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Computational geometry  a survey
 IEEE TRANSACTIONS ON COMPUTERS
, 1984
"... 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 ofalgorithms. This newly emerged area of activities has found numerous applications in various other disciplines, such as computeraided de ..."
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Cited by 21 (3 self)
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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 ofalgorithms. This newly emerged area of activities has found numerous applications in various other disciplines, such as computeraided design, computer graphics, operations research, pattern recognition, robotics, and statistics. Five major problem areasconvex hulls, intersections, searching, proximity, and combinatorial optimizationsare discussed. Seven algorithmic techniques incremental construction, planesweep, locus, divideandconquer, geometric transformation, pruneandsearch, and dynamizationare each illustrated with an example.Acollection of problem transformations to establish lower bounds for geometric problems in the algebraic computation/decision model is also included.
On the Complexity of Optimization Problems for 3Dimensional Convex Polyhedra and Decision Trees
 Comput. Geom. Theory Appl
, 1995
"... We show that several wellknown optimization problems involving 3dimensional convex polyhedra and decision trees are NPhard or NPcomplete. One of the techniques we employ is a lineartime method for realizing a planar 3connected triangulation as a convex polyhedron, which may be of independent i ..."
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Cited by 14 (0 self)
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We show that several wellknown optimization problems involving 3dimensional convex polyhedra and decision trees are NPhard or NPcomplete. One of the techniques we employ is a lineartime method for realizing a planar 3connected triangulation as a convex polyhedron, which may be of independent interest. Key words: Convex polyhedra, approximation, Steinitz's theorem, planar graphs, art gallery theorems, decision trees. 1 Introduction Convex polyhedra are fundamental geometric structures (e.g., see [20]). They are the product of convex hull algorithms, and are key components for problems in robot motion planning and computeraided geometric design. Moreover, due to a beautiful theorem of Steinitz [20, 38], they provide a strong link between computational geometry and graph theory, for Steinitz shows that a graph forms the edge structure of a convex polyhedra if and only if it is planar and 3connected. Unfortunately, algorithmic problems dealing with 3dimensional convex polyhedra ...
Algorithms for Minimum Volume Enclosing Simplex in R³
 SIAM J. Comput
, 1999
"... We develop a combinatorial algorithm for determining a minimum volume simplex enclosing a set of points in R 3 . If the convex hull of the points has n vertices, then our algorithm takes (n 4 ) time. Combining our exact but slow algorithm with a simple but crude approximation technique, we al ..."
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Cited by 13 (0 self)
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We develop a combinatorial algorithm for determining a minimum volume simplex enclosing a set of points in R 3 . If the convex hull of the points has n vertices, then our algorithm takes (n 4 ) time. Combining our exact but slow algorithm with a simple but crude approximation technique, we also develop an "approximation algorithm. The algorithm computes in O(n + 1=" 6 ) time a simplex whose volume is within (1 + ") factor of the optimal, for any " > 0. 1 Introduction Approximating a geometric body by a combinatorially simpler shape is a problem with many applications. In computer graphics and robotics, for instance, checking for collision between complex geometric models is frequently a computational bottleneck. Therefore, collision detection packages commonly use simple bounding objects, such as axisaligned bounding boxes [4, 14, 16], discrete oriented polytopes [9, 13], or spheres [10], to quickly eliminate pairs whose bounding objects are collisionfree. Since interse...
Translating a Convex Polygon to Contain a Maximum Number of Points (Extended Abstract)
 PROC. 7TH CANADIAN CONF. ON COMPUTATIONAL GEOMETRY
, 1995
"... ..."
Shortest Paths Help Solve Geometric Optimization Problems in Planar Regions
 SIAM J. Comput
"... The goal of this paper is to show that the concept of the shortest path inside a polygonal region contributes to the design of efficient algorithms for certain geometric optimization problems involving simple polygons: computing optimum separators, maximum area or perimeter inscribed triangles, a mi ..."
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Cited by 6 (0 self)
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The goal of this paper is to show that the concept of the shortest path inside a polygonal region contributes to the design of efficient algorithms for certain geometric optimization problems involving simple polygons: computing optimum separators, maximum area or perimeter inscribed triangles, a minimum area circumscribed concave quadrilateral, or a maximum area contained triangle. The structure for our algorithms is as follows: a) decompose the initial problem into a lowdegree polynomial number of optimization problems; b) solve each individual subproblem in constant time using standard methods of calculus, basic methods of numerical analysis, or linear programming. These same optimization techniques can be applied to splinegons (curved polygons). To do this, we first develop a decomposition technique for curved polygons which we substitute for triangulation in creating equally efficient curved versions of the algorithms for the shortestpath tree, rayshooting and twopoint shortes...
