# Folding Every Point on a Polygon Boundary to a Point
## 4 The Number of Arcs of the Safe Region  
In this section, we consider the complexity of the boundary of the safe region; in other words, we count the exact number of arcs of the safe region. From previous sections, we derive that a side of the safe region is a parabolic arc with point \(f\) as its focus. We also see that some edge of \(P\) may not contribute to the resulting safe region, i.e., a parabola associated with it does not touch the safe region. It is natural to ask for the number of arcs of the safe region. Assuming that the polygon \(P\) is fixed, the number of arcs of the safe region depends on the focus \(f\). We denote explicitly by \(R_{f}\) a safe region with point \(f\) as its focus. As in [4], this section analyzes the number of arcs of safe region \(R_{f}\), i.e., \(|A(R_{f})|\), where \(A(R_{f})\) is the set of arcs of \(R_{f}\). Figure 7(a) shows two safe regions \(R_{f_{2}}\) with query point \(f_{2}\) and \(R_{f_{3}}\) with query point \(f_{3}\). [PERSON] et al. [4] consider the same problem for the case where each side of \(P\) is folded onto a line. They show that the straight skeleton of \(P\) plays an important role in determining the number of sides of the resulting region. This is true for our case as well. As an example, Figure 8b shows an inscribed circle \(C\) which can be determined using the straight skeleton and two safe regions shown previously. We remark that \(f_{2}\
ot\in C\) but \(f_{3}\in C\). We start by defining useful notations related to straight skeletons and event circles. A _straight skeleton_[16] of a polygon \(P\), denoted by \(S(P)\), is a subset of \(P\) such that for each point \(u\in S(P)\), there exist at least two points on \(\delta P\) with the same distance to \(u\). More intuitively, we may see the skeleton as a Voronoi diagram of line segments where each site is an edge of the polygon. The straight skeleton \(S(P)\) partitions \(P\) into regions, referred to as _faces_. Thus, under the Voronoi interpretation, each face is bounded by exactly one polygon edge as other edges of \(S(P)\). We refer to a face that is bounded by polygon edge \(e_{i}\) as face \(F_{i}\). We also note that a face is also a convex polygon. See Figure 9 for an illustration. The skeleton may be viewed as a tree, where each non-leaf node ensures at least three equidistant points on \(\delta P\). A non-leaf node of \(S(P)\) is referred to as an _event point_. A circle centered at an event point and tangential to the nearest edge of the polygon is called an _event circle_ of \(S(P)\). Let \(\mathcal{C}\) be the set of all event circles of \(S(P)\), and let \(\mathcal{C}_{e}\subseteq\mathcal{C}\) be a set of event circles tangential to edge \(e\). We also denote by \(\operatorname{int}(C)\) an interior of circle \(\mathcal{C}\), i.e., the set \(\{(x,y):(x-x_{0})^{2}+(y-y_{0})^{2}<r^{2}\}\) for a circle with center \((x_{0},y_{0})\) and radius \(r\). The goal of this section is to show that, under the fixed polygon \(P\), the structure of \(A(R_{f})\) is governed by event circles of straight skeleton \(S(P)\) of \(P\). We start by analyzing the case when the safe region intersects with the skeleton faces. The following lemma directly follows from the [PERSON] interpretation of the straight skeleton. Lemma 6: _For each event circle \(C\) with event point \(c\), \(c\) is adjacent to face \(F_{i}\) if and only if its corresponding polygon edge \(e_{i}\) is tangential to \(C\)._  
The following two lemmas provide basic properties for our main theorem in this section. Figure 8: Safe regions with 2 and 3 parabola arcs. (**a**) Safe regions with query points \(f_{2}\) and \(f_{3}\); (**b**) An event circle determines the numbers of arcs. Figure 9: The straight skeleton of polygon \(P\), with each face colored differently. Points \(c_{1},\dots,c_{6}\) are event points. **Lemma 7**. : _For a particular focus \(f\), \(\operatorname{int}(R_{f})\) intersects with skeleton face \(F_{i}\) adjacent to edge \(e_{i}\) iff the associated parabola \(p_{i}\) is part of the arcs of \(R_{f}\)._  
Proof.