Polyhedral Terrain Research Articles

Finding shortest gentle paths on polyhedral terrains by the method of multiple shooting

The problem of finding shortest θ-gentle paths can be stated as follows: given two points p,q on a polyhedral terrain and a slope parameter θ∈(0,π/2), the objective is to find a path joining p and q on the terrain which is shortest such that the slope of the path does not exceed θ. In this paper, we introduce some geometric and analysis properties of such paths and answer the question of whether known results of classical shortest paths hold for shortest θ-gentle paths. An algorithm for approximately computing such shortest θ-gentle paths on terrains is presented, where an approximate shortest θ-gentle path joining two points is a θ-gentle path whose length is the infimum of a sequence of that of θ-gentle paths in which they are decreasing. We also show that the sequence of lengths of paths obtained by the proposed algorithm is convergent. The algorithm is implemented in C＋＋ using CGAL and Open GL in some specific circumstances.

Vertex Fault-Tolerant Geometric Spanners for Weighted Points

Given a set [Formula: see text] of [Formula: see text] points, a weight function [Formula: see text] to associate a non-negative weight to each point in [Formula: see text], a positive integer [Formula: see text], and a real number [Formula: see text], we present algorithms for computing a spanner network [Formula: see text] for the metric space [Formula: see text] induced by the weighted points in [Formula: see text]. The weighted distance function [Formula: see text] on the set [Formula: see text] of points is defined as follows: for any [Formula: see text], [Formula: see text] is equal to [Formula: see text] if [Formula: see text], otherwise, [Formula: see text] is [Formula: see text]. Here, [Formula: see text] is the Euclidean distance between [Formula: see text] and [Formula: see text] if points in [Formula: see text] are in [Formula: see text], otherwise, it is the geodesic (Euclidean) distance between [Formula: see text] and [Formula: see text]. The following are our results: (1) When the weighted points in [Formula: see text] are located in [Formula: see text], we compute a [Formula: see text]-vertex fault-tolerant [Formula: see text]-spanner network of size [Formula: see text]. (2) When the weighted points in [Formula: see text] are located in the relative interior of the free space of a polygonal domain [Formula: see text], we detail an algorithm to compute a [Formula: see text]-vertex fault-tolerant [Formula: see text]-spanner network with [Formula: see text] edges. Here, [Formula: see text] is the number of simple polygonal holes in [Formula: see text]. (3) When the weighted points in [Formula: see text] are located on a polyhedral terrain [Formula: see text], we propose an algorithm to compute a [Formula: see text]-vertex fault-tolerant [Formula: see text]-spanner network, and the number of edges in this network is [Formula: see text].

Acrophobic guard watchtower problem

In the acrophobic guard watchtower problem for a polyhedral terrain, a square axis-aligned platform is placed on the top of a tower whose bottom end-point lies on the surface of the terrain. As in the standard watchtower problem, the objective is to minimize the height (i.e., the length) of the watchtower such that every point on the surface of the terrain is weakly visible from the platform placed on the top of the tower. In this paper, we show that in R2 the problem can be solved in O(n) time, and in R3 it takes O(nlog⁡n) time, where n is the total number of vertices of the terrain.

Vertex fault-tolerant spanners for weighted points in polygonal domains

Given a set S of n points, a weight function w to associate a non-negative weight to each point in S, a positive integer [Formula: see text], and a real number [Formula: see text], we devise the following algorithms to compute a k-vertex fault-tolerant spanner network [Formula: see text] for the metric space induced by the weighted points in S: (1). When the points in S are located in a simple polygon, we present an algorithm to compute G with multiplicative stretch [Formula: see text], and the number of edges in G (size of G) is [Formula: see text]. (2) When the points in S are located in the free space of a polygonal domain [Formula: see text] with h number of obstacles, we present an algorithm to compute G with multiplicative stretch [Formula: see text] and size [Formula: see text]. (3) When the points in S are located on a polyhedral terrain, we devise an algorithm to compute G with multiplicative stretch [Formula: see text] and size [Formula: see text].

