A Geometric Approach to Three-dimensional Graph Drawing

This paper introduces a novel approach to the drawing of threedimensional graphs: the geometric approach. Graphs are decomposed into building elements that can easily be drawn as a geometric shape. Those elements are linked together in order to form the final layout. The proposed algorithm, Geo3D, has been implemented in the Java language and has been used to produce VRML models of the input graphs. Introduction The most common approach to graph drawing is to set the position of each vertex of the graph so that the final layout would satisfy some predefined aesthetics [1]. For example, force-directed algorithms use the spring model in order to minimize edge crossing and to distribute nodes evenly; other algorithms build a layout minimizing a cost function that depends on edge distance, node distribution etc. [2] The first algorithms for the drawing of graphs in three dimensions applied the above approaches, successfully used in the two-dimensional domain, to the extended space [4] [5] [6] [7]. But different results could be achieved if we consider the drawing of three-dimensional graphs in the context for which it is needed. 2D graph drawing is usually aimed at producing a good hard copy, where the quality of the result is more important than the time taken to obtain it. 3D graphs are generally to be displayed on screen, in an interactive environment where the user can navigate through the graph in order to inspect the interesting parts, where the feeling of immersion makes the user aware of the relative positions of all the elements. In such applications the usual drawing parameters are of less importance and new layout strategies become possible, also taking into consideration the increased degrees of freedom. The approach followed in this paper builds the layout as a structure made of geometric primitives. The graph is split into polygons, trees and chains that are put one next to the other. The algorithm is driven by some simple heuristics that lead to a better result. It does not directly take into consideration any aesthetic quality, it does not iterate in order to achieve the best result, it does not compute any global or local function on which to direct the layout process. Nonetheless, the final drawing has some interesting properties. The goal of achieving an aesthetically pleasing layout is still satisfied, also because the added dimension makes edge crossing and node superposition less likely. Another property, mostly ignored in other algorithms, contributes to the effectiveness of the result: shape identity, i.e. the layout persistence through different executions of the layout process. The algorithm is also fast because all the mathematical operations performed are limited to simple 3D transformations and the layout is built incrementally without modifying the positions already defined.