# A Framework for Drawing Planar Graphs with Curves and Polylines

Goodrich, Michael T. and Wagner, Cristopher G. (1998) A Framework for Drawing Planar Graphs with Curves and Polylines. In: Graph Drawing 6th International Symposium, GD’ 98, August 13-15, 1998 , pp. 153-166(Official URL: http://dx.doi.org/10.1007/3-540-37623-2_12).

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## Abstract

We describe a unified framework of aesthetic criteria and complexity measures for drawing planar graphs with polylines and curves. This framework includes several visual properties of such drawings, including aspect ratio, vertex resolution, edge length, edge separation, and edge curvature, as well as complexity measures such as vertex and edge representational complexity and the area of the drawing. In addition to this general framework, we present algorithms that operate within this framework. Specifically, we describe an algorithm for drawing any n-vertex planar graph in an O(n) \times O(n) grid using polylines that have at most two bends per edge and asymptotically-optimal worst-case angular resolution.More significantly, we show how to adapt this algorithm to draw any n-vertex planar graph using cubic Bézier curves, with all vertices and control points placed within an O(n) \times O(n) integer grid so that the curved edges achieve a curvilinear analogue of good angular resolution. All of our algorithms run in O(n) time.

Item Type: Conference Paper 10.1007/3-540-37623-2_12 M Methods > M.600 PlanarP Styles > P.600 Poly-line > P.600.700 OrthogonalP Styles > P.540 PlanarM Methods > M.999 OthersP Styles > P.300 CurvedG Algorithms and Complexity > G.070 Area / Edge LengthG Algorithms and Complexity > G.999 OthersG Algorithms and Complexity > G.210 BendsD Aesthetics > D.001 General http://gdea.informatik.uni-koeln.de/id/eprint/263

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