# Towards an Automated Design Evaluation Method for Wire Arc Additive Manufacturing
## 3 Methods
### Definition of Geometry Indicators Based on the 3D-MAT  
First of all, the radius derived from the 3D-MAT can be used to measure corner radii since two cases can occur: when moving on the medial surface into the corner, either the radius will go to zero for sharp edges or the corner's curvature will coincide with the inscribed sphere.  
For thickness measurement, however, the length of the secant \(s=\|\mathbf{p}-\mathbf{b}\|\) between the two tangent points \(\mathbf{p}\) and \(\mathbf{b}\) of an inscribed sphere is more precise than the circle's diameter, as [PERSON] et al. point out [18], since the diameter would overestimate the thickness in relation to the angle enclosed by the surface normals (see Figure 3). Therefore, both indicators are used.  
Additionally, the angle between the normal vector \(\mathbf{n}_{b}\) to base point \(\mathbf{b}\) and the secant length can be used to calculate the angle between two related faces according to Equation (1) (see Figure 3) [18].  
\[\alpha=2\arccos\left(\frac{(\mathbf{p}-\mathbf{b})\cdot\mathbf{n}_{b}}{s}\right) \tag{1}\]  
Figure 3: Dimensions of the 3D-MAT: \(\mathbf{p}\), \(\mathbf{b}\): tangent points; \(\mathbf{n}_{p}\), \(\mathbf{n}_{b}\): normal vectors; \(\alpha\): wall angle; \(s\): secant; \(g_{s}\): 2D gradient.  
Figure 2: Medial axis and medial surface as dual representation of a geometry delivering its local symmetry axis or plane. The medial axis and the medial surface are defined by the set of all midpoints of maximally inscribed circles and spheres, respectively.  
Finally, the gradient of the radius function \(g_{r}(\mathbf{x})=\|\text{grad}\,r(\mathbf{x})\|\) can be defined as a fourth geometry indicator in correspondence to Heuvers' ratios of neighboring circles. To compute the geometry indicators, the surface is discretized with a triangular mesh. Thus, every surface triangle of a watertight geometry has three adjacent triangles listed in the adjacency matrix \(a(i,j)\): for a triangle with index \(i\), the \(i\)-th row lists the three adjacent triangles in the columns \(j=1\ldots 3\). Consequently, the discrete gradient is obtained by Equation (2).  
\[g_{r}(i)=\sqrt{\sum_{j=1}^{3}\biggl{(}\frac{r(i)-r(a(i,j))}{\|\mathbf{x}(i)- \mathbf{x}(a(i,j))\|}\biggr{)}^{2}} \tag{2}\]  
While a similar metric could be derived of the tangent of the half wall angle \(\tan\frac{\alpha}{2}\), which in fact is the gradient of the radius function in the direction of the medial axis, the formulation based on the radius function has the advantage of considering radius changes in all directions.  
In summary, four indicators can be derived from the 3D-MAT to describe the geometry, which are the radius \(r\), the secant length \(s\), an angle \(\alpha\), and the gradient of the radius function \(g_{r}\). While Heuvers' circles are usually inscribed in the geometry, the 3D-MAT can be calculated for the inside and the outside (inverse direction of surface normals) of the geometry, thus providing information about convex as well as concave parts. To differentiate the indicators derived from the inside 3D-MAT and the outside 3D-MAT, the indices \((\_)i\) and \((\_)o\), respectively, are used in the following.