Orthogonal Polyhedra Research Articles

Application of the Model of Potential Containers for Packing Complex Shape Objects

The optimization problem of packing objects of arbitrary shape with generalization in dimension is considered. The purpose of this work is to develop and study algorithms that provide fast optimized placement of a large number of objects of arbitrary shape. To achieve this goal, the problem was set to improve the model of potential containers, originally developed for packing orthogonal objects in the form of rectangles or parallelepipeds. To quickly get layouts, it is proposed to switch from a polygonal representation of objects under placing to their discrete representation after voxelization. Obviously, when placing objects of complex shape are voxelized with a high degree of detail, a large number of potential containers will be formed, which will reduce the speed of solving the problem. This article presents the developed algorithms optimized by the number of potential containers processed to increase the speed of geometric design of a layout. Computational experiments have shown that the highest speed of the formation placement schemes of orthogonal polyhedrons is provided by using algorithms based on the application of the set-theoretic intersection operation. A method of removing unused potential containers is proposed, which allows several times to increase the speed of object placement. It is shown that the effectiveness of its application increases with an increase in the number of objects to be placed. A method for creating a container of complex shape form a polygonal model by applying an orthogonal polyhedron of geometric constraints to it is presented. All presented algorithms are implemented programmatically for the case of an arbitrary dimension of the problem being solved.

Методы формирования ортогональных многогранников для задач раскроя и упаковки объектов произвольной геометрии

The article deals with the problem of packing objects of arbitrary geometry. Modern methods of designing irregular packing schemes use a mathematical model based on phi-functions and a hodograph vector function of dense placement. These methods make it possible to obtain exact solutions, but they are time-consuming and very sensitive to the dimension of the problem being solved and the degree of detail of the geometry of vector objects. The use of a discrete representation of placed objects in the form of orthogonal polyhedra can signifi increase the speed of construction a packing, which makes the problem of adequately transforming the shape of placed objects (vector models in the two-dimensional case and polygonal models in the three- dimensional case) relevant. The aim of the study is to systematize methods that provide the formation of orthogonal polyhedra of various dimensions for describing objects and containers of arbitrary geometry. Methods for creating orthogonal polyhedra based on set-theoretic operations (addition, subtraction and intersection), analytical modeling using a set of functions and relational operators, as well as voxelization of fl and volumetric object models are considered. The use of set-theoretic operations is best suited for the manual creation of orthogonal polyhedra with relatively simple geometry. The method of analytical modeling is intended for the formation of voxelized objects based on geometric fi es described by a set of analytically specifi functions. The application of various relational operators to obtain orthogonal polyhedra that describe the contour, internal and external regions of analytical given objects is shown. An algorithm for creating a container in the form of an orthogonal polyhedron based on a given vector model is proposed, which makes it possible to solve problems of irregular packing of objects inside containers of arbitrary shape. All the methods presented in the article are programmatically implemented with a generalization in terms of dimension and are applicable to solving any types of cutting and packing problems.

Development of algorithms for the formation and placement of N-dimensional orthogonal polyhedrons into containers of complex geometric shape

