The θ5-graph is a spanner

Given a set of points in the plane, we show that the θ-graph with 5 cones is a geometric spanner with spanning ratio at most 50+225≈9.960. This is the first constant upper bound on the spanning ratio of this graph. The upper bound uses a constructive argument that gives a (possibly self-intersecting) path between any two vertices, of length at most 50+225 times the Euclidean distance between the vertices. We also give a lower bound on the spanning ratio of 12(115−17)≈3.798.