Solving geometric problems with the rotating calipers
01 Jan 1983-
TL;DR: This paper shows that the diameter of a convex n-sided polygon could be computed in O(n) time using a very elegant and simple procedure which resembles rotating a set of calipers around the polygon once, and that this simple idea can be generalized in two ways.
Abstract: Shamos [1] recently showed that the diameter of a convex n-sided polygon could be computed in O(n) time using a very elegant and simple procedure which resembles rotating a set of calipers around the polygon once. In this paper we show that this simple idea can be generalized in two ways: several sets of calipers can be used simultaneously on one convex polygon, or one set of calipers can be used on several convex polygons simultaneously. We then show that these generalizations allow us to obtain simple O(n) algorithms for solving a variety of problems defined on convex polygons. Such problems include (1) finding the minimum-area rectangle enclosing a polygon, (2) computing the maximum distance between two polygons, (3) performing the vector-sum of two polygons, (4) merging polygons in a convex hull finding algorithms, and (5) finding the critical support lines between two polygons. Finding the critical support lines, in turn, leads to obtaining solutions to several additional problems concerned with visibility, collision, avoidance, range fitting, linear separability, and computing the Grenander distance between sets.
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Book•
15 Jun 2011TL;DR: This book is the first to cover geometric approximation algorithms in detail, and topics covered include approximate nearest-neighbor search, shape approximation, coresets, dimension reduction, and embeddings.
Abstract: Exact algorithms for dealing with geometric objects are complicated, hard to implement in practice, and slow. Over the last 20 years a theory of geometric approximation algorithms has emerged. These algorithms tend to be simple, fast, and more robust than their exact counterparts. This book is the first to cover geometric approximation algorithms in detail. In addition, more traditional computational geometry techniques that are widely used in developing such algorithms, like sampling, linear programming, etc., are also surveyed. Other topics covered include approximate nearest-neighbor search, shape approximation, coresets, dimension reduction, and embeddings. The topics covered are relatively independent and are supplemented by exercises. Close to 200 color figures are included in the text to illustrate proofs and ideas.
410 citations
01 Jan 1999
TL;DR: An efficient O(n+1/?4.5-time algorithm for computing a (1+?)-approximation of the minimum-volume bounding box of n points in R3.
Abstract: We present an efficient O(n+1/?4.5-time algorithm for computing a (1+?)-approximation of the minimum-volume bounding box of n points in R3. We also present a simpler algorithm whose running time is O(nlogn+n/?3). We give some experimental results with implementations of various variants of the second algorithm.
302 citations
TL;DR: The state of the art of computational geometry is surveyed, a discipline that deals with the complexity of geometric problems within the framework of the analysis of algorithms.
Abstract: We survey the state of the art of computational geometry, a discipline that deals with the complexity of geometric problems within the framework of the analysis of algorithms. This newly emerged area of activities has found numerous applications in various other disciplines, such as computer-aided design, computer graphics, operations research, pattern recognition, robotics, and statistics. Five major problem areas—convex hulls, intersections, searching, proximity, and combinatorial optimizations—are discussed. Seven algorithmic techniques—incremental construction, plane-sweep, locus, divide-and-conquer, geometric transformation, prune-and-search, and dynamization—are each illustrated with an example. A collection of problem transformations to establish lower bounds for geo-metric problems in the algebraic computation/decision model is also included.
271 citations
01 Nov 2012
TL;DR: Chopper as discussed by the authors decomposes a large 3D object into smaller parts so that each part fits into the 3D printing volume, and these parts can then be assembled to form the original object.
Abstract: 3D printing technology is rapidly maturing and becoming ubiquitous. One of the remaining obstacles to wide-scale adoption is that the object to be printed must fit into the working volume of the 3D printer. We propose a framework, called Chopper, to decompose a large 3D object into smaller parts so that each part fits into the printing volume. These parts can then be assembled to form the original object. We formulate a number of desirable criteria for the partition, including assemblability, having few components, unobtrusiveness of the seams, and structural soundness. Chopper optimizes these criteria and generates a partition either automatically or with user guidance. Our prototype outputs the final decomposed parts with customized connectors on the interfaces. We demonstrate the effectiveness of Chopper on a variety of non-trivial real-world objects.
253 citations
07 Nov 1983
TL;DR: A new framework for geometric computing is presented in which planar curves are formed by the motions of objects that have both position and orientation, which makes it possible to recast large parts of computational geometry, such as convexity, in a new and advantageous light.