The article is devoted to problems related with the development of algorithms for creating, optimizing and placing of orthogonal polyhedrons of arbitrary dimension. Under N-dimensional orthogonal polyhedron is understood a composite object represented as a set of N-dimensional parallelepipeds with a fixed position relative to each other, which is considered as a whole. The need to solve the problem of placing a set of objects in the form of orthogonal polyhedrons of arbitrary dimension arises when solving a large number of practical problems in the automation of design and production. In particular, solving the layout problems of equipment and space, problems of cutting materials, problems of designing very large-scale integrated circuits, and optimizing the distribution of resources in computing and production systems, as well as many other actual problems, can be reduced to solving the problem of packing orthogonal polyhedrons. To create a new N-dimensional orthogonal polyhedron, the operations of addition, subtraction and multiplication of orthogonal objects are implemented. With the aim to increase the placement speed of orthogonal polyhedrons which were created with using the voxel model, an algorithm for partitioning of an orthogonal polyhedron into many non-overlapping each other as large as possible large orthogonal objects is proposed. In the scientific literature, there are no solutions to the problem of partitioning an orthogonal polyhedron of arbitrary dimension. The proposed partitioning algorithm is based on the usage of a model of potential containers developed to solve the orthogonal packing problems of arbitrary dimension. To obtain a container of arbitrary shape, a possibility of setting its geometric constraints with an orthogonal polyhedron is implemented. The article presents an algorithm developed for packing an N-dimensional orthogonal polyhedrons inside containers of arbitrary geometric shape, based on the application of multiplication orthogonal polyhedrons operations to obtain the set of points of permissible placement of each considered object. The examples of solving the problems of cutting the industrial materials into objects of irregular shapes, as well as solving the problems of placement complex objects created with using the voxel model into containers of various geometric shapes (including sphere and cone) are given. The effectiveness of the application of the developed algorithms is shown on the example of solving the problem of generation a dense layout of details on a 3D printer platform.

Solving the Problem of Packing Objects of Complex Geometric Shape into a Container of Arbitrary Dimension

The article is devoted to algorithms developed for solving the problem of placement orthogonal polyhedrons of arbitrary dimension into a container. To describe all free areas of a container of complex geometric shape is applied the developed model of potential containers. Algorithms for constructing orthogonal polyhedrons and their subsequent placement are presented. The decomposition algorithm intended to reduce the number of orthogonal objects forming an orthogonal polyhedron is described in detail. The proposed placement algorithm is based on the application of intersection operations to obtain the areas of permissible placement of each considered object of complex geometric shape. Examples of packing sets of orthogonal polyhedrons and voxelized objects into containers of various geometric shapes are given. The effectiveness of application of all proposed algorithms is presented on an example of solving practical problems of rational placement of objects produced by 3D printing technology. The achieved layouts exceed the results obtained by the Sinter module of the software Materialise Magics both in speed and density.

Tight Bounds for Illuminating and Covering of Orthotrees with Vertex Lights and Vertex Beacons

We consider two variants of the Art Gallery Problem: illuminating orthotrees with a minimum set of vertex lights, and covering orthotrees with a minimum set of vertex beacons. An orthotree P is a simply connected orthogonal polyhedron that is the union of a set S of cuboids glued face to face such that the graph whose vertices are the cuboids of S, two of which are adjacent if they share a common face, is a tree. A point p illuminates a point $$q \in P$$ if the line segment $$\ell$$ joining them is contained in P. A beacon b is a point in P that pulls other points in P towards itself similarly to the way a magnet attracts ferrous particles. We say that a beacon bcoversp if when b starts pulling p, p does not get stuck at a point of P before it reaches b. This happens, for instance if p reaches a point $$p'$$ such that there is an $$\epsilon >0$$ such that any point in P at distance at most $$\epsilon$$ from $$p'$$ is farther away from $$p'$$ than q (there is another pathological case that we will not detail in this abstract). In this paper we prove that any orthotree P with n vertices can be illuminated using at most $$\lfloor n/8 \rfloor$$ light sources placed at vertices of P, and that all of the points in P can always be covered with at most $$\lfloor n/12 \rfloor$$ vertex beacons. Both bounds are tight.

Optimally guarding 2-reflex orthogonal polyhedra by reflex edge guards

Let an orthogonal polyhedron be the union of a finite set of boxes in R3 (i.e., cuboids with edges parallel to the coordinate axes), whose surface is a connected 2-manifold. We study the NP-complete problem of guarding a non-convex orthogonal polyhedron having reflex edges in just two directions (as opposed to three, in the general case) by placing the minimum number of edge guards on reflex edges only.We show that⌊r−g2⌋+1 reflex edge guards are sufficient, where r is the number of reflex edges and g is the polyhedron's genus. This bound is tight for g=0. We thereby generalize a classic planar Art Gallery theorem of O'Rourke, which states that the same upper bound holds for vertex guards in an orthogonal polygon with r reflex vertices and g holes.Then we give a similar upper bound in terms of m, the total number of edges in the polyhedron. We prove that⌊m−48⌋+g reflex edge guards are sufficient, whereas the previous best known bound was ⌊11m/72+g/6⌋−1 edge guards (not necessarily reflex).We also consider the setting in which guards are open (i.e., they are segments without the endpoints), proving that the same results hold even in this more challenging case.Finally, we show how to compute guard locations matching the above bounds in O(nlog⁡n) time.

