9 research outputs found
Detecting and decomposing self-overlapping curves
AbstractA curve is self-overlapping if it can be divided by nontrivial line segments into simple curves. We show how to test whether a given curve is self-overlapping, and also how to construct sets of line segments that demonstrate that the curve is self-overlapping. We also describe several interesting topological properties of self-overlapping curves
An isoperimetric inequality in the universal cover of the punctured plane
AbstractWe find the largest Ï” (approximately 1.71579) for which any simple closed path α in the universal cover R2âZ2Ë of R2âZ2, equipped with the natural lifted metric from the Euclidean two-dimensional plane, satisfies L(α)â„Ï”A(α), where L(α) is the length of α and A(α) is the area enclosed by α. This generalizes a result of Schnell and Segura Gomis, and provides an alternative proof for the same isoperimetric inequality in R2âZ2
ConTesse: Accurate Occluding Contours for Subdivision Surfaces
This paper proposes a method for computing the visible occluding contours of
subdivision surfaces. The paper first introduces new theory for contour
visibility of smooth surfaces. Necessary and sufficient conditions are
introduced for when a sampled occluding contour is valid, that is, when it may
be assigned consistent visibility. Previous methods do not guarantee these
conditions, which helps explain why smooth contour visibility has been such a
challenging problem in the past. The paper then proposes an algorithm that,
given a subdivision surface, finds sampled contours satisfying these
conditions, and then generates a new triangle mesh matching the given occluding
contours. The contours of the output triangle mesh may then be rendered with
standard non-photorealistic rendering algorithms, using the mesh for visibility
computation. The method can be applied to any triangle mesh, by treating it as
the base mesh of a subdivision surface.Comment: Accepted to ACM Transactions on Graphics (TOG
ConTesse: Accurate Occluding Contours for Subdivision Surfaces
International audienceThis paper proposes a method for computing the visible occluding contours of subdivision surfaces. The paper first introduces new theory for contour visibility of smooth surfaces. Necessary and sufficient conditions are introduced for when a sampled occluding contour is valid, that is, when it may be assigned consistent visibility. Previous methods do not guarantee these conditions, which helps explain why smooth contour visibility has been such a challenging problem in the past. The paper then proposes an algorithm that, given a subdivision surface, finds sampled contours satisfying these conditions, and then generates a new triangle mesh matching the given occluding contours. The contours of the output triangle mesh may then be rendered with standard non-photorealistic rendering algorithms, using the mesh for visibility computation. The method can be applied to any triangle mesh, by treating it as the base mesh of a subdivision surface
Drawing graphs for cartographic applications
Graph Drawing is a relatively young area that combines elements of graph theory, algorithms, (computational) geometry and (computational) topology. Research in this field concentrates on developing algorithms for drawing graphs while satisfying certain aesthetic criteria. These criteria are often expressed in properties like edge complexity, number of edge crossings, angular resolutions, shapes of faces or graph symmetries and in general aim at creating a drawing of a graph that conveys the information to the reader in the best possible way. Graph drawing has applications in a wide variety of areas which include cartography, VLSI design and information visualization. In this thesis we consider several graph drawing problems. The first problem we address is rectilinear cartogram construction. A cartogram, also known as value-by-area map, is a technique used by cartographers to visualize statistical data over a set of geographical regions like countries, states or counties. The regions of a cartogram are deformed such that the area of a region corresponds to a particular geographic variable. The shapes of the regions depend on the type of cartogram. We consider rectilinear cartograms of constant complexity, that is cartograms where each region is a rectilinear polygon with a constant number of vertices. Whether a cartogram is good is determined by how closely the cartogram resembles the original map and how precisely the area of its regions describe the associated values. The cartographic error is defined for each region as jAcÂĄAsj=As, where Ac is the area of the region in the cartogram and As is the specified area of that region, given by the geographic variable to be shown. In this thesis we consider the construction of rectilinear cartograms that have correct adjacencies of the regions and zero cartographic error. We show that any plane triangulated graph admits a rectilinear cartogram where every region has at most 40 vertices which can be constructed in O(nlogn) time. We also present experimental results that show that in practice the algorithm works significantly better than suggested by the complexity bounds. In our experiments on real-world data we were always able to construct a cartogram where the average number of vertices per region does not exceed five. Since a rectangle has four vertices, this means that most of the regions of our rectilinear car tograms are in fact rectangles. Moreover, the maximum number vertices of each region in these cartograms never exceeded ten. The second problem we address in this thesis concerns cased drawings of graphs. The vertices of a drawing are commonly marked with a disk, but differentiating between vertices and edge crossings in a dense graph can still be difficult. Edge casing is a wellknown methodâused, for example, in electrical drawings, when depicting knots, and, more generally, in information visualizationâto alleviate this problem and to improve the readability of a drawing. A cased drawing orders the edges of each crossing and interrupts the lower edge in an appropriate neighborhood of the crossing. One can also envision that every edge is encased in a strip of the background color and that the casing of the upper edge covers the lower edge at the crossing. If there are no application-specific restrictions that dictate the order of the edges at each crossing, then we can in principle choose freely how to arrange them. However, certain orders will lead to a more readable drawing than others. In this thesis we formulate aesthetic criteria for a cased drawing as optimization problems and solve these problems. For most of the problems we present either a polynomial time algorithm or demonstrate that the problem is NP-hard. Finally we consider a combinatorial question in computational topology concerning three types of objects: closed curves in the plane, surfaces immersed in the plane, and surfaces embedded in space. In particular, we study casings of closed curves in the plane to decide whether these curves can be embedded as the boundaries of certain special surfaces. We show that it is NP-complete to determine whether an immersed disk is the projection of a surface embedded in space, or whether a curve is the boundary of an immersed surface in the plane that is not constrained to be a disk. However, when a casing is supplied with a self-intersecting curve, describing which component of the curve lies above and which below at each crossing, we can determine in time linear in the number of crossings whether the cased curve forms the projected boundary of a surface in space. As a related result, we show that an immersed surface with a single boundary curve that crosses itself n times has at most 2n=2 combinatorially distinct spatial embeddings and we discuss the existence of fixed-parameter tractable algorithms for related problems