# Folding Every Point on a Polygon Boundary to a Point
## 2 Preliminaries
### Two Parabolas  
We define _semi-confocal parabolas_ as a family of parabolas that share the same focus, but their directrixes do not need to have the same directional angle. It follows that semi-confocal parabolas are projected parabolas from multiple cutting planes that cut the same cone. We state two important facts on intersections of two parabolas which, under the conic interpretation, are straightforward.  
Lemma 1: _Two semi-confocal parabolas do not intersect iff their directional angles are the same._  
Proof: Two parabolas with the same directional angle are produced from two parallel cutting planes in the conic view, which never intersect.  
Figure 4: Interpretation of a parabola as a projection of a conic section. (**a**) Mapping a tilted parabola to the \(xy\)-plane; (**b**) Projection of a parabola on the \(xy\)-plane; (**c**) Two cutting planes; (**d**) Angle bisector and two intersections.  
**Lemma 2**: _If two semi-confocal parabolas intersect, then they intersect at two points which also lie on an angle bisector of their directrizes._  
Take conic section interpretation. Their curves on the surface of the cone must also lie on their cutting planes, in which the intersection of the planes is a straight line. This line piece goes through the cone exactly two times. See Figure 4c,d.  
It follows from Lemma 2 that two intersecting semi-confocal parabolas \(p_{i},p_{j}\) produce a safe region \(R=H(p_{i})\cap H(p_{j})\) which is a bounded convex region. Moreover, since we only deal with parabolas whose directrizes are from boundary edges of a convex polygon, the non-intersecting case never occurs.  
Given two parabolas \(p_{i}\) and \(p_{j}\), we can compute their intersections using analytical techniques in \(O(1)\) time as follows. We reduce the problem of finding intersection of two parabolas to the problem of finding intersection of a parabola and a line. Without loss of generality, we consider \(p_{i}\) in the upright form. From Lemma 2, we find an angle bisector \(b\) such that it divides the inner angle between edges \(e_{i}\) and \(e_{j}\). Then, we find the resulting intersections of \(p_{i}\) and \(b\) using quadratic equations.  
We remark particularly on the structure of the safe region \(R\).  
Consider each parabola \(p_{i}\) in the upright form. We can partition this parabola using the two intersection points into three arcs: the _left arc_, the _central arc_, and the _right arc_, where the left arc corresponds to the half-parabola unbounded to \(-\infty\) and the right arc corresponds to the half-parabola unbounded to \(+\infty\), and the central arc lies between the two intersection points (see Figure 5). Under this notation, we note that the left arc of \(p_{i}\) intersects \(p_{j}\) only once at the intersection point where it also intersects the right arc of \(p_{j}\), and vice versa.  
In our analysis, where there are parabolas \(p_{1},p_{2},\ldots,p_{n}\), we refer to the two intersection points between parabolas \(p_{i}\) and \(p_{j}\) as \(q_{ij}\) and \(q_{ji}\). To distinguish between these two points, imagine if one traverses counterclockwise on the boundary of \(H(p_{i})\cap H(p_{j})\), then one would see an arc of \(p_{i}\), the intersecting point \(q_{ij}\), the arc of \(p_{j}\), and then the intersection point \(q_{ji}\). The counterclockwise definitions of these points are crucial to our proof of Lemma 4. See Figure 4d.