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Let $(X, Vertex(X))$ be a Polyhedral surface (defined like in Polthier) , $x_0 \in X$ a vertex. Let $B_\epsilon(x_0)$ the euclidean ball centred at $x_0$ with radius $\epsilon$. \epsilon$,$\epsilon > max length(e), e \in edge(X) $. Define$\mathcal{B}_ \epsilon(x_0)$the intersection of$B_\epsilon(x_0)$with$X$to be the euclidean neighborhood of$x_0$on$X$. Define the$boundary$as the set of all vertices$ x \in \mathcal{B}_ \epsilon(x_0)$satisfying the following condition (1) : the function$ (d ( x,x_0) - \epsilon ) $changes sign, that is, there exist$x_+ \in Adjacent(x)$such that$(d ( x_+,x_0) - \epsilon ) > 0$, ( i.e. that lays outside$\mathcal{B}_\epsilon(x0)$) and at least two$x_- \in Adjacent(x)$such that$(d ( x_-,x_0) - \epsilon ) < 0$. (i.e. that lays inside$\mathcal{B}_\epsilon(x_0)$).$d$being the euclidean distance,$x \in Vertex(X)$,$Adjacent(x)$vertices adjacent to x ( connected to x by an edge in X ) Is there any algorithm to optimize the search for such x on$X$? I tried the$NN$algorithm with Fixed radius to search for$\mathcal{B}_\epsilon(x_0)$. Is there any algorithm to optimize the search for the boundary of$\mathcal{B}_\epsilon(x_0)$? I tried to define an alogrithm that starts from$x_{max}$(a point of maximum of$d(-,x_0)$in$\mathcal{B}_\epsilon(x_0) : d(y,x_0) \leq d(x_{max},x_0) , y \neq x, y \in \mathcal{B}_\epsilon(x_0)$) and define boundary points by adjacency with check condition given in (1). This shoud give a closed path$x_{max} \leadsto x_{max}$that minimizes the distance from the boundary of$B_\epsilon(x0)$. Also, may I use someway the graph structure on$X$? 3 added 38 characters in body Let$(X, Vertex(X))$be a Polyhedral surface (defined like in Polthier) ,$x_0 \in X$a vertex. Let$B_\epsilon(x_0)$the euclidean ball centred at$x_0$with radius$\epsilon$. Define$\mathcal{B}_ \epsilon(x_0)$the intersection of$B_\epsilon(x_0)$with$X$to be the euclidean neighborhood of$x_0$on$X$. Define the$boundary$as the set of all vertices$ x \in X$\mathcal{B}_ \epsilon(x_0)$ satisfying the following condition (1) : the function $(d ( x,x_0) - \epsilon )$ changes sign,

that is, there exist

$x_+ \in Adjacent(x)$ such that $(d ( x_+,x_0) - \epsilon ) > 0$, ( i.e. that lays outside $\mathcal{B}_\epsilon(x0)$) and

at least two $x_- \in Adjacent(x)$ such that $(d ( x_-,x_0) - \epsilon ) < 0$. (i.e. that lays inside $\mathcal{B}_\epsilon(x_0)$).

$d$ being the euclidean distance, $x \in Vertex(X)$ , $Adjacent(x)$ vertices adjacent to x ( connected to x by an edge in X )

Is there any algorithm to optimize the search for such x on $X$?

I tried the $NN$ algorithm with Fixed radius to search for $\mathcal{B}_\epsilon(x_0)$.

Is there any algorithm to optimize the search for the boundary of $\mathcal{B}_\epsilon(x_0)$?

I tried to define an alogrithm that starts from $x_{max}$ (a point of maximum of $d(-,x_0)$ in $\mathcal{B}_\epsilon(x_0) : d(y,x_0) \leq d(x_{max},x_0) , y \neq x, y \in \mathcal{B}_\epsilon(x_0)$ ) and define boundary points by adjacency with check condition given in (1). This shoud give a closed path $x_{max} \leadsto x_{max}$ that minimizes the distance from the boundary of $B_\epsilon(x0)$.

Also, may I use someway the graph structure on $X$?

2 added 172 characters in body

Let (X, Vertex(X)) $(X, Vertex(X))$ be a Polyhedral surface (defined like in Polthier) , x0 vertex $x_0 \in XX$ a vertex. Let B_\epsilon(x0) $B_\epsilon(x_0)$ the euclidean ball centred at x0 $x_0$ with radius \epsilon. $\epsilon$. Define \mathcal{B}_ $\mathcal{B}_ \epsilon(x0) epsilon(x_0)$ the intersection of B_\epsilon(x0) $B_\epsilon(x_0)$ with (X,Vertex(X)) $X$ to be the euclidean neighborhood of x0 $x_0$ on X. $X$. Define the boundary

{ $boundary$ as the set of all vertices $x \in vertex X$ satisfying the following condition (X) 1) : the function $(d ( x,x0x,x_0) - \epsilon ) change$ changes sign} (1) ,

that is, there exist

  an x+ $x_+ \in Adjacent(x) Adjacent(x)$ such that (d $(d ( x+,x0x_+,x_0) - \epsilon ) > 0> 0$, ( i.e. that lays outside \mathcal{B}_\epsilon(x0) $\mathcal{B}_\epsilon(x0)$) andan x- $x_- \in Adjacent(x) Adjacent(x)$ such that (d $(d ( x-,x0x_-,x_0) - \epsilon ) < 00$. (i.e. that lays inside \mathcal{B}_\epsilon(x0) d $\mathcal{B}_\epsilon(x_0)$).$d$ being the euclidean distance, $x \in vertex(X) Vertex(X)$ , Adjacent(x) vertexes $Adjacent(x)$ vertices adjacent to x ( connected to x by an edge \in X )Is there any algorithm to optimize the search for such x on X? $X$? I tried the NN $NN$ algorithm with Fixed radius to search for \mathcal{B}_\epsilon(x0). $\mathcal{B}_\epsilon(x_0)$. Is there any algorithm to optimize the search for the boundary of \mathcal{B}_\epsilon(x0)?$\mathcal{B}_\epsilon(x_0)$?I tried to define an alogrithm that starts from x_max $x_{max}$ (the \ a point of maximum of d $d(-,x_0)$ in \mathcal{B}_\epsilon(x0) $\mathcal{B}_\epsilon(x_0) : d(y,x0d(y,x_0) \leq d(x_max,x0d(x_{max},x_0) , y \neq x, y \in \mathcal{B}_\epsilon(x0) mathcal{B}_\epsilon(x_0)$ ) and define boundary This shoud give a closed path x_max -> x_max $x_{max} \leadsto x_{max}$ that minimize minimizes the distance from the boundary of B_\epsilon(x0). Can $B_\epsilon(x0)$. Also, may I use someway the graph structure on X? $X$?

 
 
 
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