Optimal algorithms for symmetry detection in two and three dimensions

Exact algorithms for detecting all rotational and involutional symmetries in point sets, polygons and polyhedra are described. The time complexities of the algorithms are shown to be (n) for polygons and (n logn) for two- and three-dimensional point sets. (n logn) time is also required for general polyhedra, but for polyhedra with connected, planar surface graphs (n) time can be achieved. All algorithms are optimal in time complexity, within constants.     

Probing a scene of nonconvex polyhedra

We show, in this paper, how the exact shapes of a class of polyhedral scenes can be computed by means of a simple sensory device issuing probes. A scene in this class consists of disjoint polyhedra with no collinear edges, no coplanar faces, and such that no edge is contained in the supporting plane of a nonincident face. The basic step of our method is a strategy for probing a single simple polygon with no collinear edges. When each probe outcome consists of a contact point and the normal to the object at the point, we present a strategy that allows us to compute the exact shape of a simple polygon with no collinear edges by means of at most3n — 3 probes, wheren is the number of edges of the polygon. This is optimal in the worst case. This strategy can be extended to probe a family of disjoint polygons. It can also be applied in planar sections of a scene of polyhedra of the class above to find out, in turn, each edge of the scene. If the scene consists ofk polyhedra with altogethern faces andm edges, we show that probes are sufficient to compute the exact shapes of the polyhedra.This work has been supported in part by the ESPRIT Basic Research Action No. 3075 (ALCOM).     

Exact and efficient construction of Minkowski sums of convex polyhedra with applications

We present an exact implementation of an efficient algorithm that computes Minkowski sums of convex polyhedra in R³. Our implementation is complete in the sense that it does not assume general position. Namely, it can handle degenerate input, and it produces exact results. We also present applications of the Minkowski-sum computation to answer collision and proximity queries about the relative placement of two convex polyhedra in R³. The algorithms use a dual representation of convex polyhedra, and their implementation is mainly based on the Arrangement package of Cgal, the Computational Geometry Algorithm Library. We compare our Minkowski-sum construction with the only three other methods that produce exact results we are aware of. One is a simple approach that computes the convex hull of the pairwise sums of vertices of two convex polyhedra. The second is based on Nef polyhedra embedded on the sphere, and the third is an output-sensitive approach based on linear programming. Our method is significantly faster. The results of experimentation with a broad family of convex polyhedra are reported. The relevant programs, source code, data sets, and documentation are available at http://www.cs.tau.ac.il/~efif/CD and a short movie [Fogel E, Halperin D. Video: Exact Minkowski sums of convex polyhedra. In: Proceedings of 21st annual ACM symposium on computational geometry. 2005. p. 382-3] that describes some of the concepts portrayed in this paper can be downloaded from http://www.cs.tau.ac.il/~efif/CD/Mink3d.avi.     

Voronoi diagrams with barriers and on polyhedra for minimal path planning

Two generalizations of the Voronoi diagram in two dimensions (E²) are presented in this paper. The first allows impenetrable barriers that the shortest path must go around. The barriers are straight line segments that may be combined into polygons and even mazes. Each region of the diagram delimits a set of points that have not only the same closest existing point, but have the same topology of shortest path. The edges of this diagram, which has linear complexity in the number of input points and barrier lines, may be hyperbolic sections as well as straight lines. The second construction considers the Voronoi diagram on the surface of a convex polyhedron, given a set of fixed source points on it. Each face is partitioned into regions, such that the shortest path to any goal point in a given region from the closest fixed source point travels over the same sequence of faces to the same closest point.This material is based upon work supported by the National Science Foundation under grants ECS-8021504 and ECS-8351942. The second author is also supported in part by a Fulbright scholarship     

A note on searching line arrangements and applications

We study the problem of searching for a vertex with a desired property in the arrangement of a set of lines in the plane. We show that this problem can be solved efficiently by modifying (and simplifying) two slope selection algorithms without using parametric search. We apply this result to a points approximation problem and obtain an optimal solution for it without using parametric search. Since this line arrangement searching problem is quite natural, our result may find other applications as well.     

Algorithms for computing the center of area of a convex polygon

The center of area of a convex polygonP is the unique pointp ^* that maximizes the minimum area overlap betweenP and any halfplane that includesp ^*. We show thatp ^* is unique and present two algorithms for its computation. The first is a combinatorial algorithm that runs in timeO (n ⁶ log² n). The second is a numerical algorithm that runs in timeO(GK(n+K)) whereK represents the number of desired bits of precision in the output coordinates andG the number of bits used to represent the coordinates of the input polygon vertices. We conclude with a discussion of implementation issues and related results.Research partially supported by the second author's NSF grant CCR-8351468, at Johns Hopkins University and Smith College.     

