Showing papers on "Polygon published in 1992"

Decimation of triangle meshes

Abstract: The polygon remains a popular graphics primitive for computer graphics application. Besides having a simple representation, computer rendering of polygons is widely supported by commercial graphics hardware and software. However, because the polygon is linear, often thousands or millions of primitives are required to capture the details of complex geometry. Models of this size are generally not practical since rendering speeds and memory requirements are proportional to the number of polygons. Consequently applications that generate large polygonal meshes often use domain-specific knowledge to reduce model size. There remain algorithms, however, where domainspecific reduction techniques are not generally available or appropriate. One algorithm that generates many polygons is marching cubes. Marching cubes is a brute force surface construction algorithm that extracts isodensity surfaces from volume data, producing from one to five triangles within voxels that contain the surface. Although originally developed for medical applications, marching cubes has found more frequent use in scientific visualization where the size of the volume data sets are much smaller than those found in medical applications. A large computational fluid dynamics volume could have a finite difference grid size of order 100 by 100 by 100, while a typical medical computed tomography or magnetic resonance scanner produces over 100 slices at a resolution of 256 by 256 or 512 by 512 pixels each. Industrial computed tomography, used for inspection and analysis, has even greater resolution, varying from 512 by 512 to 1024 by 1024 pixels. For these sampled data sets, isosurface extraction using marching cubes can produce from 500k to 2,000k triangles. Even today’s graphics workstations have trouble storing and rendering models of this size. Other sampling devices can produce large polygonal models: range cameras, digital elevation data, and satellite data. The sampling resolution of these devices is also improving, resulting in model sizes that rival those obtained from medical scanners. This paper describes an application independent algorithm that uses local operations on geometry and topology to reduce the number of triangles in a triangle mesh. Although our implementation is for the triangle mesh, it can be directly applied to the more general polygon mesh. After describing other work related to model creation from sampled data, we describe the triangle decimation process and its implementation. Results from two different geometric modeling applications illustrate the strengths of the algorithm.

Re-tiling polygonal surfaces

Abstract: This paper presents an automatic method of creating surface models at several levels of detail from an original polygonal description of a given object. Representing models at various levels of detail is important for achieving high frame rates in interactive graphics applications and also for speeding-up the off-line rendering of complex scenes. Unfortunately, generating these levels of detail is a time-consuming task usually left to a human modeler. This paper shows how a new set of vertices can be distributed over the surface of a model and connected to one another to create a re-tiling of a surface that is faithful to both the geometry and the topology of the original surface. The main contributions of this paper are: 1) a robust method of connecting together new vertices over a surface, 2) a way of using an estimate of surface curvature to distribute more new vertices at regions of higher curvature and 3) a method of smoothly interpolating between models that represent the same object at different levels of detail. The key notion in the re-tiling procedure is the creation of an intermediate model called the mutual tessellation of a surface that contains both the vertices from the original model and the new points that are to become vertices in the re-tiled surface. The new model is then created by removing each original vertex and locally re-triangulating the surface in a way that matches the local connectedness of the initial surface. This technique for surface retessellation has been successfully applied to iso-surface models derived from volume data, Connolly surface molecular models and a tessellation of a minimal surface of interest to mathematicians.

Recent Results in Art Galleries

Abstract: Two points in a polygon are called visible if the straight line segment between them lies entirely inside the polygon. The art gallery problem for a polygon P is to find a minimum set of points G in P such that every point of P is visible from some point of G. This problem has been shown to be NP-hard by Lee and Lin [71]. However, Chvatal showed that the number of points of G will never exceed [n/3] for a simple polygon of n sides [21]. This latter result is referred to as the art gallery theorem

Searching for a mobile intruder in a polygonal region

Abstract: The problem of searching for a mobile intruder in a simple polygon by a single mobile searcher is considered. This paper investigates the capabilities of searchers having different degrees of visibility by introducing the searcher having k flashlights whose visibility is limited to k rays emanating from his position, and the searcher having a point light source who can see in all directions simultaneously. This paper presents necessary and sufficient conditions for a polygon to be searchable by various searchers. The paper also introduces a class of polygons for which the searcher having two flashlights is as capable as the searcher having a point light source, and it gives a simple necessary and sufficient condition for such polygons to be searchable by the searcher having two flashlights. The complexity of generating a search schedule under some of these conditions is also discussed. Many of the results are proved using chord systems that represent the visibility relations among the vertices and edges o...

