# Fast Equipartition of Complex 2D Shapes with Minimal Boundaries +
## 4 2D-SEP Instances and Properties
### Intrinsic Boundaries' Length  
Hereafter, we study an important 2D-SEP property, the sequence of the total intrinsic boundaries' length of the optimal 2D-SEP solution \(R^{*}(S,N)\) of shape \(S\) into \(N\) regions as \(N\) increases. According to the problem definition, it holds that  
\[\lim_{N\rightarrow\infty}L(R^{*}(S,N))=\infty \tag{6}\]  
Figure 6: Suboptimal solutions of the 2D-SEP for unit area (**a**,**b**) square and (**c**,**d**) circle under \(N\in\{3,5\}\). The intrinsic boundary length (\(L\)) is reported in the caption of each shape.  
Figure 7: Optimal solutions of the 2D-SEP for unit area (**a**–**d**) square and (**e**–**h**) circle with \(N\in\{2,3,4,5\}\). The intrinsic boundary length (\(L\)) is reported in the caption of each shape.  
since the area of its region, as \(N\) tends to \(\infty\), it tends to zero, so the total intrinsic boundaries' length tends to \(\infty\), covering the whole space of the shape \(S\). When \(N=2\), the \(L(R^{*}(S,2))\) depends on shape \(S\). The following inequality provides the upper bound of the \(L(R^{*}(S,2))\)  
\[0\leq L(R^{*}(S,2))\leq\frac{Per(S)}{2} \tag{7}\]  
where \(Per(S)\) denotes the perimeter of shape \(S\). The proof is trivial for convex shapes by getting the suboptimal solution that the intrinsic boundaries are line segments \(L(R^{-}/(S,2))\leq L(R^{*}(S,2))\). This solution always exists for convex shapes [7], and it satisfies the inequality (7).  
In basic shapes of Section 4.2, for low values of \(N\), it holds that  
\[L(R^{*}(S,N))<L(R^{*}(S,N+1)). \tag{8}\]  
The fact that \(L(R^{*}(S,N))\) may be a monotonically increasing sequence for some shapes is also supported by the inequalities (6) and (7), and it is true for several simple shapes and at least for low values of \(N\).  
However, there exist some shapes in which even for low values of \(N\) (e.g., \(N=2\)), the inequality (8) is not true. Figure 8 depicts a shape that consists of three successive circles of radius, where  
\[L(R^{*}(S,2))\approx\frac{Per(S)}{3\cdot\pi}-4\cdot\varepsilon>L(R^{*}(S,3))=2 \cdot\varepsilon.\]  
The red and gray lines show the intrinsic boundary for the optimal segmentation into two and three regions, respectively. Since in this constructed example \(\varepsilon\) can be set almost zero, it holds that  
\[L(R^{*}(S,2))>>L(R^{*}(S,3)).\]  
Similarly, with the case of three circles, in the corresponding shape with successive circles \(N+1\), it holds that  
\[L(R^{*}(S,N))>>L(R^{*}(S,N+1))=N\cdot\varepsilon.\]  
In Section 6.5, we study the sequence \(L(R(S,N))\) of the solutions derived by the proposed methods. Since the sequence may be monotonically increasing for low values of \(N\) and simple shapes, it can measure the complexity of the dataset. Furthermore, if we compare the derived sequence \(L(R(S,N))\) for different methods, conclusions can be drawn about the robustness of the methods, as \(L(R(S,N))\) is expected to increase.  
Figure 8: A special case of shape where it holds that \(L(R^{*}(S,2))>L(R^{*}(S,3))=2\cdot\varepsilon\). The red and green lines show the intrinsic boundary for the optimal segmentation into two and three regions, respectively.