Raising Roofs, Crashing Cycles, and Playing Pool: Applications of a Data Structure for Finding Pairwise Interactions

Abstract

The straight skeleton of a polygon is a variant of the medial axis introduced by Aichholzer et al., defined by a shrinking process in which each edge of the polygon moves inward at a fixed rate. We construct the straight skeleton of an n-gon with r reflex vertices in time O(n 1+ε + n 8/11+ε r 9/11+ε ), for any fixed ε > 0, improving the previous best upper bound of O(nr log n). Our algorithm simulates the sequence of collisions between edges and vertices during the shrinking process, using a technique of Eppstein for maintaining extrema of binary functions to reduce the problem of finding successive interactions to two dynamic range query problems: (1) maintain a changing set of triangles in R 3 and answer queries asking which triangle is first hit by a query ray, and (2) maintain a changing set of rays in R 3 and answer queries asking for the lowest intersection of any ray with a query triangle. We also exploit a novel characterization of the straight skeleton as a lower envelope of triangles in R 3 . The same time bounds apply to constructing non-self-intersecting offset curves with mitered or beveled corners, and similar methods extend to other problems of simulating collisions and other pairwise interactions among sets of moving objects.

Bim-Based Indoor Path Planning Considering Obstacles

“…Most of the existing methods for extracting the path network are based on medial Axis Transformation (MAT) and Visibility Graph (VG) (Yang and Worboys, 2015). Straight-Medial Axis Transformation (S-MAT) is proposed by Eppstein and Erickson(1999) and it results to be a good representation of natural human behaviour, though, it may not represent accessibility within buildings accurately (Mortari et al, 2013). In open space, it is obviously the weakness of S-MAT that the generated path is distort.…”

Untitled

“…As the parameter s varies, the vertices of the adjoint polytopes trace out a skeleton of straight line segments (compare Figure 2 and Lemma 1.12). In computational geometry there are similar constructions such as the medial axis and the straight skeleton [Aichholzer et al 1995;Eppstein and Erickson 1999], which are of importance in many applications from geography to computer graphics. "Roof constructions" such as M(P) in Proposition 1.14 are also intensively studied in this context (compare Figure 4).…”

Straight Skeletons of Three-Dimensional Polyhedra

“…The straight line segments traced out by vertices during this offset process define the straight skeleton. Introduced in 1995 by Aichholzer et al [1,2], the two-dimensional straight skeleton has since found many applications, including surface folding [11], offset curve construction [15], interpolation of three-dimensional surfaces from cross-section contours [4], automated interpretation of geographic data [17], polygon decomposition [24], and graph drawing [3]. Compared to other well-known types of skeleton, the straight skeleton is more complex to compute [7,15], but its simple geometric form, comprised exclusively of line segments, offers advantages in applications.…”