GeoInformatica 6:4, 345±361, 2002 # 2002 Kluwer Academic Publishers. Manufactured in The Netherlands. Modeling Costs of Turns in Route Planning STEPHAN WINTER Institute for Geoinformation, Technical University Vienna, Gusshausstrasse 27±29, 1040 Vienna, Austria E-mail: winter@geoinfo.tuwien.ac.at Received August 7, 2001; Revised April 2, 2002; Accepted July 17, 2002 Abstract In this paper a model that handles costs of turns in route planning is de®ned, investigated and discussed. Costs are traditionally attached to edges in a graph. For some important route planning problems other costs can be identi®ed, namely costs that appear when leaving one edge and entering the next. Examples are turn restrictions, the turning angle, or the simple necessity to turn. Such costs cannot be stored as attributes of nodes or edges in the graph, and they cannot be handled correctly by shortest path algorithms without modi®cations. Turn costs can be represented by a pseudo-dual graph in a way that shortest path algorithms run without modi®cations. Although the idea is not new, it has not found much interest in the literature. The pseudo-dual graph is de®ned here in a new way, it is systematically investigated, and some practical applications are shown. Concentrating strictly on topology, it turns out that the pseudo-dual graph is conceptually cleaner and more ef®cient in route planning than alternative, currently used ways to deal with turn costs. The discussed applications are from the ®eld of pedestrian navigation, which gave rise to this research. Keywords: shortest path algorithm, turn costs, turn restrictions, route planning, pedestrian navigation, location- based services 1. Introduction In this paper a model for costs related to the continuation of travel in a node, called turn costs, is investigated. Turn costs require some effort to be treated in representation and analysis of travel networks. When a traveler is reaching the end of an edge, she makes a selection among continuing edges before she proceeds. The selection can be made based on different criteria. For instance, she can select the edge that requires the least angle turn, the edge with the least costs in terms of waiting time, or the edge which continues the actual street. All these examples show turn costs describing binary relationships between (consecutive) edges of a graph. That means, such costs cannot be stored with the edges or nodes in the graph directly. The relationship can be stored in extra tables if the turn costs cannot be calculated on the ¯y from geometry. Current navigation data standards use exception tables to represent turn restrictions [14], [25]. Using such extra tables in route ®nding algorithms would decrease performance. Furthermore, the existence of relations like turn restrictions requires