Steiner Trees on Sets of Three Points in λ-Geometry (λ =3m)

摘要

We show a method to determine a Steiner Minimum Tree (SMT) and a necessary and sufficient condition that an SMT is a full Steiner tree for three given points in λ-geometry (λ =3m, m is a positive integer). The λ-geometry allows only orientations with angles iπ/λ (i and λ (λ≥ 2) are integers), and fill up the gap between the rectilinear geometry (λ = 2) and the Euclidean geometry (λ = ∞). An SMT in λ-geometry (λ = 3m) has a similar property to that in the Euclidean geometry. The method to determine an SMT in λ-geometry is an extension of the well-known method in the Euclidean geometry. The Steiner point in λ-geometry is any point in the intersection area with a parallelogram and a Steiner locus. Then there are infinite candidate locations of the Steiner point. The Steiner point in the Euclidean geometry is that in λ-geometry (λ = 3m).