Optimal Placement of Convex Polygons to Maximize Point Containment
, 1996
"... Given a convex polygon P with m vertices and a set S of n points in the plane, we consider the problem of finding a placement of P that contains the maximum number of points in S. We allow both translation and rotation. Our algorithm is selfcontained and utilizes the geometric properties of the co ..."
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Cited by 6 (1 self)
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Given a convex polygon P with m vertices and a set S of n points in the plane, we consider the problem of finding a placement of P that contains the maximum number of points in S. We allow both translation and rotation. Our algorithm is selfcontained and utilizes the geometric properties of the containing regions in the parameter space of transformations. The algorithm requires O(nk 2 m 2 log(mk)) time and O(n +m) space, where k is the maximum number of points contained. This provides a linear improvement over the best previously known algorithm when k is large (\Theta(n)) and a cubic improvement when k is small. We also show that the algorithm can be extended to solve bichromatic and general weighted variants of the problem. 1 Introduction A planar rigid motion ae is an affine transformation of the plane that preserves distance (and therefore angles and area also). We say that a polygon P contains a set S of points if every point in S lies on P or in the interior of P . In th...
Computing Shortest Transversals
, 1991
"... We present an O(n log² n) time and O(n) space algorithm for computing the shortest line segment that intersects a set of n given line segments or lines in the plane. If the line segments do not intersect the algorithm may be trimmed to run in O(n log n) time. Furthermore, in combination with linear ..."
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Cited by 4 (3 self)
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We present an O(n log² n) time and O(n) space algorithm for computing the shortest line segment that intersects a set of n given line segments or lines in the plane. If the line segments do not intersect the algorithm may be trimmed to run in O(n log n) time. Furthermore, in combination with linear programming the algorithm will also find the shortest line segment that intersects a set of n isothetic rectangles in the plane in O(n log k) time, where k is the combinatorial complexity of the space of transversals and k 4n. These results find application in: (1) linefitting between a set of n data ranges where it is desired to obtain the shortest lineoffit, (2) finding the shortest line segment from which a convex nvertex polygon is weakly externally visible, and (3) determining the shortest lineofsight between two edges of a simple nvertex polygon, for which O(n) time algorithms are also given. All the algorithms are based on the solution to a new fundamental geometric optim...
ON COMPUTING ENCLOSING ISOSCELES TRIANGLES AND RELATED PROBLEMS
 INTERNATIONAL JOURNAL OF COMPUTATIONAL GEOMETRY & APPLICATIONS
, 2009
"... Given a set of n points in the plane, we show how to compute various enclosing isosceles triangles where different parameters such as area or perimeter are optimized. We then study a 3dimensional version of the problem where we enclose a point set with a cone of fixed apex angle α. ..."
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Cited by 3 (2 self)
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Given a set of n points in the plane, we show how to compute various enclosing isosceles triangles where different parameters such as area or perimeter are optimized. We then study a 3dimensional version of the problem where we enclose a point set with a cone of fixed apex angle α.
Minimum Enclosures with Specified Angles
, 1994
"... Given a convex polygon P , an menvelope is a convex msided polygon that contains P . Given any convex polygon P , and any sequence of m 3 angles A = hff 1 ; ff 2 ; : : : ; ff m i, we consider the problem of computing the minimum area menvelope for P whose counterclockwise sequence of exterior an ..."
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Cited by 2 (0 self)
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Given a convex polygon P , an menvelope is a convex msided polygon that contains P . Given any convex polygon P , and any sequence of m 3 angles A = hff 1 ; ff 2 ; : : : ; ff m i, we consider the problem of computing the minimum area menvelope for P whose counterclockwise sequence of exterior angles is given by A. We show that such envelopes can be computed in O(nm log m) time. The main result on which the correctness of the algorithm rests is a flushness condition stating that for any locally minimum enclosure with specified angles, one of its sides must be collinear with one of the sides of P . The support of the National Science Foundation under Grant CCR8908901, the Bureau of the Census under grant JSA 915, and the University of the District of Columbia under a Faculty Senate Summer Research Grant is gratefully acknowledged, as is the help of Sandy German in preparing this paper. A preliminary version of this paper appeared in Vision Geometry , R.A. Melter and A.Y. Wu, Edit...
Minimum Enclosing Area Triangle with a Fixed Angle
, 2010
"... Given a set S of n points in the plane and a fixed angle 0 < ω < π, we show how to find all triangles of minimum area with angle ω that enclose S in O(n log n) time. ..."
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Cited by 1 (1 self)
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Given a set S of n points in the plane and a fixed angle 0 < ω < π, we show how to find all triangles of minimum area with angle ω that enclose S in O(n log n) time.