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Update interpolation_math.md
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DanielVandH authored Apr 11, 2024
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Expand Up @@ -26,7 +26,7 @@ In particular, any point in $V_{\boldsymbol x_i}$ is closer to $\boldsymbol x_i$

# Natural Neighbours

See that the tiles of the tessellation in the figure above intersect along a line, called the _Voronoi facet_, that we denote by $\mathcal F_{ij} = \mathcal V_i \cap \mathcal V_j$. Whenever $\mathcal F_{ij} \neq \emptyset$, we say that $\boldsymbol x_i$ and $\boldsymbol x_j$ are _natural neighbours_ in $\boldsymbol X$. We denote the set of natural neighbours to a point $\boldsymbol x \in \boldsymbol X$ by $N(\boldsymbol x) \subseteq \boldsymbol X$, and we denote the corresponding indices by $N_i = \{j : \boldsymbol x_j \in N(\boldsymbol x_j)\}$.
See that the tiles of the tessellation in the figure above intersect along a line, called the _Voronoi facet_, that we denote by $\mathcal F_{ij} = \mathcal V_i \cap \mathcal V_j$. Whenever $\mathcal F_{ij} \neq \emptyset$, we say that $\boldsymbol x_i$ and $\boldsymbol x_j$ are _natural neighbours_ in $\boldsymbol X$. We denote the set of natural neighbours to a point $\boldsymbol x \in \boldsymbol X$ by $N(\boldsymbol x) \subseteq \boldsymbol X$, and we denote the corresponding indices by $N_i = \{j : \boldsymbol x_j \in N(\boldsymbol x_i)\}$.

# Natural Neighbour Coordinates

Expand Down Expand Up @@ -353,4 +353,4 @@ where $\lambda_i = 1-t$, $\lambda_j = t$, $\ell = \|x_i - \boldsymbol x_j\|$, an

This extrapolation will not always be perfect, but it is good enough until we implement more sophisticated methods. If you want to disable this approach, just use the `project = false` keyword argument when evaluating your interpolant.

Similarly, if you have points defining a boundary of some domain that isn't necessarily convex, the function `identify_exterior_points` may be useful to you, provided you have represented your boundary as defined [here in DelaunayTriangulation.jl](https://danielvandh.github.io/DelaunayTriangulation.jl/dev/boundary_handling/#Boundary-Specification). See the Switzerland example in the sidebar for more information.
Similarly, if you have points defining a boundary of some domain that isn't necessarily convex, the function `identify_exterior_points` may be useful to you, provided you have represented your boundary as defined [here in DelaunayTriangulation.jl](https://danielvandh.github.io/DelaunayTriangulation.jl/dev/boundary_handling/#Boundary-Specification). See the Switzerland example in the sidebar for more information.

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