Determining the separation of preprocessed polyhedra — A unified approach

Exploring Different Coherence Dimensions to Answer Proximity Queries for Convex Polyhedra

“…Moreover, the bulk of the available results are relative to distance computation, and then provide the natural benchmarks against which to compare new techniques. (In principle, it would be possible to consider Dobkin and Kirkpatrick's approach [7], but we only know about experiments based on its instantiation to compute distances.) H-Walk [1] combines the advantages of the algorithms proposed by Dobkin and Kirkpatrick [7] and by Lin and Canny [5].…”

“…(In principle, it would be possible to consider Dobkin and Kirkpatrick's approach [7], but we only know about experiments based on its instantiation to compute distances.) H-Walk [1] combines the advantages of the algorithms proposed by Dobkin and Kirkpatrick [7] and by Lin and Canny [5]. Lin and Canny introduced the key idea of exploiting the coherence, by observing that, when a query is asked frequently, the two closest features (points, edges or faces) can be updated more efficiently by starting from the former pair of closest features.…”

“…Since the early models of robot workspaces, there has been a steady interest in the proximity-query algorithms [6], which are generally thought of as being crucial for motion planning and for other applications in such fields as simulation, animation, virtual reality. The best asymptotic bound in the worst-case for various proximity problems on convex polytopes is O(log 2 n) [7], and can be attained by exploiting the hierarchical representations of polyhedra. However, several algorithms proposed in the literature apply to convex polyhedra, e.g.…”

Collision detection algorithms for motion planning

“…Moreover, convex polyhedra can be properly preprocessed, as described in [17], to make the complexity of intersection detection drop to O(log n log m). Preprocessing takes O(n + m) time to build a hierarchical representation of two polyhedra with n and m vertices.…”

“…This is the idea behind Dobkin and Kirkpatrick's hierarchical polyhedral representation, already mentioned in Section 2. Using their representation leads to distance computation in optimal worst-case O(log n log m) time [17]. Every step of the closest points search procedure corresponds to a level in the construction of the hierarchical representation.…”