Optimal facility location problem on polyhedral terrains using descending paths

We study the descending facility location (DFL) problem on the surface of a triangulated terrain. A path from a point s to a point t on the surface of a terrain is descending if the heights of the subsequent points along the path from s to t are in a monotonically non-increasing order [1]. We are given a set D={d1,d2,⋯,dn} of n demand points on the surface of a triangulated terrain W and our objective is to find a set F (of points), of minimum cardinality, on the surface of the terrain such that for each demand point d∈D there exists a descending path from at least one facility f∈F to d. We present an O((n+m)log⁡m) time algorithm for solving the DFL problem, where m is the number of vertices in the triangulated terrain. We achieve this by reducing the DFL problem to a graph problem called the directed tree covering(DTC)problem. In the DTC problem, we have a directed tree B=(V,E) with a set of marked nodes M⊆V. The objective is to compute a set C⊆V of minimum cardinality, such that for every node v∈M, either v∈C or there exists a node c∈C such that v is reachable from c. We prove that the DFL problem can be reduced to DTC problem in O((m+n)log⁡m) time. The DTC problem thereafter can be solved in O(|V|) time. We also prove that the general version of the DTC problem, called the directed graph covering(DGC)problem is NP-hard on directed bipartite graphs and hard to approximate within (1−ϵ)ln⁡|M|-factor, for every ϵ>0, where |M| is the size of the set of marked nodes. We also prove that for the DGC problem, an O(log⁡|M|) factor approximation is possible and this approximation factor is tight.

Distributed Multi-agent Deployment for Full Visibility of 1.5D and 2.5D Polyhedral Terrains

This paper presents deployment strategies to achieve full visibility of 1.5D and 2.5D polyhedral environments for a team of mobile robots. Agents may only communicate if they are within line-of-sight. In 1.5D polyhedral terrains we achieve this by algorithmically determining a set of locations that the robots can occupy in a distributed fashion. We characterize the time of completion of the resulting algorithm, which is dependent on the number of peaks and the initial condition. In 2.5D polyhedral terrains we achieve full visibility by asynchronously deploying groups of agents who utilize graph coloring and may start from differential initial conditions. We characterize the total number of agents needed for deployment as a function of the environment properties and allow the algorithm to activate additional agents if necessary. We provide lower and upper bounds for the time of completion as a function of the number of vertices in a planar graph representing the environment. We illustrate our results in simulation and an implementation on a multi-agent robotics platform.

Geodesic Spanners for Points on a Polyhedral Terrain

Let $S$ be a set of $n$ points on a polyhedral terrain $\mathcal{T}$ in $\mathbb{R}^3$, and let $\varepsilon>0$ be a fixed constant. We prove that $S$ admits a $(2+\varepsilon)$-spanner with $O(n\log n)$ edges with respect to the geodesic distance. This is the first spanner with constant spanning ratio and a near-linear number of edges for points on a terrain. On our way to this result, we prove that any set of $n$ weighted points in $\mathbb{R}^d$ admits an additively weighted $(2+\varepsilon)$-spanner with $O(n)$ edges; this improves the previously best known bound on the spanning ratio (which was $5+\varepsilon$) and almost matches the lower bound.

Straight Skeletons and Mitered Offsets of Polyhedral Terrains in 3D

Straight Skeletons and Mitered Offsets of Polyhedral Terrains in 3D

Computing approximately shortest descending paths on convex terrains via multiple shooting

Given a polyhedral terrain and two points p, q on the terrain, a path from p to q on the terrain is descending if z-coordinate of a point v never increases while we move v along the path from p to q. The problem of finding shortest descending paths on polyhedral terrains was posed by de Berg and van Kreveld (Algorithmica 18:306–323, 1997). In this paper, the multiple shooting approach proposed by Hoai et al. (J Comp Appl Math 317:235–246, 2017) is applied for approximately computing shortest descending paths on convex polyhedral terrains. Three factors of the approach consisting of surface partition, straightness condition, and update of shooting points are presented. We also show that if the straightness condition is satisfied then a local shortest descending path is obtained. Proposed algorithm is implemented in C++. Numerical results indicate that once a local solution is obtained it is close to a global one.

