CiteSeerX - Document Details (Isaac Councill, Lee Giles): This note considers the complexity of a... more CiteSeerX - Document Details (Isaac Councill, Lee Giles): This note considers the complexity of a free region in the configuration space of a polygonal robot translating amidst polygonal obstacles in the plane. Specifically, given polygonal sets P and Q with k and n vertices, ...
Proceedings of the seventh international conference on 3D Web technology, 2002
Because of the convenience of a text-based format 3D content is often published in form of a gzip... more Because of the convenience of a text-based format 3D content is often published in form of a gzipped file that contains an ASCII description of the scene graph. While compressed image, audio, and video data is kept in seperate binary files, polygonal data is usually included uncompressed into the ASCII description, as there is no widely-accepted standard for compressed polygon meshes. In this paper we show how to incorporate compression of polygonal data into a purely text-based scene graph description. Our scheme codes polygon meshes as ASCII strings that compress well with standard compression schemes such as gzip. The coder is lossless when only the position and texture coordinate indices are coded. If loss is acceptable, positions and texture coordinates can be quantized and delta coded, which reduces the file size further. The gzipped scene graph description files decrease by a factor of two (six) in size when the polygon meshes they contain are coded with the lossless (lossy) ASCII coder. Furthermore we describe in detail a proof-of-concept implementation that uses the Shout3D [18] pure java API-a plugin-less Web3D player that downloads all required java classes on demand. Our prototype is an extremely lightweight implementation of the decoder that can be distributed at minimal additional cost. The size of the compiled decoder class is less than 6KB by itself and less than 3KB if included into a compressed archive of java class files. It makes no use of specific features of the Shout3D API. Hence, our method will work for any scene graph API that allows (a) to extend the node set and (b) to store the scene graph as ASCII.
ACM SIGGRAPH 2005 Sketches on - SIGGRAPH '05, 2005
Current mesh compression schemes encode triangles and vertices in an order derived from systemati... more Current mesh compression schemes encode triangles and vertices in an order derived from systematically traversing the connectivity graph. These schemes struggle with gigabyte-sized mesh input where the construction and the usage of the data structures that support topological traversal queries become I/O-inefficient and require large amounts of temporary disk space. Furthermore they expect the entire mesh as input. Since meshes cannot be compressed until their generation is complete, they have to be stored at least once in uncompressed form. We radically depart from the traditional approach to mesh compression and propose a scheme that incrementally encodes a mesh in the order it is given to the compressor using only minimal memory resources. This makes the compression process essentially transparent to the user and practically independent of the mesh size. This is especially beneficial for compressing large meshes, where previous approaches spend significant memory, disk, and I/O resources on pre-processing, whereas our scheme starts compressing after receiving the first few triangles.
We show t h a t a v ertex-based data structure that keeps only 6 pointers per vertex can store tr... more We show t h a t a v ertex-based data structure that keeps only 6 pointers per vertex can store triangulations, navigate them, and maintain them under swap operations. By comparison, edge-based structures such as the winged-edge take 18{24 pointers per vertex, and triangle-based structures take 12 pointers per vertex.
Proceedings of the fifth annual symposium on Computational geometry - SCG '89, 1989
We consider arrangements of curves that intersect palrwise in at most k points. We show that a cu... more We consider arrangements of curves that intersect palrwise in at most k points. We show that a curve can sweep any such arrangement and maintain the k-intersection property if and only if k equals 1 or 2. We apply thii result to an eclectic set of problems: finding boolean formula for polygons with curved edges, counting triangles and digons in arrangements of pseudocircles, and finding extension curves for arrangements. We also discuss implementing the sweep.
We apply van Emde Boas-type strati ed trees to point location problems in rectangular subdivision... more We apply van Emde Boas-type strati ed trees to point location problems in rectangular subdivisions in 2 and 3 dimensions. In a subdivision with n rectangles having integer coordinates from 0; U ? 1], we locate an integer query point in O((log logU) d) query time using O(n) space when d 2 or O(n loglog U) space when d = 3. Applications and extensions of this \ xed universe" approach include spatial point location using logarithmic time and linear space in rectilinear subdivisions having arbitrary coordinates, point location in c-oriented polygons or fat triangles in the plane, point location in subdivisions of space into \fat prisms," and vertical ray shooting among horizontal \fat objects." Like other results on strati ed trees, our algorithms run on a RAM model and make use of perfect hashing.
