# The Potential of Focusing Acoustic Retroreflectors for Architectural Surface Treatment
## Appendix A Parabolic Dihedra and Trihedra  
The concept behind the parabolic trihedra used as physically realizable focusing retroreflectors can be understood as an extension of the two-dimensional case of parabolic dihedra. A conventional dihedron forms a corner reflector, which is a non-focusing retroreflector in two-dimensional space. A dihedral focusing retroreflector can be formed by extracting segments from two co-focused parabolas, \(y=ax^{2}\) and its vertically mirrored counterpart \(y=-ax^{2}+1/(2a)\) (Figure 1). Diverging rays emitted by a source located at the focal point (red, source-to-vertex distance \(=1/(2a)\)) are made parallel by the first reflection, and then made to converge by the second, achieving perfect focusing retroreflection. Similarly, soft-focusing retroreflection is achieved for sources elsewhere on or near the Y axis (e.g., blue source and rays).  
For simplicity, \(y=ax^{2}\) in XOY can be represented in X'O'Y' using coordinate transformation rules following Equation (A1).  
\[y=\begin{cases}\frac{\sqrt{2}\sqrt{-\sqrt{2}ax+1}-ax+\sqrt{2}}{a}\\ \frac{-\sqrt{2}\sqrt{-\sqrt{2}ax+1}-ax+\sqrt{2}}{a}\end{cases},\;\;x\leq\frac {\sqrt{2}}{2a}\] (A1)  
The extracted segment can then be written as Equation (A2),  
\[y=\begin{cases}\frac{-\sqrt{2}\sqrt{-\sqrt{2}ax+1}-ax+\sqrt{2}}{a},\;\;0\leq x \leq l\end{cases}\] (A2)  
where \(l\) is the edge length of the base dihedron (prior to curvature).  
In three dimensions, an exact focusing solution for a trihedron is not known, but an effective solution is achieved by warping the trihedron similarly to the parabolic dihedron. Figure 2 illustrates how each trihedron face can follow a pair of intersecting parabolas.  
Shown as thick curves in Figure 11, the two segments of intersecting parabolas X'ZZ' and Y'ZZ' can be, respectively written as Equations (A3) and (A4):  
\[z=\left\{\frac{-\sqrt{2}\sqrt{-\sqrt{2}ax+1}-ax+\sqrt{2}}{a}\right.,\ 0\leq x\leq l\] (A3)  
\[z=\left\{\frac{-\sqrt{2}\sqrt{-\sqrt{2}ay+1}-ay+\sqrt{2}}{a}\right.,\ 0\leq y\leq l\] (A4)  
where \(l\) is the edge length of the base trihedron. Then, one segment is swept over the other, i.e., moved along the other to form a double-parabolic surface (shown as the pink surface), the equation of which can be written as Equation (A5):  
\[z=\left\{\frac{-\sqrt{2}\sqrt{-\sqrt{2}ax+1}-\sqrt{2}\sqrt{-\sqrt{2}ay+1}-ax-ay+ 2\sqrt{2}}{a},0\leq x,y\leq l\right.\] (A5)  
Similarly, two other double-parabolic surfaces (the blue and green surfaces) can be formed using parabolas, and these three surfaces together construct a focusing trihedral retroreflector.  
Unlike a parabolic dihedron which reflects sound twice and achieves perfect focusing retroreflection for sources at its focal point, a parabolic trihedron reflects sound three times and thus focuses short for sources located at its parabolic focal point. The performance of a focusing trihedron with different source-to-vertex distances was investigated using ray-based simulation, and it was found that the best result, i.e., the highest degree of ray convergence back to the source, is achieved when \(1/\left(2a\right)=1.15\times\) the distance. The parameters of the parabolic trihedra in Section 2.2 were determined using this equation to maximize their performance