Recent results in art galleries (geometry)

Abstract: Two points in a polygon are called if the straight line between them lies entirely inside the polygon. The art gallery problem for a polygon P is to find a minimum set of points G in P such that every point in P is visible from some point of G. The author provides an introduction to art gallery theorems and surveys the recent results of the field. The emphasis is on the results rather than the techniques. Several new problems that have the same geometric flavor as art gallery problems are also examined. >

An Equal-Area Map Projection For Polyhedral Globes

Abstract: Numerous polyhedral shapes have been proposed as approximations for globes, and the projection most often used is the Gnomonic, with considerable scale and area distortion. Complicated conformal projections have been designed, but an equal-area projection has been used only once, for the icosahedron. The Lambert Azimuthal Equal-Area projection can be modified to provide an exactly fitting, perfectly equal-area projection for any polyhedral globe that has regular polygons, but is most satisfactory for the dodecahedron with 12 pentagons and for the truncated icosahedron with 20 hexagons and 12 pentagons. On the application to the truncated icosahedron, the angular deformation does not exceed 3.75°, and the scale variation is less than 3.3 percent. These advantages are at the expense of increased interruptions at the polygon edges when the polyhedral globe is unfolded. On a propose de nombreuses formes polyedriques comme approximation de globes et la projection gnomonique est la plus souvent utilisee, avec d...

Watchman routes under limited visibility

Abstract: We consider the motion planning problem for a point constrained to move along a smooth closed convex path of bounded curvature. The workspace of the moving point is bounded by a convex polygon with m vertices, containing an obstacle in a form of a simple polygon with n vertices. We present an O(m+n) time algorithm finding the path, going around the obstacle, whose curvature is the smallest possible.

The Pentagram Map

Abstract: We consider the pentagram map on the space of plane convex pentagons obtained by drawing a pentagon's diagonals, recovering unpublished results of Conway and proving new ones. We generalize this to a “pentagram map” on convex polygons of more than five sides, showing that iterated images of anyinitial polygon converge exponentially fast to a point. We conjecture that the asymptotic behavior of this convergence is the same as under a projective transformation. Finally, we show a connection between the pentagram map and a certain flow defined on parametrized curves.

Fast object-precision shadow generation for area light sources using BSP trees

Abstract: This paper introduces an efficient object-precision shadow generation algorithm for static polygonal environments directly illuminated by convex area light sources. Penumbra and umbra regions are calculated analytically and represented as a pair of BSP trees for each light source. As the trees are built, convex scene polygons are filtered down the trees, and split into fragments that are wholly lit, in penumbra, or in umbra. The illumination due to the light source is calculated at selected points within the wholly lit and penumbra regions by contour integration with the visible parts of the light source. We use a fast analytic algorithm to compute the fragments of the area light source visible from a point in penumbra. Rendering is done using hardwaresupported linear interpblated shading on a 3D graphics workstation. Because the scene itself is represented as a BSP tree, visible-surface determination may be performed by using either workstation-supported hardware (e.g., a z-buffer) or software BSP-tree traversal. We provide sample images created by our implementation, including timings and polygon counts. CR

Map-based localization using the panoramic horizon

Abstract: Presents an approach to solve the localization problem, in which an observer is given a topographic map of an area and dropped off at an unknown location. The solution to this problem requires establishing correspondences between viewer-centered observable features and their location on the map. The feature the authors select is the panoramic horizon curve, defined as the sky-ground boundary perceived by the observer as he performs a full 360/spl deg/ in place. In the authors' approach, they first precompute, offline, these horizon curves at a set of locations on a grid, from the topological map. These curves are approximated by polygons with different line fitting tolerances to gain robustness to noise in the authors' representation. These polygons are grouped into overlapping super segments, which are then encoded and stored in a table. The online computation consists of acquiring the panoramic view and extracting (with human help) the horizon curve. This curve is approximated by a polygon and the resulting super segments, used as indices in the data base, allow one to retrieve candidate locations. The best candidate is selected during a verification step which applies geometric constraints. This process uses local features and can therefore tolerate significant occlusion likely to occur in real environments. The authors illustrate the performance of the approach on results obtained from real data.

