1,523 research outputs found

    BSP-fields: An Exact Representation of Polygonal Objects by Differentiable Scalar Fields Based on Binary Space Partitioning

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    The problem considered in this work is to find a dimension independent algorithm for the generation of signed scalar fields exactly representing polygonal objects and satisfying the following requirements: the defining real function takes zero value exactly at the polygonal object boundary; no extra zero-value isosurfaces should be generated; C1 continuity of the function in the entire domain. The proposed algorithms are based on the binary space partitioning (BSP) of the object by the planes passing through the polygonal faces and are independent of the object genus, the number of disjoint components, and holes in the initial polygonal mesh. Several extensions to the basic algorithm are proposed to satisfy the selected optimization criteria. The generated BSP-fields allow for applying techniques of the function-based modeling to already existing legacy objects from CAD and computer animation areas, which is illustrated by several examples

    Algorithmic and Combinatorial Results on Fence Patrolling, Polygon Cutting and Geometric Spanners

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    The purpose of this dissertation is to study problems that lie at the intersection of geometry and computer science. We have studied and obtained several results from three different areas, namely–geometric spanners, polygon cutting, and fence patrolling. Specifically, we have designed and analyzed algorithms along with various combinatorial results in these three areas. For geometric spanners, we have obtained combinatorial results regarding lower bounds on worst case dilation of plane spanners. We also have studied low degree plane lattice spanners, both square and hexagonal, of low dilation. Next, for polygon cutting, we have designed and analyzed algorithms for cutting out polygon collections drawn on a piece of planar material using the three geometric models of saw, namely, line, ray and segment cuts. For fence patrolling, we have designed several strategies for robots patrolling both open and closed fences

    Approximation Schemes for Partitioning: Convex Decomposition and Surface Approximation

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    We revisit two NP-hard geometric partitioning problems - convex decomposition and surface approximation. Building on recent developments in geometric separators, we present quasi-polynomial time algorithms for these problems with improved approximation guarantees.Comment: 21 pages, 6 figure

    An Overview of Rendering from Volume Data --- including Surface and Volume Rendering

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    Volume rendering is a title often ambiguously used in science. One meaning often quoted is: `to render any three volume dimensional data set'; however, within this categorisation `surface rendering'' is contained. Surface rendering is a technique for visualising a geometric representation of a surface from a three dimensional volume data set. A more correct definition of Volume Rendering would only incorporate the direct visualisation of volumes, without the use of intermediate surface geometry representations. Hence we state: `Volume Rendering is the Direct Visualisation of any three dimensional Volume data set; without the use of an intermediate geometric representation for isosurfaces'; `Surface Rendering is the Visualisation of a surface, from a geometric approximation of an isosurface, within a Volume data set'; where an isosurface is a surface formed from a cross connection of data points, within a volume, of equal value or density. This paper is an overview of both Surface Rendering and Volume Rendering techniques. Surface Rendering mainly consists of contouring lines over data points and triangulations between contours. Volume rendering methods consist of ray casting techniques that allow the ray to be cast from the viewing plane into the object and the transparency, opacity and colour calculated for each cell; the rays are often cast until an opaque object is `hit' or the ray exits the volume

    Segment Visibility Counting Queries in Polygons

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    Let PP be a simple polygon with nn vertices, and let AA be a set of mm points or line segments inside PP. We develop data structures that can efficiently count the number of objects from AA that are visible to a query point or a query segment. Our main aim is to obtain fast, O(polylognmO(\mathop{\textrm{polylog}} nm), query times, while using as little space as possible. In case the query is a single point, a simple visibility-polygon-based solution achieves O(lognm)O(\log nm) query time using O(nm2)O(nm^2) space. In case AA also contains only points, we present a smaller, O(n+m2+εlogn)O(n + m^{2 + \varepsilon}\log n)-space, data structure based on a hierarchical decomposition of the polygon. Building on these results, we tackle the case where the query is a line segment and AA contains only points. The main complication here is that the segment may intersect multiple regions of the polygon decomposition, and that a point may see multiple such pieces. Despite these issues, we show how to achieve O(lognlognm)O(\log n\log nm) query time using only O(nm2+ε+n2)O(nm^{2 + \varepsilon} + n^2) space. Finally, we show that we can even handle the case where the objects in AA are segments with the same bounds.Comment: 27 pages, 13 figure

    A measure of non-convexity in the plane and the Minkowski sum

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    In this paper a measure of non-convexity for a simple polygonal region in the plane is introduced. It is proved that for "not far from convex" regions this measure does not decrease under the Minkowski sum operation, and guarantees that the Minkowski sum has no "holes".Comment: 5 figures; Discrete and Computational Geometry, 201

    Algorithms for Imprecise Trajectories

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    Planar Visibility: Testing and Counting

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    In this paper we consider query versions of visibility testing and visibility counting. Let SS be a set of nn disjoint line segments in R2\R^2 and let ss be an element of SS. Visibility testing is to preprocess SS so that we can quickly determine if ss is visible from a query point qq. Visibility counting involves preprocessing SS so that one can quickly estimate the number of segments in SS visible from a query point qq. We present several data structures for the two query problems. The structures build upon a result by O'Rourke and Suri (1984) who showed that the subset, VS(s)V_S(s), of R2\R^2 that is weakly visible from a segment ss can be represented as the union of a set, CS(s)C_S(s), of O(n2)O(n^2) triangles, even though the complexity of VS(s)V_S(s) can be Ω(n4)\Omega(n^4). We define a variant of their covering, give efficient output-sensitive algorithms for computing it, and prove additional properties needed to obtain approximation bounds. Some of our bounds rely on a new combinatorial result that relates the number of segments of SS visible from a point pp to the number of triangles in sSCS(s)\bigcup_{s\in S} C_S(s) that contain pp.Comment: 22 page
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