Compacting Voxelized Polyhedra via Tree Stacking

AbstractVolume compaction is a geometric problem that aims to reduce the volume of a polyhedron via shape transform. Compactable structures are easier to transport and in some cases easier to manufacture, therefore, they are commonly found in our daily life (e.g. collapsible containers) and advanced technology industries (e.g., the recent launch of 60 Starlink satellites compacted in a single rocket by SpaceX). It is known in the literature that finding a universal solution to compact an arbitrary 3D shape is computationally challenging. Previous approaches showed that stripifying mesh surface can lead to optimal compaction, but the resulting structures were often impractical. In this paper, we propose an algorithm that cuts the 3D orthogonal polyhedron, tessellated by thick square panels, into a tree structure that can be transformed into compact piles by folding and stacking. We call this process tree stacking. Our research found that it is possible to decompose the problem into a pipeline of several solvable local optimizations. We also provide an efficient algorithm to check if the solution exists by avoiding the computational bottleneck of the pipeline. Our results show that tree stacking can efficiently generate stackable structures that have better folding accuracy and similar compactness comparing to the most compact stacking using strips.

Minimizing the solid angle sum of orthogonal polyhedra

In this paper we characterize the set of orthogonal polyhedra in R3 that minimize the sum of their internal solid angles. To prove our result, we first generalize the well known result that any orthogonal polygon in the plane with n vertices has (n+4)/2 convex and (n−4)/2 reflex vertices; see [1,2]. We prove that an orthogonal polyhedron of genus g with n vertices, k of which have degree greater than or equal to 4, has (n+8−8g+3k)/2 convex vertices and (n−8+8g−3k)/2 reflex vertices. We also prove that the sum of the solid angles of an orthogonal polyhedron is at least (n−4+4g)π, and at most (3n−24−4g)π. These inequalities are minimized and maximized, respectively, when g=0.

Unfolding Genus-2 Orthogonal Polyhedra with Linear Refinement

We show that every orthogonal polyhedron of genus $$g \le 2$$ can be unfolded without overlap while using only a linear number of orthogonal cuts (parallel to the polyhedron edges). This is the first result on unfolding general orthogonal polyhedra beyond genus-0. Our unfolding algorithm relies on the existence of at most 2 special leaves in what we call the “unfolding tree” (which ties back to the genus), so unfolding polyhedra of genus 3 and beyond requires new techniques.

Improved Algorithms for Grid-Unfolding Orthogonal Polyhedra

An unfolding of a polyhedron is a single connected planar piece without overlap resulting from cutting and flattening the surface of the polyhedron. Even for orthogonal polyhedra, it is known that edge-unfolding, i.e., cuts are performed only along the edges of a polyhedron, is not sufficient to guarantee a successful unfolding in general. However, if additional cuts parallel to polyhedron edges are allowed, it has been shown that every orthogonal polyhedron of genus zero admits a grid-unfolding with quadratic refinement. Using a new unfolding technique developed in this paper, we improve upon the previous result by showing that linear refinement suffices. For 1-layer orthogonal polyhedra of genus [Formula: see text], we show a grid-unfolding algorithm using only [Formula: see text] additional cuts, affirmatively answering an open problem raised in a recent literature. Our approach not only requires fewer cuts but yields much simpler algorithms.

