Basic metric $\rho _b$

$\{\mathcal{X}^{(*)};\rho_b\}$ defines the metric space, where $\mathcal{X}^{(*)}$ is the set of all patches with any number of triangles. The binary operation $\rho _b$ takes two patches and returns a natural number in the range $[0,\infty)$ . $\rho_b:\mathcal{X}^{(*)}\times\mathcal{X}^{(*)}\rightarrow \mathbb{N}^0$ ; defined as:

$$\rho_b(x,y)=\vert\lfloor x\rceil-\lfloor y\rceil \vert\\$$ (1)

$\lfloor\;\rceil$ is a function defined in $\mathcal{X}$ over $\mathbb{N}^0$ and for some $x\in\mathcal{X}^{(k)}$ , $\lfloor x\rceil$ , returns the size of patch $x$ , and is exactly the sum of all cells of all triangles in the patch, the amount is given by expression 2, where $k$ is the number of triangles in each patch:

$$\lfloor x\rceil=\sum_k (\frac{n_k(n_k+1)}{2}+2n_k+1)\\$$ (2)

Figure 2: Basic metric on patches