International Journal of Computational Geometry & Applications, 1999
We give a logarithmic-time algorithm to compute the shortest segment joining two convex n-gons A ... more We give a logarithmic-time algorithm to compute the shortest segment joining two convex n-gons A and B while avoiding another convex n-gon C. Our algorithm uses a tentative prune-and-search technique on standard representations of the polygons as arrays or balanced binary search trees.
We show that there is a constant α > 0 such that, for any set P of n ≥ 5 points in general positi... more We show that there is a constant α > 0 such that, for any set P of n ≥ 5 points in general position in the plane, a crossing-free geometric graph on P that is chosen uniformly at random contains, in expectation, at least (1 2 + α)M edges, where M denotes the number of edges in any triangulation of P. From this we derive (to our knowledge) the first non-trivial upper bound of the form c n • tr(P) on the number of crossing-free geometric graphs on P ; that is, at most a fixed exponential in n times the number of triangulations of P. (The trivial upper bound of 2 M • tr(P), or c = 2 M/n , follows by taking subsets of edges of each triangulation.) If the convex hull of P is triangular, then M = 3n − 6, and we obtain c < 7.98. Upper bounds for the number of crossing-free geometric graphs on planar point sets have so far applied the trivial 8 n factor to the bound for triangulations; we slightly decrease this bound to O(343.11 n).
We prove a generalization of the famous Ham Sandwich Theorem for the plane. Given gn red points a... more We prove a generalization of the famous Ham Sandwich Theorem for the plane. Given gn red points and gm blue points in the plane in general position, there exists an equitable subdivision of the plane into g disjoint convex polygons, each of which contains n red points and m blue points. For g = 2 this problem is equivalent to the Ham Sandwich Theorem in the plane. We also present an efficient algorithm for constructing an equitable subdivision.
DOI to the publisher's website. • The final author version and the galley proof are versions of t... more DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the "Taverne" license above, please follow below link for the End User Agreement:
We present efficient geometric algorithms for simplifying polygonal paths in R 2 and R 3 that hav... more We present efficient geometric algorithms for simplifying polygonal paths in R 2 and R 3 that have angle constraints, improving by nearly a linear factor over the graph-theoretic solutions based on known techniques. The algorithms we present match the time bounds for their unconstrained counterparts. As a key step in our solutions, we formulate and solve an off-line ball exclusion search problem, which may be of interest in its own right.
The Terrain Resources Inventory Mapping TRIM data standard in BC, Canada, includes speci cations ... more The Terrain Resources Inventory Mapping TRIM data standard in BC, Canada, includes speci cations for river data and elevation data that are typically met by i n terpretation of stereo orthophotos. It does not specify that breaklines for watersheds be interpreted from the photos, thus we m ust extract them from the data. We seek a system of watersheds that, while not exact due to errors in the data, is at least self-consistent h a ving no overlapping watersheds and no unclassi ed points. We build a TIN terrain model using a Delaunay triangulation that combines both river and elevation data. Based on the standard de nition of water ow along steepest-descent paths, we create an algorithm that identi es ridges and channels in the TIN, and extracts watersheds. Several unexpected geometric con gurations, which w e h a v e not seen in the literature, follow from the standard de nitions. These must be correctly computed to obtain a consistent system of watersheds.
Do convex obstacles in the plane always leave 3 separate escape routes? Here, an escape route is ... more Do convex obstacles in the plane always leave 3 separate escape routes? Here, an escape route is a locally geodesic path that avoids the obstacles; escape routes are separate if they have no point in common but their origin. We answer this question, posed at FWCG '09 by Al-Jubeh, Ishaque and Tóth, in the affirmative and show how to efficiently compute the routes.
