SCG '90最新文献

{"title":"On simultaneous inner and outer approximation of shapes","authors":"R. Fleischer, K. Mehlhorn, G. Rote, E. Welzl, C. Yap","doi":"10.1145/98524.98572","DOIUrl":"https://doi.org/10.1145/98524.98572","url":null,"abstract":"For compact Euclidean bodies <italic>P, Q</italic>, we define <italic>λ</italic>(<italic>P, Q</italic>) to be smallest ratio <italic>r/s</italic> where <italic>r</italic> > 0, <italic>s</italic> > 0 satisfy <italic>sQ′</italic> ⊆ <italic>P</italic> ⊆ <italic>rQ″</italic>. Here <italic>sQ</italic> denotes a scaling of <italic>Q</italic> by factor <italic>s</italic>, and <italic>Q′</italic>, <italic>Q″</italic> are some translates of <italic>Q</italic>. This function <italic>λ</italic> gives us a new distance function between bodies which, unlike previously studied measures, is invariant under affine transformations. If homothetic bodies are identified, the logarithm of this function is a metric. (Two bodies are <italic>homothetic</italic> if one can be obtained from the other by scaling and translation).\u0000For integer <italic>&kgr;</italic> ≥ 3, define <italic>λ</italic>(<italic>&kgr;</italic>) to be the minimum value such that for each convex polygon <italic>P</italic> there exists a convex <italic>&kgr;</italic>-gon <italic>Q</italic> with <italic>λ</italic>(<italic>P, Q</italic>) ≤ <italic>λ</italic>(<italic>&kgr;</italic>). Among other results, we prove that 2.118… ≤ <italic>λ</italic>(3) ≤ 2.25 and λ(<italic>&kgr;</italic>) = 1 + &THgr;(<italic>&kgr;</italic><supscrpt>-2</supscrpt>). We give an <italic>&Ogr;</italic>(<italic>n</italic><supscrpt>2</supscrpt> log<supscrpt>2</supscrpt> <italic>n</italic>) time algorithm which for any input convex <italic>n</italic>-gon <italic>P</italic>, finds a triangle <italic>T</italic> that minimizes <italic>λ</italic>(<italic>T, P</italic>) among triangles. But in linear time, we can find a triangle <italic>t</italic> with <italic>λ</italic>(<italic>t, P</italic>) ≤ 2.25.\u0000Our study is motivated by the attempt to reduce the complexity of the polygon containment problem, and also the motion planning problem.\u0000 In each case, we describe algorithms which will run faster when certain implicit <italic>slackness</italic> parameters of the input are bounded away from 1. These algorithms illustrate a new algorithmic paradigm in computational geometry for coping with complexity.","PeriodicalId":113850,"journal":{"name":"SCG '90","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1990-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131002561","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Geometric computations with algebraic varieties of bounded degree","authors":"C. Bajaj","doi":"10.1145/98524.98557","DOIUrl":"https://doi.org/10.1145/98524.98557","url":null,"abstract":"The set of solutions to a collection of polynomial equations is referred to as an algebraic set. An algebraic set that cannot be represented as the union of two other distinct algebraic sets, neither containing the other, is said to be irreducible. An irreducible algebraic set is also known as an algebraic variety. This paper deals with geometric computations with algebraic varieties. The main results are algorithms to (1) compute the degree of an algebraic variety, (2) compute the rational parametric equations (a rational map from points on a hyperplane) for implicitly defined algebraic varieties of degrees two and three. These results are based on sub-algorithms using multi-polynomial resultants and multi-polynomial remainder sequences for constructing a one-to-one projection map of an algebraic variety to a hypersurface of equal dimension, as well as, an inverse rational map from the hypersurface to the algebraic variety. These geometric computations arise naturally in geometric modeling, computer aided design, computer graphics, and motion planning, and have been used in the past for special cases of algebraic varieties, i.e. algebraic curves and surfaces.","PeriodicalId":113850,"journal":{"name":"SCG '90","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1990-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125723911","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A computational geometry workbench","authors":"A. Knight, J. May, J. McAffer, T. Nguyen, J. Sack","doi":"10.1145/98524.98602","DOIUrl":"https://doi.org/10.1145/98524.98602","url":null,"abstract":"We are constructing a workbench for computational geometry. This is intended to provide a framework for the implementation, testing, demonstration and application of algorithms in computational geometry. The workbench is being written in Smalltalk/V using an Apple Macintosh II.\u0000The object-oriented model used in Smalltalk is well-suited to algorithms manipulating geometric objects. In addition, the programming environment can be easily extended, and provides excellent graphics facilities, data abstraction, encapsulation, and incremental modification.\u0000We have completed the design and implementation of the workbench platform, insofar as such a system can ever be considered complete. Among the features of the system are:an interactive graphical environment, including operations for creation and editing of geometric figures, and for the operation of algorithm on these figures\u0000the system supports:high-level representation-independent geometric objects (points, lines, polygons,…)\u0000geometric data structures (segment trees, range trees,…)\u0000non-geometric data structures (finger trees, splay trees, heaps, …)\u0000“standard” algorithmic tools in as general a form as possible. Algorithms currently available in the system include Tarjan and van Wyk's triangulation of a simple polygon, Fortune's Voronoi diagram,\u0000Preparata's chain decomposition, and Melkman's convex hull algorithm.