1,523 research outputs found
BSP-fields: An Exact Representation of Polygonal Objects by Differentiable Scalar Fields Based on Binary Space Partitioning
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
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
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
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
Let be a simple polygon with vertices, and let be a set of
points or line segments inside . We develop data structures that can
efficiently count the number of objects from that are visible to a query
point or a query segment. Our main aim is to obtain fast,
), query times, while using as little space as
possible. In case the query is a single point, a simple
visibility-polygon-based solution achieves query time using
space. In case also contains only points, we present a smaller,
-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 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 query time
using only space. Finally, we show that we can
even handle the case where the objects in are segments with the same
bounds.Comment: 27 pages, 13 figure
A measure of non-convexity in the plane and the Minkowski sum
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
Planar Visibility: Testing and Counting
In this paper we consider query versions of visibility testing and visibility
counting. Let be a set of disjoint line segments in and let
be an element of . Visibility testing is to preprocess so that we can
quickly determine if is visible from a query point . Visibility counting
involves preprocessing so that one can quickly estimate the number of
segments in visible from a query point .
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,
, of that is weakly visible from a segment can be
represented as the union of a set, , of triangles, even though
the complexity of can be . 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 visible from a point to the number of triangles in that contain .Comment: 22 page
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