Consider the Delaunay triangulation T of a set P of points in the plane. The spanning ratio of T ... more Consider the Delaunay triangulation T of a set P of points in the plane. The spanning ratio of T , i.e. the maximum ratio between the length of the shortest path between this pair on the graph of the triangulation and their Euclidean distance. It has long been conjectured that the spanning ratio of T can be at most π/2. We show in this note that there exist point sets in convex position with a spanning ratio > 1.5810 and in general position with a spanning ratio > 1.5846, both of which are strictly larger than π/2 ≈ 1.5708. Furthermore, we show that any set of points drawn independently from the same distribution will, with high probability, have a spanning ratio larger than π/2.
We consider the problem of converting boundary representations of polyhedral objects into constru... more We consider the problem of converting boundary representations of polyhedral objects into constructive-solid-geometry (CSG) representations. The CSG representations for a polyhedron P are based on the half-spaces supporting the faces of P. For certain kinds of polyhedra ...
Proceedings of the fourteenth annual symposium on Computational geometry - SCG '98, 1998
No polynomial-time algorithm is kno-ivn to compute t,he minimum weight triangulation (MWT) of a f... more No polynomial-time algorithm is kno-ivn to compute t,he minimum weight triangulation (MWT) of a finite planar point set. In this paper xve present efficient implementat,ions of the LMT-skeleton heurist'ic, xvhich identifies edges that must be, and cannot be, in an MWT. For uniformly distributed points, v:e can compute the esact MWT of tens of thousands of points in minutes. These results are obtained by improving the asymptot,ic time and memory usage of the LMTskeleton heurist.ic and of filters that identify initial candidate edges, and also by bucketing and further t,uning for evenly distributed points. Further details and an implementation as a macro for the IPE dran;ng prog7cam are available on the web:
Consider the Delaunay triangulation T of a set P of points in the plane as a Euclidean graph, in ... more Consider the Delaunay triangulation T of a set P of points in the plane as a Euclidean graph, in which the weight of every edge is its length. It has long been conjectured that the dilation in T of any pair p, p ∈ P , which is the ratio of the length of the shortest path from p to p in T over the Euclidean distance pp , can be at most π/2 ≈ 1.5708. In this paper, we show how to construct point sets in convex position with dilation > 1.5810 and in general position with dilation > 1.5846. Furthermore, we show that a sufficiently large set of points drawn independently from any distribution will in the limit approach the worst-case dilation for that distribution.
CiteSeerX - Document Details (Isaac Councill, Lee Giles): This note considers the complexity of a... more CiteSeerX - Document Details (Isaac Councill, Lee Giles): This note considers the complexity of a free region in the configuration space of a polygonal robot translating amidst polygonal obstacles in the plane. Specifically, given polygonal sets P and Q with k and n vertices, ...
Proceedings of the seventh international conference on 3D Web technology, 2002
Because of the convenience of a text-based format 3D content is often published in form of a gzip... more Because of the convenience of a text-based format 3D content is often published in form of a gzipped file that contains an ASCII description of the scene graph. While compressed image, audio, and video data is kept in seperate binary files, polygonal data is usually included uncompressed into the ASCII description, as there is no widely-accepted standard for compressed polygon meshes. In this paper we show how to incorporate compression of polygonal data into a purely text-based scene graph description. Our scheme codes polygon meshes as ASCII strings that compress well with standard compression schemes such as gzip. The coder is lossless when only the position and texture coordinate indices are coded. If loss is acceptable, positions and texture coordinates can be quantized and delta coded, which reduces the file size further. The gzipped scene graph description files decrease by a factor of two (six) in size when the polygon meshes they contain are coded with the lossless (lossy) ASCII coder. Furthermore we describe in detail a proof-of-concept implementation that uses the Shout3D [18] pure java API-a plugin-less Web3D player that downloads all required java classes on demand. Our prototype is an extremely lightweight implementation of the decoder that can be distributed at minimal additional cost. The size of the compiled decoder class is less than 6KB by itself and less than 3KB if included into a compressed archive of java class files. It makes no use of specific features of the Shout3D API. Hence, our method will work for any scene graph API that allows (a) to extend the node set and (b) to store the scene graph as ASCII.
