Sweeping Arrangements of Non-Piercing Regions in the Plane

Let Γ be a finite set of Jordan curves in the plane. For any curve γ ∈ Γ, we denote the bounded region enclosed by γ as γ̃. We say that Γ is a non-piercing family if for any two curves α, β ∈ Γ, α̃ \β̃ is a connected region. A non-piercing family of curves generalizes a family of 2-intersecting curves in which each pair of curves intersect in at most two points. Snoeyink and Hershberger (“Sweeping Arrangements of Curves”, SoCG’89) proved that if we are given a family Γ of 2-intersecting curves and a sweep curve γ ∈ Γ, then the arrangement can be swept by γ while always maintaining the 2-intersecting property of the curves. We generalize the result of Snoeyink and Hershberger to the setting of non-piercing curves. We show that given an arrangement of non-piercing curves Γ, and a sweep curve γ ∈ Γ, the arrangement can be swept by γ so that the arrangement remains non-piercing throughout the process. We also give a shorter and simpler proof of the result of Snoeyink and Hershberger, and give an eclectic set of applications.