\u0000\u0000 tools for the animation of geometric algorithms\u0000high-level graphical and symbolic debugging facilities\u0000portability, due to the separation of the machine-independent code and the machine-dependent user-interface.\u0000automatic handling of basic operations (device-independent graphics, storage management) allowing the implementor to focus on algorithmic issues\u0000\u0000Our group is currently working on extensions in two directions:implementing additional algorithms from two-dimensional computational geometry\u0000providing the framework for implementations of three-dimensional algorithms\u0000\u0000We are also conducting comparison studies of different algorithms and data structures, including a comparison of different triangulation and convex hull algorithms for large input sizes and an empirical test of the dynamic optimality conjecture of Sleator and Tarjan using both Splay and Finger trees in the Tarjan and van Wyk triangulation.\u0000The workbench is being demonstrated during this symposium.","PeriodicalId":113850,"journal":{"name":"SCG '90","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1990-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128461658","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Finding compact coordinate representations for polygons and polyhedra","authors":"V. Milenkovic, L. Nackman","doi":"10.1145/98524.98579","DOIUrl":"https://doi.org/10.1145/98524.98579","url":null,"abstract":"A standard technique in solid modeling is to represent planes (or lines) by explicit equations and to represent vertices and edges implicitly by means of combinatorial information. Numerical problems that arise from using floating-point arithmetic to implement operations on solids can be avoided by using exact arithmetic. Since the execution time of exact arithmetic operators increases with the number of bits required to represent the operands, it is important to avoid increasing the number of bits required to represent the plane (or line) equation coefficients. Set operations on solids do not increase the number of bits required. However, rotating a solid greatly increases the number of bits required, thus adversely affecting efficiency. One proposed solution to this problem is to round the coefficients of each plane (or line) equation without altering the combinatorial information. We show that such rounding is NP-complete.","PeriodicalId":113850,"journal":{"name":"SCG '90","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1990-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133747645","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The combinatorial complexity of hyperplane transversals","authors":"S. Cappell, J. Goodman, J. Pach, R. Pollack, M. Sharir, R. Wenger","doi":"10.1145/98524.98542","DOIUrl":"https://doi.org/10.1145/98524.98542","url":null,"abstract":"We show that the maximum combinatorial complexity of the space of hyperplane transversals to a family of <italic>n</italic> separated and strictly convex sets in R<italic><supscrpt>d</supscrpt></italic> is &THgr;(<italic>n<supscrpt>⌊d/2⌋</supscrpt></italic>), which generalizes results of Edelsbrunner and Sharir in the plane. As a key step in the argument, we show that the space of hyperplanes tangent to <italic>&kgr;</italic> ≤ <italic>d</italic> separated and strictly convex sets in R<italic><supscrpt>d</supscrpt></italic> is a topological (<italic>d</italic> - <italic>&kgr;</italic>)-sphere.","PeriodicalId":113850,"journal":{"name":"SCG '90","volume":"16 1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1990-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131594127","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An O(n2log n) time algorithm for the MinMax angle triangulation","authors":"H. Edelsbrunner, T. Tan, R. Waupotitsch","doi":"10.1145/98524.98535","DOIUrl":"https://doi.org/10.1145/98524.98535","url":null,"abstract":"<italic>We show that a triangulation of a set of n points in the plane that minimizes the maximum angle can be computed in time O(n</italic><supscrpt>2</supscrpt> log <italic>n) and space O(n). In the same amount of time and space we can also handle the constrained case where edges are prescribed. The algorithm iteratively improves an arbitrary initial triangulation and is fairly easy to implement.