ACM SIGGRAPH 2005 Sketches on - SIGGRAPH '05, 2005
Current mesh compression schemes encode triangles and vertices in an order derived from systemati... more Current mesh compression schemes encode triangles and vertices in an order derived from systematically traversing the connectivity graph. These schemes struggle with gigabyte-sized mesh input where the construction and the usage of the data structures that support topological traversal queries become I/O-inefficient and require large amounts of temporary disk space. Furthermore they expect the entire mesh as input. Since meshes cannot be compressed until their generation is complete, they have to be stored at least once in uncompressed form. We radically depart from the traditional approach to mesh compression and propose a scheme that incrementally encodes a mesh in the order it is given to the compressor using only minimal memory resources. This makes the compression process essentially transparent to the user and practically independent of the mesh size. This is especially beneficial for compressing large meshes, where previous approaches spend significant memory, disk, and I/O resources on pre-processing, whereas our scheme starts compressing after receiving the first few triangles.
We show t h a t a v ertex-based data structure that keeps only 6 pointers per vertex can store tr... more We show t h a t a v ertex-based data structure that keeps only 6 pointers per vertex can store triangulations, navigate them, and maintain them under swap operations. By comparison, edge-based structures such as the winged-edge take 18{24 pointers per vertex, and triangle-based structures take 12 pointers per vertex.
Proceedings of the fifth annual symposium on Computational geometry - SCG '89, 1989
We consider arrangements of curves that intersect palrwise in at most k points. We show that a cu... more We consider arrangements of curves that intersect palrwise in at most k points. We show that a curve can sweep any such arrangement and maintain the k-intersection property if and only if k equals 1 or 2. We apply thii result to an eclectic set of problems: finding boolean formula for polygons with curved edges, counting triangles and digons in arrangements of pseudocircles, and finding extension curves for arrangements. We also discuss implementing the sweep.
We apply van Emde Boas-type strati ed trees to point location problems in rectangular subdivision... more We apply van Emde Boas-type strati ed trees to point location problems in rectangular subdivisions in 2 and 3 dimensions. In a subdivision with n rectangles having integer coordinates from 0; U ? 1], we locate an integer query point in O((log logU) d) query time using O(n) space when d 2 or O(n loglog U) space when d = 3. Applications and extensions of this \ xed universe" approach include spatial point location using logarithmic time and linear space in rectilinear subdivisions having arbitrary coordinates, point location in c-oriented polygons or fat triangles in the plane, point location in subdivisions of space into \fat prisms," and vertical ray shooting among horizontal \fat objects." Like other results on strati ed trees, our algorithms run on a RAM model and make use of perfect hashing.
International Journal of Computational Geometry & Applications, 1999
We give a logarithmic-time algorithm to compute the shortest segment joining two convex n-gons A ... more We give a logarithmic-time algorithm to compute the shortest segment joining two convex n-gons A and B while avoiding another convex n-gon C. Our algorithm uses a tentative prune-and-search technique on standard representations of the polygons as arrays or balanced binary search trees.
We show that there is a constant α > 0 such that, for any set P of n ≥ 5 points in general positi... more We show that there is a constant α > 0 such that, for any set P of n ≥ 5 points in general position in the plane, a crossing-free geometric graph on P that is chosen uniformly at random contains, in expectation, at least (1 2 + α)M edges, where M denotes the number of edges in any triangulation of P. From this we derive (to our knowledge) the first non-trivial upper bound of the form c n • tr(P) on the number of crossing-free geometric graphs on P ; that is, at most a fixed exponential in n times the number of triangulations of P. (The trivial upper bound of 2 M • tr(P), or c = 2 M/n , follows by taking subsets of edges of each triangulation.) If the convex hull of P is triangular, then M = 3n − 6, and we obtain c < 7.98. Upper bounds for the number of crossing-free geometric graphs on planar point sets have so far applied the trivial 8 n factor to the bound for triangulations; we slightly decrease this bound to O(343.11 n).
We prove a generalization of the famous Ham Sandwich Theorem for the plane. Given gn red points a... more We prove a generalization of the famous Ham Sandwich Theorem for the plane. Given gn red points and gm blue points in the plane in general position, there exists an equitable subdivision of the plane into g disjoint convex polygons, each of which contains n red points and m blue points. For g = 2 this problem is equivalent to the Ham Sandwich Theorem in the plane. We also present an efficient algorithm for constructing an equitable subdivision.
