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106
Sharp features on multiresolution subdivision surfaces
 In Proceedings of Pacific Graphics
, 2001
"... In this paper we describe a method for creating sharp features and trim regions on multiresolution subdivision surfaces along a set of userdefined curves. Operations such as engraving, embossing, and trimming are important in many surface modeling applications. Their implementation, however, is non ..."
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Cited by 23 (4 self)
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In this paper we describe a method for creating sharp features and trim regions on multiresolution subdivision surfaces along a set of userdefined curves. Operations such as engraving, embossing, and trimming are important in many surface modeling applications. Their implementation, however, is nontrivial due to computational, topological, and smoothness constraints that the underlying surface has to satisfy. The novelty of our work lies in the ability to create sharp features anywhere on a surface and in the fact that the resulting representation remains within the multiresolution subdivision framework. Preserving the original representation has the advantage that other operations applicable to multiresolution subdivision surfaces can subsequently be applied to the edited model. We also introduce an extended set of subdivision rules for CatmullClark surfaces that allows the creation of creases along diagonals of control mesh faces. 1
New Results on Shortest Paths in Three Dimensions
 Proc. 20th Annual ACM Symposium on Computational Geometry
, 2004
"... We revisit the problem of computing shortest obstacleavoiding paths among obstacles in three dimensions. We prove new hardness results, showing, e.g., that computing Euclidean shortest paths among sets of “stacked ” axisaligned rectangles is NPcomplete, and that computing L1shortest paths among ..."
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Cited by 21 (0 self)
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We revisit the problem of computing shortest obstacleavoiding paths among obstacles in three dimensions. We prove new hardness results, showing, e.g., that computing Euclidean shortest paths among sets of “stacked ” axisaligned rectangles is NPcomplete, and that computing L1shortest paths among disjoint balls is NPcomplete. On the positive side, we present an efficient algorithm for computing an L1shortest path between two given points that lies on or above a given polyhedral terrain. We also give polynomialtime algorithms for some versions of stacked polygonal obstacles that are “terrainlike ” and analyze the complexity of shortest path maps in the presence of parallel halfplane “walls.”
ResourceConstrained Geometric Network Optimization (Extended Abstract)
"... We study a variety of geometric network optimization problems on a set of points, in which we are given a resource bound, B, on the total length of the network, and our objective is to maximize the number of points visited (or the total "value" of points visited). In particular, we resolve the wel ..."
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Cited by 21 (1 self)
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We study a variety of geometric network optimization problems on a set of points, in which we are given a resource bound, B, on the total length of the network, and our objective is to maximize the number of points visited (or the total "value" of points visited). In particular, we resolve the wellpublicized open problem on the approximability of the rooted "orienteering problem" for the case in which the sites are given as points in the plane and the network required is a cycle. We obtain a 2approximation for this problem. We also obtain approximation algorithms for variants of this problem in which the network required is a tree (3approximation) or a path (2approximation). No prior approximation bounds were known for any of these problems. We also obtain improved approximation algorithms for geometric instances of the unrooted orienteering problem, where we obtain a 2approximation for both the cycle and tree versions of the problem on points in the plane, as well as a ...
Interactive decal compositing with discrete exponential maps
 ACM Trans. Graph
, 2006
"... Figure 1: A clay elephant statue (left) was modeled using sketchbased implicitsurface modeling software. Then, a lapped base texture and 25 feature textures were extracted from 22 images taken with a digital camera and composited on the surface. Photography, image creation, and texture positioning ..."
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Cited by 20 (6 self)
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Figure 1: A clay elephant statue (left) was modeled using sketchbased implicitsurface modeling software. Then, a lapped base texture and 25 feature textures were extracted from 22 images taken with a digital camera and composited on the surface. Photography, image creation, and texture positioning was completed in under an hour. A method is described for texturing surfaces using decals, images placed on the surface using local parameterizations. Decal parameterizations are generated with a novel O(N logN) discrete approximation to the exponential map which requires only a single additional step in Dijkstra’s graphdistance algorithm. Decals are dynamically composited in an interface that addresses many limitations of previous work. Tools for image processing, deformation/featurematching, and vector graphics are implemented using direct surface interaction. Exponential map decals can contain holes and can also be combined with conformal parameterization to reduce distortion. The exponential map approximation can be computed on any point set, including meshes and sampled implicit surfaces, and is relatively stable under resampling. The decals stick to the surface as it is interactively deformed, allowing the texture to be preserved even if the surface changes topology. These properties make exponential map decals a suitable approach for texturing animated implicit surfaces.
