ANGLES OF PLANAR TRIANGULAR GRAPHS

We give a characterization of all the planar drawings of a triangular graph through a system of equations and inequalities relating its angl es; we also discuss minimality properties of the characterization. The characterization can be used: (1) to decide in linear time whether a given distribution of angles between the edges of a planar triangular graph can result in a planar drawing; (2) to reduce the problem of max imizing the minimum angle in a planar straight-line drawing of a plana r triangular graph to a nonlinear optimization problem purely on a spa ce of angles; (3) to give a characterization of the planar drawings of a triconnected graph through a system of equations and inequalities r elating its angles; (4) to give a characterization of Delaunay triangu lations through a system of equations and inequalities relating its an gles; (5) to give a characterization of all the planar drawings of a t riangular graph through a system of equations and inequalities relatin g the lengths of its edges; in turn, this result allows us to give a n ew characterization of the disc-packing representations of planar tria ngular graphs.