**3. Improvement of the Auxiliary Surface Layout**

As mentioned in [19,20], MAS becomes less accurate when the auxiliary surfaces are conformal to boundaries encompassing wedges. In particular, satisfaction of the boundary condition at midpoints (MPs) (see Figure 1) close to the tips is no longer sufficiently good. To overcome this complication, the auxiliary surface, defined by radius ρ*aux* and using expressions analogous to (1-3), may be deformed so that ASs not only approach the tips closely, but become denser in the tip neighborhood as well. ASs may approach CPs following several patterns. In this work, two basic patterns were tested. Let *M* be the number of ASs to be moved. Let ρ*<sup>m</sup>* be the polar radius of the *m*th AS (*m* = 1, 2, ... , *M*), and let *g* be the maximum polar radius difference between the *m*th AS and the *m*th CP. Finally, let *s* be the proximity factor, defined in [0, 1], so that 0 corresponds to no approach and 1 corresponds to the maximum approach (resulting in coincident ASs and CPs). Then, the schemes proposed for the auxiliary surface deformation are defined in (8) as follows:

$$
\rho\_m' = \rho\_m + \operatorname{gs}\left(\frac{m}{M}\right)^\nu,\tag{8}
$$

where ν = 1 for simple and ν = 2 for progressive reach. The deformation effect is graphically described in Figure 3.

Furthermore, in a way similar to [20], ASs and CPs, accordingly, should become denser close to the wedge tip. Again, there is no unique way to accomplish this. In this work, the scheme implemented multiplies the polar angle ϕ*<sup>m</sup>* of the *m*th AS location by a factor *Dm*, 0 < *Dstart* ≤ *Dm* ≤ 1, where *Dstart* is user-defined. For example, in the first quadrant of the 'eye', *Dm* is defined as being close to 0 for ASs near the wedge tip, and close to 1 for ASs close to the vertical axis. For progressive densification, the scheme proposed is: ϕ *<sup>m</sup>* = ϕ*mD*<sup>2</sup> *<sup>m</sup>*. Moreover, additional ASs may be superimposed to the already existing ones close to the tips if necessary. The combined effect of the tip approach and densification is depicted in Figure 4.

**Figure 3.** Polar radius decrease from 1.5 to 1 according to the proposed schemes.

**Figure 4.** Deformation of the auxiliary surface, combined with densification in the vicinity of the tips: all inner ASs are allowed to approach the CPs, whereas only 1/4 of the outer ASs are allowed to do so.