Origin of giant Martian polygons

Abstract: Extensive areas of the Martian northern plains in Utopia and Acidalia planitiae are characterized by 'polygonal terrane'. Polygonal terrane consists of material cut by complex troughs defining a pattern resembling mudcracks, columnar joints, or frost-wedge polygons on earth. However, the Martian polygons are orders of magnitude larger than these potential earth analogues, leading to severe mechanical difficulties for genetic models based on simple analogy arguments. Plate-bending and finite element models indicate that shrinkage of desiccating sediment or cooling volcanics accompanied by differential compaction over buried topography can account for the stresses responsible for polygon troughs as well as the large size of the polygons. Although trough widths and depths relate primarily to shrinkage, the large scale of the polygonl pattern relates to the spacing between topographic elevations on the surface buried beneath polygonal terrane material. Geological relationships favor a sedimentary origin for polygonal terrane material, but our model is not dependent on the specific genesis. Our analysis also suggests that the polygons must have formed at a geologically rapid rate.

Finding minimum areak-gons

Abstract: Given a setP ofn points in the plane and a numberk, we want to find a polygon[Figure not available: see fulltext.] with vertices inP of minimum area that satisfies one of the following properties: (1)[Figure not available: see fulltext.] is a convexk-gon, (2)[Figure not available: see fulltext.] is an empty convexk-gon, or (3)[Figure not available: see fulltext.] is the convex hull of exactlyk points ofP. We give algorithms for solving each of these three problems in timeO(kn3). The space complexity isO(n) fork=4 andO(kn2) fork?5. The algorithms are based on a dynamic programming approach. We generalize this approach to polygons with minimum perimeter, polygons with maximum perimeter or area, polygons containing the maximum or minimum number of points, polygons with minimum weight (for some weights added to vertices), etc., in similar time bounds.

A qualitative comparison study of data structures for large line segment databases

Abstract: A qualitative comparative study is performed of the performance of three popular spatial indexing methods - the R-tree, R+-tree, and the PMR quadtree-in the context of processing spatial queries in large line segment databases. The data is drawn from the TIGER/Line files used by the Bureau of the Census to deal with the road networks in the US. The goal is not to find the best data structure as this is not generally possible. Instead, their comparability is demonstrated and an indication is given as to when and why their performance differs. Tests are conducted with a number of large datasets and performance is tabulated in terms of the complexity of the disk activity in building them, their storage requirements, and the complexity of the disk activity for a number of tasks that include point and window queries, as well as finding the nearest line segment to a given point and an enclosing polygon.

Multi-layer atmospheric fading in real-time computer image generator

Abstract: The effect of multi-layer atmospheric scattering on the visibility F of each point P on a visible surface of each polygon in a display scene, in a computer image generation (CIG) system, is provided by determining the effective average reciprocal half-fading distance between viewpoint and the viewed point, knowing the altitudes at which each of the different scattering layers start and accounting for any transitional slopes therebetween. The total reduced visibility of that viewed point P is a function of the average reciprocal half-fading distance for that point and of the total range between the viewed point P and the viewpoint VP.

The two guards problem

Abstract: Given a simple polygon in the plane with two distinguished vertices, s and g, is it possible for two guards to simultaneously walk along the two boundary chains from s to g in such a way that they are always mutually visible? We decide this question in time O (n log n) and in linear space, where n is the number of edges of the polygon. Moreover, we compute a walk of minimum length within time O(n log n+k), where k is the size of the output, and we prove that this is optimal.

