Ai Arthur van Goethem

Topologically safe curved schematisation

Abstract Traditionally schematised maps make extensive use of curves. However, automated methods for schematisation are mostly restricted to straight lines. We present a generic framework for topology-preserving curved schematisation that allows a choice of quality measures and curve types. The framework fits a curve to every part of the input. It uses Voronoi diagrams to ensure that curves fitted to disjoint parts do not intersect. The framework then employs a dynamic program to find an optimal schematisation using the fitted curves. Our fully-automated approach does not need critical points or salient features. We illustrate our framework with Bézier curves and circular arcs.

Travel-Time Maps: Linear Cartograms with Fixed Vertex Locations

Linear cartograms visualize travel times between locations, usually by deforming the underlying map such that Euclidean distance corresponds to travel time. We introduce an alternative model, where the map and the locations remain fixed, but edges are drawn as sinusoid curves. Now the travel time over a road corresponds to the length of the curve. Of course the curves might intersect if not placed carefully. We study the corresponding algorithmic problem and show that suitable placements can be computed efficiently. However, the problem of placing as many curves as possible in an ideal, centered position is NP-hard. We introduce three heuristics to optimize the number of centered curves and show how to create animated visualizations.

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.

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.

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.

Exploring Curved Schematization of Territorial Outlines

Hand-drawn schematized maps traditionally make extensive use of curves. However, there are few automated approaches for curved schematization; most previous work focuses on straight lines. We present a new algorithm for area-preserving curved schematization of territorial outlines. Our algorithm converts a simple polygon into a schematic crossing-free representation using circular arcs. We use two basic operations to iteratively replace consecutive arcs until the desired complexity is reached. Our results are not restricted to arcs ending at input vertices. The method can be steered towards different degrees of “curviness”: we can encourage or discourage the use of arcs with a large central angle via a single parameter. Our method creates visually pleasing results even for very low output complexities. To evaluate the effectiveness of our design choices, we present a geometric evaluation of the resulting schematizations. Besides the geometric qualities of our algorithm, we also investigate the potential of curved schematization as a concept. We conducted an online user study investigating the effectiveness of curved schematizations compared to straight-line schematizations. While the visual complexity of curved shapes was judged higher than that of straight-line shapes, users generally preferred curved schematizations. We observed that curves significantly improved the ability of users to match schematized shapes of moderate complexity to their unschematized equivalents.

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 \(\chi \). Each cell s in the grid is assigned a subset of colors \(\chi _s \subseteq \chi \) and should be partitioned such that for each color \(c\in \chi _s\) 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 \(\chi = \{\text {red},\text {blue}\}\). 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.

Circles in the Water: Towards Island Group Labeling

Many algorithmic results are known for automated label placement on maps. However, algorithms to compute labels for groups of features, such as island groups, are largely missing. In this paper we address this issue by presenting new, efficient algorithms for island label placement in various settings. We consider straight-line and circular-arc labels that may or may not overlap a given set of islands. We concentrate on computing the line or circle that minimizes the maximum distance to the islands, measured by the closest distance. We experimentally test whether the generated labels are reasonable for various real-world island groups, and compare different options. The results are positive and validate our geometric formalizations.