Casting with Skewed Ejection Direction

Casting is a manufacturing process in which liquid is poured into a cast (mold) that has a cavity with the shape of the object to be manufactured. The liquid then hardens, after which the cast is removed. We address geometric problems concerning the removal of the cast. A cast consists of two parts, one of which retracts in a given direction carrying the object with it. Afterwards, the object will be ejected from the retracted cast part. In this paper we give necessary and sufficient conditions to test the feasibility of the cast part retraction and object ejection, where retraction and ejection directions need not be the same. For polyhedral objects, we show that the test can be performed in O(n² log²n) time and the cast parts can be constructed within the same time bound. The complexity of the cast parts constructed is worst-case optimal. We also give a polynomial-time algorithm for finding a feasible pair of retraction and ejection directions for a given polyhedral object.     

Staircase visibility and computation of kernels

Let be some set of orientations, that is, . We consider the consequences of defining visibility based on curves that are monotone with respect to the orientations in . We call such curves -staircases. Two points p andq in a polygonP are said to -see each other if an -staircase fromp toq exists that is completely contained inP. The -kernel of a polygonP is then the set of all points which -see all other points. The -kernel of a simple polygon can be obtained as the intersection of all {}-kernels, with . With the help of this observation we are able to develop an algorithm to compute the -kernel of a simple polygon, for finite . We also show how to compute theexternal -kernel of a polygon in optimal time . The two algorithms are combined to compute the ( -kernel of a polygon with holes in time .This work was supported by the Deutsche Forschungsgemeinschaft under Grant No. Ot 64/5-4 and the Natural Sciences and Engineering Research Council of Canada and Information Technology Research Centre of Ontario.     

Recognition of largest empty orthoconvex polygon in a point set

An algorithm for computing the maximum area empty isothetic orthoconvex polygon among a set of n points on a 2D rectangular region, is presented. The worst-case time and space complexities of the proposed algorithm are O(n³) and O(n²) respectively.     

Normal eulerian clique-covering and hamiltonicity

A connected graph is hamiltonian if it contains a cycle which goes through all vertices exactly once. Determining if a graph is hamiltonian is known as an NP-complete problem and no satisfactory characterization for these graphs has been found.Since the seminal work of Dirac in 1952 many sufficient conditions were found. In 1974, Goodman and Hedetniemi gave such a condition based on the existence of a clique-covering of the graph. This condition was recently generalized using the notion of eulerian clique-covering. In addition, an algorithm able to find a normal eulerian clique-covering for a large class of graphs was also introduced. A normal clique-covering has additional properties, making the search for such a covering easier than in the general case.In this article, we prove several properties of normal clique-coverings. In particular we prove that there exists an eulerian clique-covering of a graph if and only if there exists a normal one. Using this result, the search for an eulerian clique-covering as a sufficient condition for hamiltonicity can be reduced to the normal case.     

Detecting symmetries of rational plane and space curves

This paper addresses the problem of determining the symmetries of a plane or space curve defined by a rational parametrization. We provide effective methods to compute the involution and rotation symmetries for the planar case. As for space curves, our method finds the involutions in all cases, and all the rotation symmetries in the particular case of Pythagorean-hodograph curves. Our algorithms solve these problems without converting to implicit form. Instead, we make use of a relationship between two proper parametrizations of the same curve, which leads to algorithms that involve only univariate polynomials. These algorithms have been implemented and tested in the Sage system.     

Efficient Collision Detection among Moving Spheres with Unknown Trajectories

Collision detection is critical for applications that demand a great deal of spatial interaction among objects. In such applications the trajectory of an object is often not known in advance either since a user is allowed to move an object at his/her will, or since an object moves under the rules that are hard to describe by exact mathematical formulas. In this paper we present a new algorithm that efficiently detects the collisions among moving spheres with unknown trajectories. We assume that the current position and velocity of every sphere can be probed at any time although its trajectory is unknown. We also assume that the magnitude of the acceleration of each sphere is bounded. Under these assumptions, we represent the bounding volume of the sphere as a moving sphere of variable radius, called a time-varying bound. Whenever the time-varying bounds of two spheres collide with each other, they are checked for actual collision. Exploiting these bounds, the previous event-driven approach for detecting the collisions among multiple moving spheres with ballistic trajectories is generalized for those with unknown trajectories. The proposed algorithm shows an interactive performance for thousands of moving spheres with unknown trajectories without any hardware help.     

