Generalized self-approaching curves Original Research Article

We consider all planar oriented curves that have the following property depending on a fixed angle ϕ. For each point B on the curve, the rest of the curve lies inside a wedge of angle ϕ with apex in B. This property restrains the curveʹs meandering, and for ϕ⩽π/2 this means that a point running along the curve always gets closer to all points on the remaining part. For all ϕ<π, we provide an upper bound c(ϕ) for the length of such a curve, divided by the distance between its endpoints, and prove this bound to be tight. A main step is in proving that the curveʹs length cannot exceed the perimeter of its convex hull, divided by 1+cos ϕ.