We introduce a new approach for drawing diagrams. Our approach is to use a technique we call confluent drawing for visualizing non-planar graphs in a planar way. This approach allows us to draw, in a crossing-free manner, graphs—such as software interaction diagrams—that would normally have many crossings. The main idea of this approach is quite simple: we allow groups of edges to be merged together and drawn as “tracks” (similar to train tracks). Producing such confluent diagrams automatically from a graph with many crossings is quite challenging, however, so we offer two heuristic algorithms to test if a non-planar graph can be drawn efficiently in a confluent way. In addition, we identify several large classes of graphs that can be completely categorized as being either confluently drawable or confluently non-drawable.

/pdf/confluent-drawings-visualizing-non-planar-diagrams-in-a-17of57sbcs.pdf

Confluent drawings: visualizing non-planar diagrams in a planar way.

Recently a Delaunay refinement algorithm has been proposed that can mesh domains as general as piecewise smooth complexes [7]. This class includes polyhedra, smooth and piecewise smooth surfaces, volumes enclosed by them, and above all non-manifolds. In contrast to previous approaches, the algorithm does not impose any restriction on the input angles. Although this algorithm has a provable guarantee about topology, certain steps are too expensive to make it practical.

/pdf/a-practical-delaunay-meshing-algorithm-for-alarge-class-of-37zqlymg81.pdf

A Practical Delaunay Meshing Algorithm for aLarge Class of Domains

A Lombardi drawing of a graph is one in which vertices are represented as points, edges are represented as circular arcs between their endpoints, and every vertex has perfect angular resolution (equal angles between consecutive edges, as measured by the tangents to the circular arcs at the vertex). We describe two algorithms that create "Lombardi-style" drawings (which we also call near-Lombardi drawings), in which all edges are still circular arcs, but some vertices may not have perfect angular resolution. Both of these algorithms take a force-directed, spring-embedding approach, with one using forces at edge tangents to produce curved edges and the other using dummy vertices on edges for this purpose. As we show, these approaches produce near-Lombardi drawings, with one being slightly better at achieving near-perfect angular resolution and the other being slightly better at balancing edge placements.

/pdf/force-directed-lombardi-style-graph-drawing-36fuw2q9kh.pdf

Force-Directed lombardi-style graph drawing

We present a method for modifying a force-directed graph drawing algorithm into an algorithm for drawing graphs with curved lines. Our method is based on embedding control points as dummy vertices so that edges can be drawn as splines. Our experiments show that our method yields aesthetically pleasing curvilinear drawing with improved angular resolution. Applying our method to the GEM algorithm on the test suite of the “Rome Graphs” resulted in an average improvement of 46% in angular resolution and of almost 6% in edge crossings.

/pdf/curvilinear-graph-drawing-using-the-force-directed-method-1jkd70627o.pdf

Curvilinear graph drawing using the force-directed method

Planar Orthogonal and Polyline Drawing Algorithms.

In this paper we address the problem of drawing planar graphs with circular arcs while maintaining good angular resolution and small drawing area. We present a lower bound on the area of drawings in which edges are drawn using exactly one circular arc. We also give an algorithm for drawing n-vertex planar graphs such that the edges are sequences of two continuous circular arcs. The algorithm runs in O(n) time and embeds the graph on the O(n) × O(n) grid, while maintaining ?(1/d(v)) angular resolution, where d(v) is the degree of vertex v. Since in this case we use circular arcs of infinite radius, this is also the first algorithm that simultaneously achieves good angular resolution, small area, and at most one bend per edge using straight-line segments. Finally, we show how to create drawings in which edges are smooth C1-continuous curves, represented by a sequence of at most three circular arcs.

/pdf/drawing-planar-graphs-with-circular-arcs-dmjifjcnlu.pdf

Drawing planar graphs with circular arcs

In this paper we address the problem of drawing planar graphs with circular arcs while maintaining good angular resolution and small drawing area. We present a lower bound on the area of drawings in which edges are drawn using exactly one circular arc. We also give an algorithm for drawing n-vertex planar graphs such that the edges are sequences of two continuous circular arcs. The algorithm runs in O(n) time and embeds the graph on the O(n) × O(n) grid, while maintaining θ(1/d(v)) angular resolution, where d(v) is the degree of vertex v. Since in this case we use circular arcs of infinite radius, this is also the first algorithm to simultaneously achieve good angular resolution, small area and at most one bend per edge using straight-line segments. Finally, we show how to create drawings in which edges are smooth C1-continuous curves, represented by a sequence of at most three circular arcs.

/pdf/drawing-planar-graphs-with-circular-arcs-2q6st6dyzf.pdf

C. C. Cheng

Papers

Drawing planar graphs with circular arcs

Drawing Planar Graphs with Circular Arcs