TwoPoint Euclidean Shortest Path Queries in the Plane (Extended Abstract)
, 1999
"... ) To appear in Proc. Tenth Annual ACMSIAM Symposium on Discrete Algorithms (SODA '99), January 1719, 1999 YiJen Chiang Joseph S. B. Mitchell y Abstract We consider the twopoint query version of the fundamental geometric shortest path problem: Given a set h of polygonal obstacles in the pla ..."
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Cited by 18 (2 self)
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) To appear in Proc. Tenth Annual ACMSIAM Symposium on Discrete Algorithms (SODA '99), January 1719, 1999 YiJen Chiang Joseph S. B. Mitchell y Abstract We consider the twopoint query version of the fundamental geometric shortest path problem: Given a set h of polygonal obstacles in the plane, having a total of n vertices, build a data structure such that for any two query points s and t we can efficiently determine the length, d(s; t), of an Euclidean shortest obstacleavoiding path, ß(s; t), from s to t. Additionally, our data structure should allow one to report the path ß(s; t), in time proportional to its (combinatorial) size. We present various methods for solving this twopoint query problem, including algorithms with o(n), O(log n+h), O(h log n), O(log 2 n) or optimal O(log n) query times, using polynomialspace data structures, with various tradeoffs between space and query time. While several results have been known for approximate twopoint Euclidean shortest p...
On the Competitive Complexity of Navigation Tasks
 Sensor Based Intelligent Robots, volume 2238 of Lecture Notes Comput. Sci
, 2002
"... A strategy S solving a navigation task T is called competitive with ratio r if the cost of solving any instance t of T does not exceed r times the cost of solving t optimally. The competitive complexity of task T is the smallest possible value r any strategy S can achieve. We discuss this notion ..."
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Cited by 18 (12 self)
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A strategy S solving a navigation task T is called competitive with ratio r if the cost of solving any instance t of T does not exceed r times the cost of solving t optimally. The competitive complexity of task T is the smallest possible value r any strategy S can achieve. We discuss this notion, and survey some tasks whose competitive complexities are known.
An Efficient Approximation Algorithm for Weighted Region Shortest Path Problem
, 2000
"... In this paper we present an approximation algorithm for solving the following optimal motion planning problem: Given a planar space composed of triangular regions, each of which is associated with a positive unit weight, and two points s and t, find a path from s to t with the minimum weight, where ..."
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Cited by 17 (5 self)
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In this paper we present an approximation algorithm for solving the following optimal motion planning problem: Given a planar space composed of triangular regions, each of which is associated with a positive unit weight, and two points s and t, find a path from s to t with the minimum weight, where the weight of a path is defined to be the weighted sum of the lengths of the subpaths within each region.
The Polygon Exploration Problem
 SIAM J. Comput
, 2002
"... We present an online strategy that enables a mobile robot with vision to explore an unknown simple polygon. We prove that the resulting tour is less than 26.5 times as long as the shortest watchman tour that could be computed o#line. ..."
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Cited by 16 (4 self)
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We present an online strategy that enables a mobile robot with vision to explore an unknown simple polygon. We prove that the resulting tour is less than 26.5 times as long as the shortest watchman tour that could be computed o#line.
Movement planning in the presence of flows
 Algorithmica
, 2001
"... This paper investigates the problem of timeoptimum movement planning in two and three dimensions for a point robot which has bounded control velocity through a set of n polygonal regions of given translational flow velocities. This intriguing geometric problem has immediate applications to macrosc ..."
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Cited by 15 (5 self)
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This paper investigates the problem of timeoptimum movement planning in two and three dimensions for a point robot which has bounded control velocity through a set of n polygonal regions of given translational flow velocities. This intriguing geometric problem has immediate applications to macroscale motion planning for ships, submarines and airplanes in the presence of significant flows of water or air. Also, it is a central motion planning problem for many of the mesoscale and microscale robots that recently have been constructed, that have environments with significant flows that affect their movement. In spite of these applications, there is very little literature on this problem, and prior work provided neither an upper bound on its computational complexity nor even a decision algorithm. It can easily be seen that an optimum path for the 2D version of this problem can consist of at least an exponential number of distinct segments through flow regions. We provide the first known computational complexity hardness result for the 3D version of this problem; we show the problem is PSPACE hard. We give the first known decision algorithm for the 2D flow path problem, but this decision algorithm has very high computational complexity. We also give the first known efficient approximation algorithms with bounded error.