Abstract: Extended Abstract 1. Introduction and summary We present a new framework for geometric computing in which planar curves are formed by the motions of objects that have both position and orientation. Informally, consider the motion of a car: at each instant the car is positioned at some point in the plane, and oriented with its hood facing in some direction. Together, the position and the orientation specify a state. In the new framework, curves and polygons are paths in this state space satisfying the constraint that wherever the tangent of the position component of the curve is defined, the orientation must be either the same as that tangent, or the opposite. In other words, the car may move either forwards or backwards, but it may not skid sideways. To distinguish these structures from classical curves and polygons, we call them tracings. Several important concepts can be defined fOf tracings, some familiar from differential topology, but others apparently new. These include the notions of winding number, degree, and sweep number. A general construction, known as fiber product, is used to define various operations on tracings, including convolution and multiplication. Each fiber product involves forming all pairs of states, one from each tracing, that satisfy a certain constraint Using these ideas, it is possible to recast large parts ofcomputational geometry, such as convexity, in a new and advantageous light This applies to proofs of theorems, as well as to descriptions of algorithms. Besides allowing us to recast old material, this kinetic approach has also led to a number of new algorithms that improve on the previously known bounds. For example, one algorithm computes the distance between two convex polygons in logarithmic time, while another tests if two convex polygons can disjointIy fit into a third one in linear time (no rotations allowed). A final attraction of the kinetic approach is that, after appropriately extending the plane into a new manifold, the two-sided plane, we can define a formal duality between points and lines that maintains the sense of left and right Informal notions of such a duality had previously been used in computational geometry, but now for the first time we can say in a precise sense that, for example, an algorithm for computing the convex hull of n points is also an algorithm for computing the intersection of n half-planes. It may be of interest to note that the origins of …
249 citations
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14,948 citations
Book•
01 Jan 1973
TL;DR: In this article, a unified, comprehensive and up-to-date treatment of both statistical and descriptive methods for pattern recognition is provided, including Bayesian decision theory, supervised and unsupervised learning, nonparametric techniques, discriminant analysis, clustering, preprosessing of pictorial data, spatial filtering, shape description techniques, perspective transformations, projective invariants, linguistic procedures, and artificial intelligence techniques for scene analysis.
Abstract: Provides a unified, comprehensive and up-to-date treatment of both statistical and descriptive methods for pattern recognition. The topics treated include Bayesian decision theory, supervised and unsupervised learning, nonparametric techniques, discriminant analysis, clustering, preprosessing of pictorial data, spatial filtering, shape description techniques, perspective transformations, projective invariants, linguistic procedures, and artificial intelligence techniques for scene analysis.
13,647 citations
TL;DR: A collision avoidance algorithm for planning a safe path for a polyhedral object moving among known polyhedral objects that transforms the obstacles so that they represent the locus of forbidden positions for an arbitrary reference point on the moving object.
Abstract: This paper describes a collision avoidance algorithm for planning a safe path for a polyhedral object moving among known polyhedral objects. The algorithm transforms the obstacles so that they represent the locus of forbidden positions for an arbitrary reference point on the moving object. A trajectory of this reference point which avoids all forbidden regions is free of collisions. Trajectories are found by searching a network which indicates, for each vertex in the transformed obstacles, which other vertices can be reached safely.
2,396 citations
"Solving geometric problems with the..." refers background in this paper
...Vector sums of polygons and polyhedra have applications in collision avoidance problems [8]....
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TL;DR: The presented algorithms use the “divide and conquer” technique and recursively apply a merge procedure for two nonintersecting convex hulls to ensure optimal time complexity within a multiplicative constant.
Abstract: The convex hulls of sets of n points in two and three dimensions can be determined with O(n log n) operations. The presented algorithms use the “divide and conquer” technique and recursively apply a merge procedure for two nonintersecting convex hulls. Since any convex hull algorithm requires at least O(n log n) operations, the time complexity of the proposed algorithms is optimal within a multiplicative constant.
731 citations
"Solving geometric problems with the..." refers background in this paper
...A typical divide-and-conquer approach to finding the convex hull of a set of n points on the plane consists of sorting the points along the x axis and subsequently merging bigger and bigger convex polygons until one final convex polygon is obtained [9]....
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...While O(n) algorithms exist for finding the bridges of two disjoint convex polygons [9], we show here that the bridges can also be computed very simply with the rotating calipers....
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TL;DR: The method is of interest in certain packing and optimum layout problems because it consists of first determining the minimal-perimeter convex polygon that encloses the given curve and then selecting the rectangle of minimum area capable of containing this polygon.
Abstract: This paper describes a method for finding the rectangle of minimum area in which a given arbitrary plane curve can be contained. The method is of interest in certain packing and optimum layout problems. It consists of first determining the minimal-perimeter convex polygon that encloses the given curve and then selecting the rectangle of minimum area capable of containing this polygon. Three theorems are introduced to show that one side of the minimum-area rectangle must be collinear with an edge of the enclosed polygon and that the minimum-area encasing rectangle for the convex polygon is also the minimum-area rectangle for the curve.
441 citations
"Solving geometric problems with the..." refers background or methods in this paper
...This problem has received attention recently in the image processing literature and has applications in certain packing and optimal layout problems [2] as well as automatic tariffing in goodstraffic [3]....
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...The algorithm presented in [2] constructs a rectangle in O(n) time for each edge of P and selects the smallest of these for a total running time of O(n2)....
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...Freeman and Shapira [2] prove the following crucial theorem for solving this problem Theorem 2....
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