Computing representative networks for braided rivers

Drainage networks on terrains have been studied extensively from an algorithmic perspective. However, in drainage networks water flow cannot bifurcate and hence they do not model braided rivers (multiple channels which split and join, separated by sediment bars). We initiate the algorithmic study of braided rivers by employing the descending quasi Morse-Smale complex on the river bed (a polyhedral terrain), and extending it with a certain ordering of bars from one river bank to the other. This allows us to compute a graph that models a representative channel network, consisting of lowest paths. To ensure that channels in this network are sufficiently different we define a sand function that represents the volume of sediment separating them. We show that in general the problem of computing a maximum network of non-crossing channels which are δ-different from each other (as measured by the sand function) is NP-hard. However, using our ordering between the river banks, we can compute a maximum δ-different network that respects this order in polynomial time. We implemented our approach and applied it to simulated and real-world braided rivers.

On the Expected Complexity of Voronoi Diagrams on Terrains

We investigate the combinatorial complexity of geodesic Voronoi diagrams on polyhedral terrains using a probabilistic analysis. Aronov et al. [2008] prove that, if one makes certain realistic input assumptions on the terrain, this complexity is Θ( n + m √ n ) in the worst case, where n denotes the number of triangles that define the terrain and m denotes the number of Voronoi sites. We prove that, under a relaxed set of assumptions, the Voronoi diagram has expected complexity O ( n + m ), given that the sites are sampled uniformly at random from the domain of the terrain (or the surface of the terrain). Furthermore, we present a construction of a terrain that implies a lower bound of Ω( nm 2/3 ) on the expected worst-case complexity if these assumptions on the terrain are dropped. As an additional result, we show that the expected fatness of a cell in a random planar Voronoi diagram is bounded by a constant.

TERRAIN VISIBILITY WITH MULTIPLE VIEWPOINTS

We study the problem of visibility in polyhedral terrains in the presence of multiple viewpoints. We consider a triangulated terrain with m > 1 viewpoints (or guards) located on the terrain surface. A point on the terrain is considered visible if it has an unobstructed line of sight to at least one viewpoint. We study several natural and fundamental visibility structures: (1) the visibility map, which is a partition of the terrain into visible and invisible regions; (2) the colored visibility map, which is a partition of the terrain into regions whose points have exactly the same visible viewpoints; and (3) the Voronoi visibility map, which is a partition of the terrain into regions whose points have the same closest visible viewpoint. We study the complexity of each structure for both 1.5D and 2.5D terrains, and provide efficient algorithms to construct them. Our algorithm for the visibility map in 2.5D terrains improves on the only existing algorithm in this setting. To the best of our knowledge, the other structures have not been studied before.

Flow computations on imprecise terrains

We study water flow computation on imprecise terrains. We consider two approaches to modeling flow on a terrain: one where water flows across the surface of a polyhedral terrain in the direction of steepest descent, and one where water only flows along the edges of a predefined graph, for example a grid or a triangulation. In both cases each vertex has an imprecise elevation, given by an interval of possible values, while its (x,y)-coordinates are fixed. For the first model, we show that the problem of deciding whether one vertex may be contained in the watershed of another is NP-hard. In contrast, for the second model we give a simple O ( n log n ) time algorithm to compute the minimal and the maximal watershed of a vertex, or a set of vertices, where n is the number of edges of the graph. On a grid model, we can compute the same in O ( n ) time.

Computing Correlation between Piecewise-Linear Functions

We study the problem of computing correlation between two piecewise-linear bivariate functions defined over a common domain, where the surfaces they define in three dimensions---polyhedral terrains---can be transformed vertically by a linear transformation of the third coordinate (scaling and translation). We present a randomized algorithm that minimizes the maximum vertical distance between the graphs of the two functions, over all linear transformations of one of the terrains, in $O(n^{4/3}\operatorname{polylog}n)$ expected time, where $n$ is the total number of vertices in the graphs of the two functions. We also present approximation algorithms for minimizing the mean distance between the graphs of univariate and bivariate functions. For univariate functions we present a $(1+\varepsilon)$-approximation algorithm that runs in $O(n (1 + \log^2 (1/\varepsilon)))$ expected time for any fixed $\varepsilon >0$. The $(1+\varepsilon)$-approximation algorithm for bivariate functions runs in $O(n/\varepsilon)$ time, for any fixed $\varepsilon >0$, provided the two functions are defined over the same triangulation of their domain.