Polygon fragmentation method of distortion correction in computer image generating systems

Abstract: A method for distortion correction of computer-generated textured images maps vertices and texture coefficients from viewer space to projector space, so that environmental objects are pre-distorted upon the projection raster in order to appear in their proper form and perspective when the raster is projected onto a curved surface in viewer space, and viewed therefrom. Distortion correction is carried out by utilizing a piecewise-linear approximation for smoothly, continuously and closely approximating the required pre-curvature. The viewing space raster is subdivided into a number of triangles, within each of which a linear approximation is applied to the image mapping: the raster faces are first subdivided along the lines of a rectangular grid and the face fragments falling within each rectangle are then subdivided along the grid diagonal. This produces face fragments which are small enough so that a linear approximation provides an accurate transformation of each fragment. Distortion maps, typically computed off line and stored in a database memory, are utilized to project the edges and texture modulation gradients from view space back into projector space, so that the edges and texture patterns can now be matched at intersections of adjacent triangles, in a manner to be substantially devoid of any abrupt changes, and thus be properly pre-distorted.

Method of computing multi-conductor parasitic capacitances for VLSI circuits

Abstract: A method of computing parasitic capacitances between multiple electrical conductors within an electric circuit computes a division of the circuit's physical layout into a plurality of windows. The parasitic capacitances associated with the conductors of each window are computed, and the results for the various windows combined into a matrix of parasitic capacitances for the overall circuit. The windows are preferably overlapped, with the capacitance values for conductor pairs located in more than one window averaged. Complex polygons are fractured into simpler shapes by extending a ray from a vertex of the polygon to intersect an opposed segment, and defining the peripheries of the simpler elements as comprising the ray and respective different portions of the original polygon's periphery. Rays may be extended in a x,y pattern from multiple vertices of the polygon until a ray is located that extends through the polygon's interior, with the fracturing performed along that ray. Fracturing preferably continues until all of the elements are reduced to Manhattan-oriented rectangles or triangles. Where one element overlaps another element in another plane, fracturing is performed along a projection of the overlapping edge on the second element to reduce inaccuracies in the approximated charge density on the overlapped element.

Shortest paths help solve geometric optimization problems in planar regions

Abstract: The goal of this paper is to show that the concept of the shortest path inside a polygonal region contributes to the design of efficient algorithms for certain geometric optimization problems involving simple polygons: computing optimum separators, maximum area or perimeter-inscribed triangles, a minimum area circumscribed concave quadrilateral, or a maximum area contained triangle. The structure for the algorithms presented is as follows: (a) decompose the initial problem into a low-degree polynomial number of optimization problems; (b) solve each individual subproblem in constant time using standard methods of calculus, basic methods of numerical analysis, or linear programming. These same optimization techniques can be applied to splinegons (curved polygons). First a decomposition technique for curved polygons is developed; this technique is substituted for triangulation in creating equally efficient curved versions of the algorithms for the shortest-path tree, ray-shooting, and two-point shortest path...

The robot localization problem in two dimensions

Abstract: We consider the following problem: given a simple polygon P and a star-shaped polygon V, find a point (or the set of points) in P from which the portion of P that is visible is congruent to V. The problem arises in the localization of robots using a range-finder—P is a map of a known environment, V is the portion visible from the robot's position, and the robot must use this information to determine its position in the map. We give a scheme that preprocesses P so that any subsequent query V is answered in optimal time O(m + log n + A), where m and n are the number of vertices in V and P, and A is the number of points in P that are valid answers (the output size). Our technique allows us to trade off smoothly between the query time and the preprocessing time or space. We also devise a data structure for output-sensitive determination of the visibility polygon of a query point inside a polygon P. We then consider a variant of the localization problem in which there is a maximum distance to which the robot can “see”—this is motivated by practical considerations, and we outline a similar solution for this case. We also show that a single localization query V can be answered in time O(mn) with no preprocessing.