3D Rectangulations and Geometric Matrix Multiplication

The problem of partitioning an orthogonal polyhedron P into a minimum number of 3D rectangles is known to be NP-hard. In this paper, we first develop a 4-approximation algorithm for the special case of the problem in which P is a 3D histogram. It runs in \(O(m \log m)\) time, where m is the number of corners in P. We then apply it to exactly compute the arithmetic matrix product of two \(n \times n\) matrices A and B with nonnegative integer entries, yielding a method for computing \(A \times B\) in \(\tilde{O}(n^2 + \min \{ r_A r_B,\, n \min \{r_A,\ r_B\}\})\) time, where \(\tilde{O}\) suppresses polylogarithmic (in n) factors and where \(r_A\) and \(r_B\) denote the minimum number of 3D rectangles into which the 3D histograms induced by A and B can be partitioned, respectively.

Constrained floorplans in 2D and 3D

A rectilinear dual of a plane graph refers to a partition of a rectangular area into nonoverlapping rectilinear polygonal modules, where each module corresponds to a vertex such that two modules have a side-contact iff their corresponding vertices are adjacent. An interesting observation is that most algorithms yielding low polygonal complexity rectilinear duals require the use of ⊤-shape modules or their extensions. To justify the intuition that ⊤-shape modules are more powerful than other 8-sided modules, we show the optimum polygonal complexity of ⊤-free rectilinear duals to be exactly 12. Our construction uses only monotone staircase modules. We continue this line of research by studying polygonal complexity and area-universality of rectilinear duals using modules of constrained shapes.Motivated by the design challenges in 3D IC technology, we also study the generalization of rectilinear duals in three dimensions by allowing each module to be an orthogonal polyhedron. Such a generalization to three dimensions opens the door for non-planar graphs to be accommodated in floorplan design. In particular, we prove that all chordal graphs admit 3D floorplans, which parallels the well-known result that all maximal plane graphs admit rectilinear duals.

PolyCut

PolyCubes, or orthogonal polyhedra, are useful as parameterization base-complexes for various operations in computer graphics. However, computing quality PolyCube base-complexes for general shapes, providing a good trade-off between mapping distortion and singularity counts, remains a challenge. Our work improves on the state-of-the-art in PolyCube computation by adopting a graph-cut inspired approach. We observe that, given an arbitrary input mesh, the computation of a suitable PolyCube base-complex can be formulated as associating, or labeling, each input mesh triangle with one of six signed principal axis directions. Most of the criteria for a desirable PolyCube labeling can be satisfied using a multi-label graph-cut optimization with suitable local unary and pairwise terms. However, the highly constrained nature of PolyCubes, imposed by the need to align each chart with one of the principal axes, enforces additional global constraints that the labeling must satisfy. To enforce these constraints, we develop a constrained discrete optimization technique, PolyCut , which embeds a graph-cut multi-label optimization within a hill-climbing local search framework that looks for solutions that minimize the cut energy while satisfying the global constraints. We further optimize our generated PolyCube base-complexes through a combination of distortion-minimizing deformation, followed by a labeling update and a final PolyCube parameterization step. Our PolyCut formulation captures the desired properties of a PolyCube base-complex, balancing parameterization distortion against singularity count, and produces demonstrably better PolyCube base-complexes then previous work.

Unfolding Orthogonal Polyhedra with Quadratic Refinement: The Delta-Unfolding Algorithm

We show that every orthogonal polyhedron homeomorphic to a sphere can be unfolded without overlap while using only polynomially many (orthogonal) cuts. By contrast, the best previous such result used exponentially many cuts. More precisely, given an orthogonal polyhedron with n vertices, the algorithm cuts the polyhedron only where it is met by the grid of coordinate planes passing through the vertices, together with Θ(n2) additional coordinate planes between every two such grid planes.