Near optimal algorithm for the shortest descending path on the surface of a convex terrain

We study the problem of finding a shortest descending path (SDP) between a pair of points, called source (s) and destination (t), on the surface of a triangulated convex terrain with n faces. A path from s to t on a polyhedral terrain is descending if the height of a point p never increases while we move p along the path from s to t. Time and space complexity requirement of our algorithm are O(μ(n)logn) and O(τ(n)), respectively. Here μ(n) and τ(n) are time and space complexity requirement for finding shortest geodesic path (SGP) between a pair of points on the surface of a convex polyhedra. The best known bounds on μ(n) and τ(n) are both O(nlogn) due to Schreiber and Sharir (2008) [11]. Earlier best known time and space complexity results of SDP on convex terrain were O(n2logn) and O(n2), respectively, and appears in Roy et al. (2007) [10]. Thus our algorithm improves both time and space complexity requirement of SDP problem by almost a linear factor over earlier best known results.

Lower Bound of Face Guards of Polyhedral Terrains

We study the problem of determining the minimum number of face guards which cover the surface of a polyhedral terrain. We show that ⌊(2n-5)/7⌋ face guards are sometimes necessary to guard the surface of an n-vertex triangulated polyhedral terrain.

Lower Bound of Face Guards of Polyhedral Terrains

Lower Bound of Face Guards of Polyhedral Terrains

SHORTEST DESCENDING PATHS: TOWARDS AN EXACT ALGORITHM

A path from s to t on a polyhedral terrain is descending if the height of a point p never increases while we move p along the path from s to t. Although a shortest path on a terrain unfolds to a straight line, a shortest descending path (SDP) does not. We give a full characterization of the bend angles of an SDP, showing that they follow a generalized form of Snell's law of refraction of light. The complexity of finding SDPs is open—only approximation algorithms are known. We reduce the SDP problem to the problem of finding an SDP through a given sequence of faces. We give polynomial time algorithms for SDPs on two special classes of terrains, but argue that the general case will be difficult.

Visibility-based conformal cooling channel generation for rapid tooling

In plastic injection moulding process, cooling channel design is an essential factor that affects the quality of the moulded parts and the productivity of the process. Non-uniform cooling or long cooling cycle time would result if a poorly designed cooling channel is adopted. Due to limitations of traditional machining processes, the cooling channel is usually formed from straight-line drilled holes and only simple shapes are allowed, regardless of the shape complexity of the part being moulded. With the advent of rapid tooling technology, cooling channels in complex shapes can now be possible. However, there are not many design methodologies for supporting this type of cooling channel. In this paper, a methodology called visibility-based cooling channel generation is proposed for automatic preliminary cooling channel design for rapid tooling. The cooling process between a mould surface and a cooling channel is considered analogous to whether they can be visible from each other. Without loss of generality, the mould surface is approximated by a polyhedral terrain and is normally offset. A number of point light sources together that can illuminate the whole polyhedral terrain are assigned to suitable terrain offset vertices. A cooling channel is then generated by connecting all the assigned light sources. When comparing the conventional verification and redesign methods by melt flow analysis, computer-aided design and, a better design of cooling channel for its mould surface results in a short time independent of the experience of mould engineer.

SMOOTHING IMPRECISE 1.5D TERRAINS

We study optimization problems for polyhedral terrains in the presence of data imprecision. An imprecise terrain is given by a triangulated point set where the height component of the vertices is specified by an interval of possible values. We restrict ourselves to terrains with a one-dimensional projection, usually referred to as 1.5-dimensional terrains, where an imprecise terrain is given by an x-monotone polyline, and the y-coordinate of each vertex is not fixed but only constrained to a given interval. Motivated mainly by applications in terrain analysis, in this paper we study five different optimization measures related to obtaining smooth terrains, for the 1.5-dimensional case. In particular, we present exact algorithms to minimize and maximize the total turning angle, as well as to minimize the maximum slope change. Furthermore, we also give approximation algorithms to minimize the largest turning angle and to maximize the smallest turning angle.