Traversing the machining graph of a pocket

A simple and linear-time algorithm is presented for solving the problem of traversing a machining graph with minimum retractions encountered in zigzag pocket machining and other applications. This algorithm finds a traversal of the machining graph of a general pocket P with N_h holes, such that the number of retractions in the traversal is no greater than OPT+N_h+N_r, where OPT is the (unknown) minimum number of retractions required by any algorithm and N_r is the number of reducible blocks in P (to be defined in the paper). When the step-over distance is small enough relative to the size of P, N_r becomes zero, and our result deviates from OPT by at most the number of holes in P, a significant improvement over the upper bound 5OPT+6N_h achieved [Proceedings of the Seventh ACM-SIAM Symposium on Discrete Algorithms, 1996; Algorithmica 2000 (26) 19]. In particular, if N_h is zero as well, i.e. when P has no holes, the proposed algorithm outputs an optimal solution. A novel computational modeling tool called block transition graph is introduced to formulate the traversal problem in a compact and concise form. Efficient algorithms are then presented for traversing this graph, which in turn gives rise to the major result.     

A feature-based method of rapidly detecting global exact symmetries in CAD models

Detecting global exact symmetries in CAD models is of great importance in the research of CAD/CAE integration. Therefore, a method is proposed in this paper to rapidly detect the global exact rotational and reflectional symmetries in feature-based CAD models. The theories of determining the symmetries of the Boolean combinations of the features are framed. Based on these theories, our approach is processed as follows. First, the features of the CAD models are classified into congruent feature sets. Next, through the study on the relationship between feature information and the symmetries of features, by using only feature information, as many symmetries of the feature sets as possible are detected. Then these feature sets are sorted into an ordered sequence. Finally, symmetries of the entire model can be derived by successively merging and verifying the symmetries of feature sets in the ordered sequence. Experimental results show that the global exact symmetries can be robustly and rapidly detected.     

Applying the improved Chen and Han’s algorithm to different versions of shortest path problems on a polyhedral surface

The computation of shortest paths on a polyhedral surface is a common operation in many computer graphics applications. There are two best known exact algorithms for the “single source, any destination” shortest path problem. One is proposed by Mitchell et al. (1987) [1]. The other is by Chen and Han (1990) [11]. Recently, Xin and Wang (2009) [9] improved the CH algorithm by exploiting a filtering theorem and achieved a practical method that outperforms both the CH algorithm and the MMP algorithm whether in time or in space.In this paper, we apply the improved CH algorithm to different versions of shortest path problems. The contributions of this paper include: (1) For a surface point p∈△v₁v₂v₃, we present an unfolding technique for estimating the distance value at p using the distances at v₁,v₂ and v₃. (2) We show that the improved CH algorithm can be naturally extended to the “multiple sources, any destination” version. Also, introducing a well-chosen heuristic factor into the improved CH algorithm will induce an exact solution to the “single source, single destination” version. (3) At the conclusion of multi-source shortest path algorithms, we can use the distance values at vertices to approximately compute the geodesic-distance-based offsets, the Voronoi diagram and the Delaunay triangulation in O(n) time. (4) By importing a precision parameter λ, we obtain a precision controlled approximant which varies from the improved CH algorithm to Dijkstra’s algorithm as λ increases from 0 to 1. Thus, an interesting relationship between them can be naturally established.     

A characterization of complete bipartite RAC graphs

A Right Angle Crossing drawing (RAC drawing for short) of a graph is such that edges can only cross at an angle of . In this paper we provide a characterization of the complete bipartite graphs that admit a straight-line RAC drawing.     

Efficient triangulation of simple polygons

This paper considers the topic of efficiently traingulating a simple polygon with emphasis on practical and easy-to-implement algorithms. It also describes a newadaptive algorithm for triangulating a simplen-sided polygon. The algorithm runs in timeO(n(1+t _o)) witht ₀<n. The quantityt ₀ measures theshape-complexity of thetriangulation delivered by the algorithm. More preciselyt ₀ is the number of obtained triangles contained in the triangulation that share zero edges with the input polygon and is, furthermore, related to the shape-complexity of theinput polygon. Although the worst-case complexity of the algorithm isO(n ²), for several classes of polygons it runs in linear time. The practical advantages of the algorithm are that it is simple and does not require sorting or the use of balanced tree structures. On the theoretical side, it is of interest because it is the first polygon triangulation algorithm where thecomputational complexity is a function of theoutput complexity. As a side benefit, we introduce a new measure of the complexity of a polygon triangulation that should find application in other contexts as well.