## 4 Basic Ideas on Scalar Subdivision Schemes  
A subdivision scheme is an iterative method that generates curves and surfaces based on successive refinements of a polygon or a mesh. The rules that determine said refinements can be formulated by linear, non-linear, or geometrical operators [29; 30; 31; 32; 33]. The case of linear rules is related to refinable functions in Wavelets Theory [34]. In this setting, the vertices of the polygon or mesh are the coefficients in a particular basis of the so-called _subdivision curve_ or _subdivision surface_. In what follows we focus on the case of the subdivision curve. Linear uniform subdivision schemes are based on the notion of _refinable function_, i.e., a function \(\varphi(t)\) satisfying a relation of the form  
\[\varphi(t)=\sum_{j\in\mathbb{Z}}a_{j}\varphi(2t-j),\quad t\in\mathbb{R},a_{j} \in\mathbb{R} \tag{5}\]and such that, given an initial set of _control points_\(\mathbf{P}^{0}=\{P_{j}^{0}\in\mathbb{R}^{m},j=0,\ldots,n-1\}\) and the periodisation \(P_{j}^{0}=P_{j+n^{\prime}}^{0}\,\forall j\in\mathbb{Z}\), the closed curve  
\[c(t)=\sum_{j\in\mathbb{Z}}P_{j}^{0}\varphi(t-j),\qquad t\in\mathbb{R} \tag{6}\]  
can also be written as  
\[c(t)=\sum_{j\in\mathbb{Z}}P_{j}^{k}\varphi\left(2^{k}t-j\right),\qquad t\in \mathbb{R}\]  
with  
\[P_{i}^{k+1}=\sum_{j\in\mathbb{Z}}a_{i-2j}P_{j}^{k},\qquad k\in\mathbb{N},\ i=0, \ldots,2^{k+1}n-1, \tag{7}\]  
where for the set \(\mathbf{P}^{k}\) of new points the periodisation modulo \(2^{k}n\) holds, i.e., \(P_{j}^{k}=P_{j+2^{k}n^{\prime}}^{k}\), \(j\in\mathbb{Z}\). The relation in (7) is known as the _subdivision rule_ and defines a _subdivision scheme_, while the coefficients \(\mathbf{a}=\{a_{j},j\in\mathbb{Z}\}\) in (5) form the so-called _subdivision mask_. Then, a subdivision scheme is an iterative method where a curve is generated by consecutive refinements of an initial polygon (see Figure 2). Regarding the convergence of this scheme, it has been proven that the sequence of polygons with vertices in \(\mathbf{P}^{k}\) converges uniformly to a smooth curve \(c(t)\), although in practice a few iterations are sufficient to produce a polygon that appears smooth to the human eye. The sampling of the curve at integer parameters is derived from (5) and reads as  
\[c(s)=\sum_{j\in\mathbb{Z}}P_{j}^{0}\varphi(s-j),\qquad t\in\mathbb{Z},s\in \mathbb{Z}. \tag{8}\]  
The set of values \(\boldsymbol{\beta}^{0}=\left\{\beta_{j}^{0}=\varphi(j),\ j\in\mathbb{Z}\right\}\) is called _first limit stencil_ and it can be obtained from a linear system of equations stemming from (5) (see Ref. [35]) or by performing the spectral analysis of the _local subdivision matrix_ (see Refs. [33; 36; 37]). In a similar way, the _second limit stencil_ and higher order stencils \(\boldsymbol{\beta}^{k-1}=\left\{\beta_{j}^{k-1}=\varphi^{(k-1)}(j),\ j\in \mathbb{Z}\right\}\), \(k\in\mathbb{N}\), \(k\geq 1\) can be determined. Hence, by (6) the \(k\)-th derivative of the curve at dyadic parameters \(t=\mathbb{Z}/2^{m}\) is  
\[c^{(k)}\big{(}\tfrac{s}{2^{m}}\big{)}=\sum_{j\in\mathbb{Z}}\beta_{s-j}^{k}P_{ j}^{m}. \tag{9}\]  
The mask and the stencils have compact support and therefore in (5) they define a function that has compact support. Even though we use indexes ranging over \(\mathbb{Z}\), we are dealing with finite masks and stencils and correspondingly we have only a finite number of non-zero elements. Figure 2: Iterations of a non-interpolatory subdivision scheme. (**a**) Control points \(\mathbf{P}^{0}\) (blue balls). (**b**) First refinement of the polygon with vertices \(\mathbf{P}^{1}\). (**c**) Second refinement with vertices \(\mathbf{P}^{2}\). (**d**) Fourth refinement and interpolated points \(\mathbf{V}^{0}\) in the limit (red squares). Given the control points \(\mathbf{P}^{0}\), let us denote with \(\mathbf{V}^{0}=\{V_{j}^{0}\in\mathbb{R}^{m},j=0,\ldots,n-1\}\) the points obtained by evaluating the subdivision curve with (8), i.e., \(V_{j}^{0}=c(j),j=0,\ldots,n-1\) (see Figure 2d).