DOI to the publisher's website. • The final author version and the galley proof are versions of t... more DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the "Taverne" license above, please follow below link for the End User Agreement:
We present efficient geometric algorithms for simplifying polygonal paths in R 2 and R 3 that hav... more We present efficient geometric algorithms for simplifying polygonal paths in R 2 and R 3 that have angle constraints, improving by nearly a linear factor over the graph-theoretic solutions based on known techniques. The algorithms we present match the time bounds for their unconstrained counterparts. As a key step in our solutions, we formulate and solve an off-line ball exclusion search problem, which may be of interest in its own right.
The Terrain Resources Inventory Mapping TRIM data standard in BC, Canada, includes speci cations ... more The Terrain Resources Inventory Mapping TRIM data standard in BC, Canada, includes speci cations for river data and elevation data that are typically met by i n terpretation of stereo orthophotos. It does not specify that breaklines for watersheds be interpreted from the photos, thus we m ust extract them from the data. We seek a system of watersheds that, while not exact due to errors in the data, is at least self-consistent h a ving no overlapping watersheds and no unclassi ed points. We build a TIN terrain model using a Delaunay triangulation that combines both river and elevation data. Based on the standard de nition of water ow along steepest-descent paths, we create an algorithm that identi es ridges and channels in the TIN, and extracts watersheds. Several unexpected geometric con gurations, which w e h a v e not seen in the literature, follow from the standard de nitions. These must be correctly computed to obtain a consistent system of watersheds.
Do convex obstacles in the plane always leave 3 separate escape routes? Here, an escape route is ... more Do convex obstacles in the plane always leave 3 separate escape routes? Here, an escape route is a locally geodesic path that avoids the obstacles; escape routes are separate if they have no point in common but their origin. We answer this question, posed at FWCG '09 by Al-Jubeh, Ishaque and Tóth, in the affirmative and show how to efficiently compute the routes.
Consider the Delaunay triangulation T of a set P of points in the plane. The spanning ratio of T ... more Consider the Delaunay triangulation T of a set P of points in the plane. The spanning ratio of T , i.e. the maximum ratio between the length of the shortest path between this pair on the graph of the triangulation and their Euclidean distance. It has long been conjectured that the spanning ratio of T can be at most π/2. We show in this note that there exist point sets in convex position with a spanning ratio > 1.5810 and in general position with a spanning ratio > 1.5846, both of which are strictly larger than π/2 ≈ 1.5708. Furthermore, we show that any set of points drawn independently from the same distribution will, with high probability, have a spanning ratio larger than π/2.
We consider the problem of converting boundary representations of polyhedral objects into constru... more We consider the problem of converting boundary representations of polyhedral objects into constructive-solid-geometry (CSG) representations. The CSG representations for a polyhedron P are based on the half-spaces supporting the faces of P. For certain kinds of polyhedra ...
Proceedings of the fourteenth annual symposium on Computational geometry - SCG '98, 1998
No polynomial-time algorithm is kno-ivn to compute t,he minimum weight triangulation (MWT) of a f... more No polynomial-time algorithm is kno-ivn to compute t,he minimum weight triangulation (MWT) of a finite planar point set. In this paper xve present efficient implementat,ions of the LMT-skeleton heurist'ic, xvhich identifies edges that must be, and cannot be, in an MWT. For uniformly distributed points, v:e can compute the esact MWT of tens of thousands of points in minutes. These results are obtained by improving the asymptot,ic time and memory usage of the LMTskeleton heurist.ic and of filters that identify initial candidate edges, and also by bucketing and further t,uning for evenly distributed points. Further details and an implementation as a macro for the IPE dran;ng prog7cam are available on the web:
Consider the Delaunay triangulation T of a set P of points in the plane as a Euclidean graph, in ... more Consider the Delaunay triangulation T of a set P of points in the plane as a Euclidean graph, in which the weight of every edge is its length. It has long been conjectured that the dilation in T of any pair p, p ∈ P , which is the ratio of the length of the shortest path from p to p in T over the Euclidean distance pp , can be at most π/2 ≈ 1.5708. In this paper, we show how to construct point sets in convex position with dilation > 1.5810 and in general position with dilation > 1.5846. Furthermore, we show that a sufficiently large set of points drawn independently from any distribution will in the limit approach the worst-case dilation for that distribution.
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