# What are perimeter and aspect ratio, in the context of 3D mesh approximate convex decomposition?

I'm trying to understand the logic behind Mamou and Ghorbel's algorithm in their paper A Simple And Efficient Approach For 3D Mesh Approximate Convex Decomposition.

I cannot understand what is the aspect ratio of a surface as defined by the author. It includes the perimeter of two vertices v and w. These two vertices represent two triangles in the original mesh as they come from the dual graph as the authors define it. So I'm guessing they are talking about the perimeter of the two triangles these vertices correspond to. What is the perimeter of two triangles? Do they mean the sum?

The algorithm proceeds like this:

1. Create a dual graph of the mesh, where each vertex corresponds to a triangle of the original mesh, and two vertices are connected by an edge iff their corresponding triangles share an edge.

2. Iteratively choose and collapse an edge, so that two vertices are merged into one. We keep track of the "ancestors" of each vertex, A(v) that have been merged into it in this way.

Here's where I get confused:

The decimation process described in the previous section is guided by a cost function describing the concavity and the aspect ratio [5] of the surface $S(v, w)$ resulting from the unification of the vertices $v$ and $w$ and their ancestors:

$$S(v, w) = A(v) ∪ A(w) ∪ {w, v}. (2)$$

As in [5], we define the aspect ratio $E_{shape}(v, w)$ of the surface S(v, w) as follows:

$$E_{shape}(v, w) = \frac{ρ2(S(v, w))}{4π × σ(S(v, w))}, (3)$$

where $ρ(S(v, w))$ and $σ(S(v, w))$ are respectively the perimeter and the area of $S(v, w)$.

• CS Stack Exchange is probably a better fit. Also, most people outside academia don't have access to papers in IEEE, ACM, etc, so including a publicly accessible link is helpful when possible. Jul 8, 2018 at 1:50

Since $S(v, w)$ is a subset of dual graph, formed from vertices v, w, and all of their ancestors from previous edge collapse operations, it corresponds to a contiguous subset of triangles in the original mesh.