Convexity-Increasing Morphs of Planar Graphs

We study the problem of convexifying drawings of planar graphs. Given any planar straight-line drawing of an internally 3-connected graph, we show how to morph the drawing to one with strictly convex faces while maintaining planarity at all times. Our morph is convexity-increasing, meaning that once an angle is convex, it remains convex. We give an efficient algorithm that constructs such a morph as a composition of a linear number of steps where each step either moves vertices along horizontal lines or moves vertices along vertical lines. Moreover, we show that a linear number of steps is worst-case optimal. To obtain our result, we use a well-known technique by Hong and Nagamochi for finding redrawings with convex faces while preserving y-coordinates. Using a variant of Tutte's graph drawing algorithm, we obtain a new proof of Hong and Nagamochi's result which comes with a better running time. This is of independent interest, as Hong and Nagamochi's technique serves as a building block in existing morphing algorithms.Comment: Preliminary version in Proc. WG 201

The Complexity of Drawing a Graph in a Polygonal Region

We prove that the following problem is complete for the existential theory of the reals: Given a planar graph and a polygonal region, with some vertices of the graph assigned to points on the boundary of the region, place the remaining vertices to create a planar straight-line drawing of the graph inside the region. This strengthens an NP-hardness result by Patrignani on extending partial planar graph drawings. Our result is one of the first showing that a problem of drawing planar graphs with straight-line edges is hard for the existential theory of the reals. The complexity of the problem is open in the case of a simply connected region. We also show that, even for integer input coordinates, it is possible that drawing a graph in a polygonal region requires some vertices to be placed at irrational coordinates. By contrast, the coordinates are known to be bounded in the special case of a convex region, or for drawing a path in any polygonal region.Comment: Appears in the Proceedings of the 26th International Symposium on Graph Drawing and Network Visualization (GD 2018

Simplifying the Representation of Radiance from Multiple Emitters

International audienceIn recent work radiance function properties and discontinuity meshing have been used to construct high quality interpolants representing radiance. Such approaches do not consider the combined effect of multiple sources and thus perform unnecessary discontinuity meshing calculations and often construct interpolants with too fine subdivision. In this research we present an extended structured sampling algorithm that treats scenes with shadows and multiple sources. We then introduce an algorithm which simplifies the mesh based on the interaction of multiple sources. For unoccluded regions an a posteriori simplification technique is used. For regions in shadow, we first compute the maximal umbral/penumbral and penumbral/light boundaries. This construction facilitates the determination of whether full discontinuity meshing is required or whether it can be avoided due to the illumination from another source. An estimate of the error caused by potential simplification is used for this decision. Thus full discontinuitymesh calculation is only incurred in regions where it is necessary resulting in a more compact representation of radiance

Hierarchical Techniques for Global Illumination Computations -- Recent Trends and Developments

Since the beginning of computer graphics, one of the primary goals has been to create convincingly realistic images of three-dimensional environments that would be impossible to distinguish from photographs of the real scene. The goal to create photo-realistic images has lead to the development of completely new software techniques for dealing with the inherent geometric and optical complexity of real world scenes. This paper gives an overview of advanced algorithms for photorealistic rendering and in particular discusses hierarchical techniques for global illumination computations