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Update ghost_triangles.md
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DanielVandH authored Oct 8, 2024
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Expand Up @@ -8,7 +8,7 @@ In this section, we will describe how ghost triangles and ghost vertices are use

Ghost vertices are negative vertices that are associated with a part of a boundary. Mathematically speaking, they are typically treated as points out at infinity, and each edge on the boundary adjoins a ghost vertex. This way, all edges have two adjoining vertices and we can associate any point in space, including points outside of the triangulation, with a triangle. As an example, if `tri` is a triangulation and `get_adjacent(tri, u, v) == -1`, then this means that `(u, v)` is an edge on the boundary of the triangulation. This number `-1` is defined as `DelaunayTriangulation.𝒢` internally.

In the case of a single contiguosu boundary, the only possible ghost vertex is `-1`. When it comes to considering a boundary with mutliple sections or multiple boundaries, then we need to have multiple ghost vertices to refer to eachs ection separately. We accomplish this by simply subtracting 1 from the current ghost vertex for each new section. For example, if the boundary node vector is
In the case of a single contiguous boundary, the only possible ghost vertex is `-1`. When it comes to considering a boundary with mutliple sections or multiple boundaries, then we need to have multiple ghost vertices to refer to eachs ection separately. We accomplish this by simply subtracting 1 from the current ghost vertex for each new section. For example, if the boundary node vector is

```julia
boundary_nodes = [
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As you can see, the outer boundary has ghost edges (shown in blue) going out to infinity, oriented with the pole of inaccessibility of the entire domain (shown in red). The ghost edges along the circular boundary are finite and simply connect with the pole of inaccessibility of the circle (shown in magenta).

For more complex domains, in particular non-convex domains, the ghost edges start to overlap and they become less useful, which unfortunately slows down point location (see [`find_triangle`](@ref)'s docstring).
For more complex domains, in particular non-convex domains, the ghost edges start to overlap and they become less useful, which unfortunately slows down point location (see [`find_triangle`](@ref)'s docstring).

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