Generalized Self-Approaching Curves
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 curve’s 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 curve’s length cannot exceed the perimeter of its convex hull, divided by 1 + cos ϕ.
KeywordsSelf-approaching curves convex hull detour arc length
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