Rotational polygon containment and minimum enclosure using only robust 2D constructions

Abstract

An algorithm and a robust floating point implementation is given for rotational polygon containment: given polygons P 1 , P 2 , P 3 ,. .. , P k and a container polygon C, find rotations and translations for the k polygons that place them into the container without overlapping. A version of the algorithm and implementation also solves rotational minimum enclosure: given a class C of container polygons, find a container C ∈ C of minimum area for which containment has a solution. The minimum enclosure is approximate: it bounds the minimum area between (1 − ε)A and A. Experiments indicate that finding the minimum enclosure is practical for k = 2, 3 but not larger unless optimality is sacrificed or angles ranges are limited (although these solutions can still be useful). Important applications for these algorithm to industrial problems are discussed. The paper also gives practical algorithms and numerical techniques for robustly calculating polygon set intersection, Minkowski sum, and range intersection: the intersection of a polygon with itself as it rotates through a range of angles. In particular, it introduces nearest pair rounding, which allows all these calculations to be carried out in rounded floating point arithmetic.

An approximate arrangement algorithm for semi-algebraic curves

“…For example, Milenkovic uses cascaded numerical geometric operations in part layout. 23,24,25 However, one can construct any algebraic expression by cascading two simple geometric constructions: (1) join two points to form a line and (2) intersect two lines. 26,27 This suggests that exact geometric cascading is as hard as exact scientific computing, which is untenable.…”

Audio Fingerprint Extraction Using an Adapted Computational Geometry Algorithm

“…In our case, the threshold is the value of the fraction between the intersected convex layers, which expresses the degree of correlation between two audio files (clips). Thus, the proposed method is superior to known methods because the threshold is satisfied with a non-linear way (Milenkovic, 1999) via degree of intersection (see Section 2.3). In this case, the absolute alignment between the amplitudes of the peaks with a minimum threshold error is not needed (as shown in Figure 10).…”

Scandinavian Thins on Top of Cake: New and Improved Algorithms for Stacking and Packing

“…The problems we defined above are different from disjoint packing problems [4, 32,48,63]: we required that the suitcase encloses any of the items one at a time (or equivalently, encloses all items "stacked" on top of each other); in a packing problem all items have to be enclosed by a suitcase simultaneously without overlap. Packing is hard when the number of items K is not fixed; even the simplest case of packing translates of axis-aligned rectangular items into an axis-aligned rectangular suitcase is weakly NP-hard, as can be seen by a simple reduction from PARTITION [32, Lemma 2.2].…”