Concave polygon

Abstract: A polygon hidden surface and hidden line removal algorithm is presented. The algorithm recursively subdivides the image into polygon shaped windows until the depth order within the window is found. Accuracy of the input data is preserved.The approach is based on a two-dimensional polygon clipper which is sufficiently general to clip a concave polygon with holes to the borders of a concave polygon with holes.A major advantage of the algorithm is that the polygon form of the output is the same as the polygon form of the input. This allows entering previously calculated images to the system for further processing. Shadow casting may then be performed by first producing a hidden surface removed view from the vantage point of the light source and then resubmitting these tagged polygons for hidden surface removal from the position of the observer. Planar surface detail also becomes easy to represent without increasing the complexity of the hidden surface problem. Translucency is also possible.Calculation times are primarily related to the visible complexity of the final image, but can range from a linear to an exponential relationship with the number of input polygons depending on the particular environment portrayed. To avoid excessive computation time, the implementation uses a screen area subdivision preprocessor to create several windows, each containing a specified number of polygons. The hidden surface algorithm is applied to each of these windows separately. This technique avoids the difficulties of subdividing by screen area down to the screen resolution level while maintaining the advantages of the polygon area sort method.

Abstract: We show that for any concave polygon that has no parallel sides and for any k, there is a k-fold covering of some point set by the translates of this polygon that cannot be decomposed into two coverings. Moreover, we give a complete classification of open polygons with this property. We also construct for any polytope (having dimension at least three) and for any k, a k-fold covering of the space by its translates that cannot be decomposed into two coverings.

Abstract: An object is captured by a set of fingers when there exists no trajectory to bring the object arbitrarily far from the fingers. Object concavity is a special geometric property that allows objects to be captured with only few fingers. In particular, certain concave objects may be captured by appropriately placing two fingers close to some pair of opposite concave sections. This paper addresses the problem of computing all configurations of the fingers that are farthest away from each other while still capable of capturing the object. We propose an O(n2lg n) algorithm for this task and present preliminary results showing efficiency of the algorithm

Abstract: In a method and apparatus for rendering a trimmed NURBS surface representing a mapping from U and V parametric coordinates to X, Y and Z geometric coordinates and having a trimming region bound by a trim polyline, the UV parametric surface is divided into contiguous U and V intervals intersecting to form UV rectangles, the trim polyline intersecting a subset of the UV rectangles to divide each of the UV rectangles of the subset into at least one polygon lying within the region and at least one polygon lying outside the trimming region. For each UV rectangle intersected by the trim polyline, the vertices of each polygon within the trimming region formed by the intersection of the trim polyline and the UV rectangle is determined, and vertex data for the vertices so determined is provided to a concave polygon processor 104 to render the polygon. For each V interval intersected by the trim polyline, polyline data stored in a global memory 102 is read by the graphics processor, and the polyline defined by the read data clipped to form a V-clipped polyline defined by V-clipped data which is stored in local memory associated with the processor. The graphics processor then processes the V interval using the V-clipped polyline data stored in local memory.