Skeleton computation of orthogonal polyhedra

AbstractSkeletons are powerful geometric abstractions that provide useful representations for a number of geometric operations. The straight skeleton has a lower combinatorial complexity compared with the medial axis. Moreover, while the medial axis of a polyhedron is composed of quadric surfaces the straight skeleton just consist of planar faces. Although there exist several methods to compute the straight skeleton of a polygon, the straight skeleton of polyhedra has been paid much less attention. We require to compute the skeleton of very large datasets storing orthogonal polyhedra. Furthermore, we need to treat geometric degeneracies that usually arise when dealing with orthogonal polyhedra. We present a new approach so as to robustly compute the straight skeleton of orthogonal polyhedra. We follow a geometric technique that works directly with the boundary of an orthogonal polyhedron. Our approach is output sensitive with respect to the number of vertices of the skeleton and solves geometric degeneracies. Unlike the existing straight skeleton algorithms that shrink the object boundary to obtain the skeleton, our algorithm relies on the plane sweep paradigm. The resulting skeleton is only composed of axis‐aligned and 45° rotated planar faces and edges.

STOKER'S THEOREM FOR ORTHOGONAL POLYHEDRA

Stoker's theorem states that in a convex polyhedron, the dihedral angles and edge lengths determine the facial angles if the graph is fixed. In this paper, we study under what conditions Stoker's theorem holds for orthogonal polyhedra, obtaining uniqueness and a linear-time algorithm in some cases, and NP-hardness in others.

Reconstructing orthogonal polyhedra from putative vertex sets

In this paper we study the problem of reconstructing orthogonal polyhedra from a putative vertex set, i.e., we are given a set of points and want to find an orthogonal polyhedron for which this is the set of vertices. This is well-studied in 2D; we mostly focus on 3D, and on the case where the given set of points may be rotated beforehand. We obtain fast algorithms for reconstruction in the case where the answer must be orthogonally convex.

Orthogonal Polyhedra in <italic>3D Time-Color Space</italic> as a Geometric Model for Representation of Video Sequences with Low or Inexistent Redundancy between Frames

Several video compression methods have in common their conciseness depends of the degree of redundancy between the frames associated to the sequences they describe. Cartoon animations are a good example of animations with an elevated redundancy and from which an elevated compression is expected. However, scientific sequences provide us with a huge set of examples where the degree of redundancy is very low or inexistent. Because of the accuracy required when these sequences are analyzed, any kind of threshold, which could elevate the redundancy degree, is prohibited. In this work, a proposed solution to the addressed problem considers the linearization of each frame. That is, a frame can be considered as a matrix but by stacking its columns on top of one another a vector is obtained. In this way each pixel can be referenced by only one coordinate in the vector. Starting from these considerations a geometric representation is built: A video sequence will be associated to a unique 3D Orthogonal Pseudo-Polyhedron (3D-OPP). Such 3D-OPP is embedded in a 3D Time-Color Space where X1-axis corresponds to the position of pixels in the linearization, the second spatial dimension (X2-axis) is associated to the color value of a pixel, and finally, X3-axis describes the displaying time of frames in the original animation. Such 3D-OPP can be compressed, manipulated, and displayed in screen by expressing it according to the Extreme Vertices Model in the n-Dimensional Space (nD-EVM). The nD-EVM shares the representation of n-Dimensional Orthogonal Pseudo-Polytopes (nD-OPPs) by considering only a subset of their vertices.

Grid Vertex-Unfolding Orthogonal Polyhedra

An edge-unfolding of a polyhedron is produced by cutting along edges and flattening the faces to a net, a connected planar piece with no overlaps. A grid unfolding allows additional cuts along grid edges induced by coordinate planes passing through every vertex. A vertex-unfolding allows faces in the net to be connected at single vertices, not necessarily along edges. We show that any orthogonal polyhedra of genus zero has a grid vertex-unfolding. (There are orthogonal polyhedra that cannot be vertex-unfolded, so some type of “gridding” of the faces is necessary.) For any orthogonal polyhedron P with n vertices, we describe an algorithm that vertex-unfolds P in O(n 2) time. Enroute to explaining this algorithm, we present a simpler vertex-unfolding algorithm that requires a 3×1×1 refinement of the vertex grid.

Unfolding Manhattan Towers

We provide an algorithm for unfolding the surface of any orthogonal polyhedron that falls into a particular shape class we call Manhattan Towers, to a nonoverlapping planar orthogonal polygon. The algorithm cuts along edges of a 4 × 5 × 1 refinement of the vertex grid.