</italic>","PeriodicalId":113850,"journal":{"name":"SCG '90","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1990-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117027325","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A trivial knot whose spanning disks have exponential size","authors":"J. Snoeyink","doi":"10.1145/98524.98555","DOIUrl":"https://doi.org/10.1145/98524.98555","url":null,"abstract":"If a closed curve in space is a trivial knot (intuitively, one can untie it without cutting) then it is the boundary of some disk with no self-intersections. In this paper we investigate the minimum number of faces of a polyhedral spanning disk of a polygonal knot with <italic>n</italic> segments. We exhibit a knot whose minimal spanning disk has exp(<italic>cn</italic>) faces, for some positive constant <italic>c</italic>.","PeriodicalId":113850,"journal":{"name":"SCG '90","volume":"239 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1990-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124076242","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the optimal bisection of a polygon (extended abstract)","authors":"E. Koutsoupias, C. Papadimitriou, M. Sideri","doi":"10.1145/98524.98565","DOIUrl":"https://doi.org/10.1145/98524.98565","url":null,"abstract":"We give a polynomial approximation sceme for subdividing a simple polygon into approximately equal parts by curves of the smallest possible total length. For convex polygons we show that an exact fast algorithm is possible. Several generalizations are shown NP-complete.","PeriodicalId":113850,"journal":{"name":"SCG '90","volume":"142 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1990-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122150415","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Stabbing and ray shooting in 3 dimensional space","authors":"M. Pellegrini","doi":"10.1145/98524.98563","DOIUrl":"https://doi.org/10.1145/98524.98563","url":null,"abstract":"In this paper we consider the following problems: given a set <italic>T</italic> of triangles in 3-space, with |<italic>T</italic>| = <italic>n</italic>,<list><item>answer the query “given a line <italic>l</italic>, does <italic>l</italic> stab the set of triangles?” (<italic>query problem</italic>).\u0000</item><item>find whether a stabbing line exists for the set of triangles (<italic>existence problem</italic>).\u0000</item><item>Given a ray <italic>&rgr;</italic>, which is the first triangle in <italic>T</italic> hit by <italic>&rgr;</italic>?\u0000</item></list>\u0000The following results are shown.<list><item>There is an &OHgr;(<italic>n</italic><supscrpt>3</supscrpt>) lower bound on the descriptive complexity of the set of all stabbers for a set of triangles.\u0000</item><item>The existence problem for triangles on a set of planes with <italic>g</italic> different plane inclinations can be solved in <italic>&Ogr;</italic>(<italic>g</italic><supscrpt>2</supscrpt><italic>n</italic><supscrpt>2</supscrpt> log <italic>n</italic>) time (Theorem 2).\u0000</item> <item>The query problem is solvable in quasiquadratic <italic>&Ogr;</italic>(<italic>n</italic><supscrpt>2+ε</supscrpt>) preprocessing and storage and logarithmic <italic>&Ogr;</italic>(log <italic>n</italic>) query time (Theorem 4).\u0000</item><item>If we are given <italic>m</italic> rays we can answer ray shooting queries in <italic>&Ogr;</italic>(<italic>m</italic><supscrpt>5/6-δ</supscrpt> <italic>n</italic><supscrpt>5/6+5δ</supscrpt> log<supscrpt>2</supscrpt> <italic>n</italic> + <italic>m</italic> log<supscrpt>2</supscrpt> <italic>n</italic> + <italic>n</italic> log <italic>n</italic> log <italic>m</italic>) randomized expected time and <italic>&Ogr;</italic>(<italic>m</italic> + <italic>n</italic>) space (Theorem 5).\u0000</item><item>In time <italic>&Ogr;</italic>((<italic>n</italic>+<italic>m</italic>)<supscrpt>5/3+4δ</supscrpt>) it is possible to decide whether two non convex polyhedra of complexity <italic>m</italic> and <italic>n</italic> intersect (Corollary 1).\u0000</item><item>Given <italic>m</italic> rays and <italic>n</italic> axis-oriented boxes we can answer ray shooting queries in randomized expected time <italic>&Ogr;</italic>(<italic>m</italic><supscrpt>3/4-δ</supscrpt> <italic>n</italic><supscrpt>3/4+3δ</supscrpt> log<supscrpt>4</supscrpt> <italic>n</italic> + <italic>m</italic> log<supscrpt>4</supscrpt> <italic>n</italic> + <italic>n</italic> log <italic>n</italic> log <italic>m</italic>) and <italic>&Ogr;</italic>(<italic>m</italic> + <italic>n</italic>) space (Theorem 6).\u0000</item></list>","PeriodicalId":113850,"journal":{"name":"SCG '90","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1990-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123099081","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Reconstructing sets from interpoint distances (extended abstract)","authors":"S. Skiena, Warren D. Smith, Paul Lemke","doi":"10.1145/98524.98598","DOIUrl":"https://doi.org/10.1145/98524.98598","url":null,"abstract":"We consider the problem of determining which point sets in some given space realise a given distance multiset. Special cases include the “turnpike problem” where the points lie on a line, and the “beltway problem” where the points lie on a loop. Of interest is the algorithmic problem of determining such point sets for a given collection of distances and the combinatorial problem of finding bounds on the maximum number of different solutions. These problems find applications in many fields, including genetics and crystallography. In this paper, we give improved combinatorial bounds for the turnpike and baltway problems in both one and higher dimensions. We present a practical algorithm which, on n points drawn at random from a real interval, finds all solutions in &Ogr; (n2logn) time with probability 1. We also prove that some variants of the problem are NP-complete.","PeriodicalId":113850,"journal":{"name":"SCG '90","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1990-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132608099","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}