Map schematization with circular arcs

We present an algorithm to compute schematic maps with circular arcs. Our algorithm iteratively replaces two consecutive arcs with a single arc to reduce the complexity of the output map and thus to increase its level of abstraction. Our main contribution is a method for replacing arcs that meet at high-degree vertices. This allows us to greatly reduce the output complexity, even for dense networks. We experimentally evaluate the effectiveness of our algorithm in three scenarios: territorial outlines, road networks, and metro maps. For the latter, we combine our approach with an algorithm to more evenly distribute stations. Our experiments show that our algorithm produces high-quality results for territorial outlines and metro maps. However, the lack of caricature (exaggeration of typical features) makes it less useful for road networks

Topologically safe curved schematization

Traditionally schematized maps make extensive use of curves. However, automated methods for schematization are mostly restricted to straight lines. We present a generic framework for topology-preserving curved schematization that allows a choice of quality measures and curve types. Our fully-automated approach does not need critical points or salient features. We illustrate our framework with Bézier curves and circular arcs

Optimal straight-line labels for island groups

Maps are used to solve a wide variety of tasks, ranging from navigation to analysis. Often, the quality of a map is directly related to the quality of its labelling. Consequently, a lot of research has focussed on the automatization of the labelling process. Surprisingly the (automated) labelling of island groups has received little attention so far. This is at least partially caused by the lack of cartographic principles. 31 Though extensive guidelines for map labelling exist, information on the labelling of groups of islands is surprisingly sparse. We define a formal framework for island labelling. The framework spawns a large series of unexplored computational geometry problems, which are interesting for the CG-community. In this paper we start by looking at a non-overlapping, straight label. We describe two algorithms for a straight-line label that is, or is not, allowed overlap with islands. Furthermore, we discus several extensions to these algorithms solving closely related problems

Accentuating focus maps via partial schematization

We present an algorithm for schematized focus maps. Focus maps integrate a high detailed, enlarged focus region continuously in a given base map. Recent methods integrate both with such low distortion that the focus region becomes hard to identify. We combine focus maps with partial schematization to display distortion of the context and to emphasize the focus region. Schematization visually conveys geographical accuracy, while not increasing map complexity. We extend the focus-map algorithm to incorporate geometric proximity relationships and show how to combine focus maps with schematization in order to cater to different use cases

Grouping time-varying data for interactive exploration

We present algorithms and data structures that support the interactive analysis of the grouping structure of one-, two-, or higher-dimensional time-varying data while varying all defining parameters. Grouping structures characterise important patterns in the temporal evaluation of sets of time-varying data. We follow Buchin et al. [JoCG 2015] who define groups using three parameters: group-size, group-duration, and inter-entity distance. We give upper and lower bounds on the number of maximal groups over all parameter values, and show how to compute them efficiently. Furthermore, we describe data structures that can report changes in the set of maximal groups in an output-sensitive manner. Our results hold in R^d for fixed d

The Painter's Problem: covering a grid with colored connected polygons

Motivated by a new way of visualizing hypergraphs, we study the following problem. Consider a rectangular grid and a set of colors χ. Each cell s in the grid is assigned a subset of colors χs⊆χ and should be partitioned such that for each color c∈χs at least one piece in the cell is identified with c. Cells assigned the empty color set remain white. We focus on the case where χ={red,blue}. Is it possible to partition each cell in the grid such that the unions of the resulting red and blue pieces form two connected polygons? We analyze the combinatorial properties and derive a necessary and sufficient condition for such a painting. We show that if a painting exists, there exists a painting with bounded complexity per cell. This painting has at most five colored pieces per cell if the grid contains white cells, and at most two colored pieces per cell if it does not