Digital geometry is the study of geometrical properties of subsets of digital images. If the digitization is sufficiently fine-grained, such properties can be regarded as approximations to the corresponding properties of the "real" sets that gave rise, by digitization, to the digital sets; but it is also important to define how the properties can be computed for the digital sets themselves. Questions of particular interest include how images and image subsets are digitized; how geometric properties are defined for digitized sets; the computational complexity of computing them--in particular, whether they can be computed using simple (e.g., local) operations; characterizing image operations that preserve them; and characterizing digital objects that could be the digitizations of real objects that have given geometric properties. Concepts that have been extensively studied include topological properties (connected components, boundaries); curves and surfaces; straightness, curvature, convexity, and elongatedness; distance, extent, length, area, surface area, volume, and moments; shape description, similarity, symmetry, and relative position; shape simplification and skeletonization.

Digital geometry

We prove existence and uniqueness results for patterns of circles with prescribed intersection angles in constant curvature surfaces. Our method is based on two new functionals--one for the Euclidean and one for the hyperbolic case. We show how Colin de Verdi`ere's, Br"agger's and Rivin's functionals can be derived from ours.

Variational principles for circle patterns and Koebe's theorem

This article provides a review of the current state of science regarding cartographic interaction, a complement to the traditional focus within cartography on cartographic rep- resentation. Cartographic interaction is defined as the dialog between a human and map, mediated through a computing device, and is essential to the research into interactive car- tography, geovisualization, and geovisual analytics. The review is structured around six fundamental questions facing a science of cartographic interaction: (1) what is cartographic interaction (e.g., digital versus analog interactions, interaction versus interfaces, stages of interaction, interactive maps versus mapping systems versus map mash-ups); (2) why pro- vide cartographic interaction (e.g., visual thinking, geographic insight, the stages of sci- ence, the cartographic problematic); (3) when should cartographic interaction be provided (e.g., static versus interactive maps, interface complexity, the productivity paradox,flexibil- ity versus constraint, work versus enabling interactions); (4) who should be provided with cartographic interaction (e.g., user-centered design, user ability, expertise, and motivation, adaptive cartography and geocollaboration); (5) where should cartographic interaction be provided (e.g., input capabilities, bandwidth and processing power, display capabilities, mobile mapping and location-based services); and (6) how should cartographic interac- tion be provided (e.g., interaction primitives, objective-based versus operator-based versus operand-based taxonomies, interface styles, interface design)? The article concludes with a summary of research questions facing cartographic interaction and offers an outlook for cartography as a field of study moving forward.

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Interactive maps: What we know and what we need to know

The development of children's ability to use spatial representations.

The author shows how his conceptual model relates to the cognitive processes that go on during vehicle navigation and driving. Conceptually, the system may include human minds as well as various vehicle navigation aids. It must be able to perform five fundamental functions: geographic database, location of vehicle and destination, route planning, instruction generation, and vehicle control. It also must provide links or communication methods whereby information and commands may flow among these components. These functions may be grouped and assigned to various people and devices involved in the system. Adoption of this model allows researchers to focus more clearly on particular components of the system and on the associated research questions. >

G. D. Theisen

Papers

What is a map