algorithm checks if the point is above or below the segment following the last encountered point of the opposite color. If it is above, that point is added to the merged upper envelope. If it of a different color than the last added point, the red and blue envelopes have crossed, so the intersection point is also added to the merged upper envelope.

The sequence of edges and vertices resulting from this algorithm is only dependent on the ordering of points, and the results of the line-point comparisons. Thus, the result can be certified with the following certificates:

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