We consider a problem in computational origami. Given a piece of paper as a convex polygon $P$ and a point $f$ located within, fold every point on a boundary of $P$ to $f$ and compute a region that is safe from folding, i.e., the region with no creases. This problem is an extended version of a problem by Akitaya, Ballinger, Demaine, Hull, and Schmidt~[CCCG'21] that only folds corners of the polygon. To find the region, we prove structural properties of intersections of parabola-bounded regions and use them to devise a linear-time algorithm. We also prove a structural result regarding the complexity of the safe region as a variable of the location of point $f$, i.e., the number of arcs of the safe region can be determined using the straight skeleton of the polygon $P$.

翻译：我们考虑一个计算折纸问题。给定一张凸多边形纸片 $P$ 及其内部一点 $f$，将 $P$ 边界上的每一点折叠至 $f$，并计算免受折叠影响的区域，即无折痕的区域。该问题是 Akitaya、Ballinger、Demaine、Hull 和 Schmidt [CCCG'21] 研究的仅折叠多边形角点问题的扩展版本。为确定该区域，我们证明了抛物线有界区域交集的结构性质，并据此设计了一个线性时间算法。此外，我们得到了关于安全区域复杂度随点 $f$ 位置变化的结构性结果，即安全区域的弧段数量可由多边形 $P$ 的直线骨架确定。