# Folding Every Point on a Polygon Boundary to a Point
## 4 The Number of Arcs of the Safe Region
: \((\Rightarrow)\) Assume that \(\operatorname{int}(R_{f})\) intersects \(F_{i}\). Since \(F_{i}\) is bounded by a polygon edge and \(R_{f}\) is contained in \(P\), we know that there are parts of the boundary of \(R_{f}\) that intersect \(F_{i}\). Consider any point \(u\) on the boundary of \(R_{f}\) inside \(F_{i}\). Clearly, \(u\) must be on an arc of some parabola, i.e., we have that:  
\[\min_{j}|\overline{uw_{j}}|=|\overline{uf}|,\]  
where \(u_{j}\) is an orthogonal projection of \(u\) onto \(e_{j}\). Since all points in \(F_{i}\) are closer to \(e_{i}\) than other edges, the minimizer of the above term is \(u_{i}\); thus, \(u\) must also lie on \(p_{i}\), i.e., \(p_{i}\) is part of arcs of \(R_{f}\). \((\Leftarrow)\) We prove by contradiction. Assume that \(\operatorname{int}(R_{f})\) does not intersect \(F_{i}\), but \(p_{i}\) is part of the boundary of \(R_{f}\). Consider any point \(u\in p_{i}\) on the boundary. Since \(\operatorname{int}(R_{f})\) is disjoint from \(F_{i}\), \(u\) is strictly in some face \(F_{j}\). In this case, we have that:  
\[|\overline{uf}|=|\overline{uw_{i}}|>|\overline{uw_{j}}|,\]  
where \(u_{i}\) and \(u_{j}\) are orthogonal projections of \(u\) onto \(e_{i}\) and \(e_{j}\), implying that \(u\
ot\in H(p_{j})\), a contradiction. **Lemma 8**. : _For \(C\in\mathcal{C}\) with event point \(c\), a safe region \(R_{f}\) strictly contains an event point \(c\), i.e., \(c\in\operatorname{int}(R_{f})\), iff \(f\in\operatorname{int}(C)\)._  
Proof. : Let \(r\) be the radius of an event circle \(C\). Project \(c\) orthogonally to a line extension of every edge \(e_{i}\), named the projected point \(c_{i}\). \((\Leftarrow)\) Assume that \(f\in\operatorname{int}(C)\), i.e., \(|\overline{cf}|<r\). We show that \(r\in H(p_{i})\) for every parabola \(p_{i}\) associated with polygon edge \(e_{i}\). This is the case when \(r\) is strictly closer to \(f\) than every other edge \(e_{i}\). Consider each edge \(e_{i}\) tangent to \(C\), we have \(|\overline{cf}|<r=|\overline{cc_{i}}|\). For edge \(e_{i}\) not tangent to \(C\), we have that \(|\overline{cc_{i}}|>r\); thus, \(|\overline{cf}|<r<|\overline{cc_{i}}|\). Hence, \(c\) is in the safe region \(R_{f}\). \((\Rightarrow)\) Assume that \(f\
ot\in\operatorname{int}(C)\). In this case, we have \(|\overline{cf}|\geq r\). Since \(C\) is an event circle, there exists edge \(e_{i}\) tangent to \(C\). For that particular edge, we have \(|\overline{cc_{i}}|=r\). Thus, \(|\overline{cf}|\geq r=|\overline{cc_{i}}|\), and \(c\) is not strictly contained in the safe region \(R_{f}\). The following theorem gives the number of boundary arcs of \(R_{f}\) as a function of event circles containing \(f\). **Theorem 2**. : _If \(f\) is strictly inside some event circle, i.e., \(f\in\operatorname{int}(C)\) for some \(C\in\mathcal{C}\), then:_  
\[|A(R_{f})|=|\{e_{i}\in E(P):\text{there exists }C\in\mathcal{C}_{e_{i}} \text{ s.t. }f\in\operatorname{int}(C)\}|\]  
_Otherwise, \(|A(R_{f})|=2\)._  
Proof. : We first assume that \(f\in\operatorname{int}(C)\) for some event circle \(C\). Consider each event circle \(C\) with event point \(c\) such that \(f\in\operatorname{int}(C)\). From Lemma 8, we know that \(c\in\operatorname{int}(R_{f})\). This also means that \(\operatorname{int}(R_{f})\) intersects every face \(F_{i}\) adjacent to event point \(c\). Lemma 6 ensures that these faces \(F_{i}\)'s correspond with edges \(e_{i}\)'s tangent to \(C\), the set of edges \(e_{i}\) such that \(C\in\mathcal{C}_{e_{i}}\). Since Lemma 7 ensures that for each face \(F_{i}\) adjacent to edge \(e_{i}\) intersecting with \(R_{f}\), the parabola \(p_{i}\) appears as an arc of \(R_{f}\), we have that for each tangent edge \(e_{i}\) of \(C\), its parabola \(p_{i}\) appears as an arc in \(R_{f}\). The lemma, in this case, follows by taking the union of all boundary edges from every event circle \(C\in\mathcal{C}\) that \(f\) is strictly inside.