Locating a field of view in which selected IC conductors are unobscured

Abstract: Methods and apparatus are disclosed for positioning the field of view in a system for IC probing or repair, where the system comprises means for supporting an IC device having multiple physical layers and multiple internal nets, and controllable positioning means for positioning the field of view relative to the device. A data set is prepared which describes each physical layer of the device as a plurality of reduced polygons. Each of the reduced polygons is associated with a physical layer of the device and with at least one net of the device. For each of a plurality of selected nets, polygons associated with the net are dilated to define the periphery of a region encompassed by the field of view when a point within the field of view is traced around the periphery of a reduced polygon. A bit plane of the dilated polygons is mapped for each net, and regions of overlap the mapped bit planes are identified. The positioning means is controlled to position the field of view relative to the device at a location which corresponds to at least one overlap region.

Hyperbolic Interpolation

Abstract: It's always a red letter day when I can figure out a new use for homogeneous coordinates. This time I'll tell you about a way to use them to interpolate various parameters properly when tiling polygons. The exact definition of "properly" comes from one of those things that homogeneous coordinates are good at-perspective.

Method and apparatus for rendering polygons

Abstract: Shading values are generated for the vertices of a polygon clipped by a view volume without performing unnecessary lighting calculations for polygons lying entirely outside the view volume. Each polygon vertex is transformed and tested to determine whether it is within the view volume. If a tested vertex lies within the view volume, lighting calculations are performed for that vertex. If the polygon lies entirely outside the view volume, no lighting calculations are performed for any of the vertices of the polygon, and only a subset of the coordinate transformation calculation is performed. Complete transformation and lighting calculations are performed for vertices lying outside the view volume only if necessary for determining by interpolation the values for new vertices formed by clipping.

Polynomial-size nonobtuse triangulation of polygons

Abstract: We describe methods for triangulating polygonal regions of the plane so that no triangle has a large angle. Our main result is that a polygon with n sides can be triangulated with O(n2) nonobtuse triangles. We also show that any triangulation (without Steiner points) of a simple polygon has a refinement with O(n4) nonobtuse triangles. Finally we show that a triangulation whose dual is a path has a refinement with only O(n2) nonobtuse triangles.

Intersection detection and separators for simple polygons

Abstract: A simple algorithm is presented for detecting whether two preprocessed simple polygons intersect one another. Given a simple polygon, A, in O(n log n) time and O(n) space we preprocess A constructing an enveloping triangulation called a scaffold. To determine whether two preprocessed polygons A and B overlap another, we start with these two envelopes and successively strip away overlapping triangles of the scaffolds until we either detect an intersection between the objects or until we have succeeded in separating them spatially. The running time of the intersection query depends on the complexity of the minimum link polygonal curve separating the two objects. Given two preprocessed simple polygons A and B, placed at arbitrary locations in the plane we can determine whether these polygons intersect one another in O(m log2n is the total number of vertices and m is the complexity of a minimum link polygonal curve separating A from B. We generalize this to the problem of computing arbitrary Boolean functions of two preprocessed polygons.

Image forming device

Abstract: PURPOSE:To make the bit map development of an image at high speed by dividing a polygon shown by vector information by a scan line passing the vertexes into the minimum number of triangles and a quadrangle whose two sides are parallel to these. CONSTITUTION:The polygon given by the vector information is divided into the minimum number of triangles and a trapezoid whose two sides are parallel to the scan line by a straight line parallel to the scan line passing the vertexes. Each side is made approximate to an integral coordinate. However, when the side is parallel to the scan line, the approximation to the side is not executed. And, an edge list is prepared and registered on a RAM. Then, it is judged whether the coordinate registered on the edge list is the starting point to paint- out, or not, and in case of the starting point, the paint-out to the next coordinate is executed. The process is finished when the paint-out is completed. Thus, a process speed is improved without a decimal arithmetic or a half-adjust.

Three dimensional graphics processing with pre-sorting of surface portions

Abstract: To effect hidden surface removal, polygons making up a three dimensional object are initially ordered using a Topological Sort into, for example, three rendering order lists corresponding to orthogonal axis. When the object is to be represented from a given view direction, the view direction is compared to the axis and the rendering list corresponding to the closest match is selected. Polygons are then rendered in list order, backward facing polygons being omitted. Performing the ordering at the polygon level, in an initial step, saves computation and consequently enables the projection and rendering processes to be speeded up to provide real time interactive three dimensional graphics. The Topological Sorting process is particularly computationally effective.