## 3 Helix Polygon
### Properties of Helix  
Constructing the Helix polygon sheds light on the fact that for \(i>2\), the angles made by vertices \(v_{2i}\) are concave and for \(i>0\), vertices \(v_{2i+1}\) and \(v_{2}\) are convex (Figure 2). In fact, the pocket of a polygon \(P\) is defined as \(CH(P)-P\) where \(CH(P)\) is the convex hull of the vertices of \(P\). The pocket of a Helix polygon with \(n\) vertices is a Helix polygon with \(n-1\) vertices (Figure 4). The pocket of a polygon \(H_{n}\) is denoted by \(P(H_{n})\). For \(i\), \(1\leq i\leq n-1\), the vertices of \(P(H_{n})\) are called \(v_{i}^{\prime}\), located on \(v_{i+1}\). For \(n>4\), the angle \(\widehat{v_{i}}\) in \(P(H_{n})\) is obtained as follows:  
\[\widehat{v_{i}}=\left\{ \begin{array}{ll}\widehat{v_{1}v_{2}v_{3}}-\widehat{ v_{1}v_{2}v_{4}}&i=1\\ \widehat{v_{1}v_{3}v_{2}}-\widehat{v_{1}v_{3}v_{5}}&i=2\\ 2\pi-\text{interior angle of $v_{i+1}$in $H_{n}$}&i\geq 3.\end{array} \right. \tag{2}\]  
Figure 4: (**a**) Helix \(H_{12}\). (**b**) Pocket of \(H_{12}\) which is a Helix with \(11\) vertices. Figure 3: Final step of Algorithm 1 for generating \(H_{12}\). We will show that polygon \(H_{n}\) for any given natural number \(n[3]\), can be defined by \(n-2\) natural vertices guards which are located on \(v_{1},v_{2},\ldots,v_{n-3},v_{n}\). The Boolean formula, \(F_{n}\), is as below:  
\[F_{n}=\sum_{i=1}^{\lceil n/2\rceil-2}A_{i}\cdot g_{2i}+A_{\lfloor n/2\rfloor-1 }\cdot g_{n} \tag{3}\]  
where \(A_{1}=g_{1}\), \(A_{i+1}=A_{i}.g_{2i+1}\) for all \(1\leq i\leq\lfloor n/2\rfloor-2\) and \(g_{i}\) is the natural vertex guard located on the vertex \(v_{i}\) of \(H_{n}\). To clarify this point, \(F_{8}\) can be written as follows:  
\[F_{8}=\sum_{i=1}^{2}A_{i}.g_{2i}+A_{5}.g_{8} \tag{4}\]  
where \(A_{1}=g_{1}\), \(A_{2}=g_{1}.g_{3}\) and \(A_{3}=g_{1}.g_{3}.g_{5}\). Thus, we have  
\[F_{8}=g_{1}.g_{2}+g_{1}.g_{3}.g_{4}+g_{1}.g_{3}.g_{5}.g_{8}\]  
Generally, we expand \(F_{n}\) as follows:  
\[F_{n}=\left\{ \begin{array}{ll}\sum_{i=1}^{k-1}(\prod_{j=0}^{i-1}g_{2j+1}).g_{ 2i}+(\prod_{j=0}^{k-2}g_{2j+1}).g_{n}&n=2k+1,k\in\mathbb{N}\\ \sum_{i=1}^{k-1}(\prod_{j=0}^{i-1}g_{2j+1}).g_{2i}+(\prod_{j=0}^{k-1}g_{2j+1}). g_{n}&n=2k+2,k\in\mathbb{N}.\end{array} \right. \tag{5}\]  
According to Lemma 1, \(F_{n}\) defined by Equation (5) describes \(H_{n}\). Lemma 1: _Helix polygon \(H_{n}\) can be defined by \(n-2\) natural vertex guards \(g_{i}\) (\(1\leq i\leq n;i\
eq n-1,n-2\)) located on \(v_{1},v_{2},\ldots v_{n-3},v_{n}\) and the correspondent Boolean formula is Formula 5._  
Proof: We will prove the lemma by induction. When \(n=3\), \(k=1\) and \(F_{3}=g_{1}.g_{3}\) clearly define triangle \(H_{3}\). For \(n=4\), \(k=1\) and \(F_{4}=g_{1}.g_{4}\) define \(H_{4}\) and it implies that Lemma 1 holds for \(n=3,4\). Now, for \(n\geq 5\), without loss of generality, assume that \(n=2k+2,k\in\mathbb{N}\). According to Property 2, \(P(H_{n})\) is a Helix polygon with \(2k+1\) vertices. By induction hypothesis, \(P(H_{n})\) can be defined as follows:  
\[F^{\prime}=\sum_{i=1}^{k-1}(\prod_{j=0}^{i-1}g_{2j+1}^{\prime}).g_{2i}^{\prime }+(\prod_{j=0}^{k-2}g_{2j+1}^{\prime}).g_{2k+1}^{\prime} \tag{6}\]  
where \(g_{i}^{\prime}\) is a natural guard on the vertex \(v_{i}^{\prime}\) of \(P(H_{n})\). According to correspondence between the vertices of \(H_{n}\) and \(P(H_{n})\), we have  
\[g_{i}^{\prime}=\left\{ \begin{array}{ll}g_{i+1}^{c}&i\geq 3\\ G_{i+1}.g_{i+1}^{c}&i=1,2.\end{array} \right. \tag{7}\]  
in which \(g^{c}\) is the complement of guard \(g\) and \(G_{2}\) and \(G_{3}\) are the guards located on \(v_{2}\) and \(v_{3}\) with the angles \(\widehat{v_{1}v_{2}v_{3}}\) and \(\widehat{v_{1}v_{3}v_{2}}\), respectively.