# Enhancement of ECE SuperPin Curved Reflex Reflector by the Use of Double Pins with Corner Cubes
## 2 Principles  
The EU ECE standard is designed to reduce injuries and deaths resulting from traffic accidents by providing adequate illumination of the roadway and by enhancing the conspicuity of motor vehicles on public roads so that their presence is perceived, in daylight, darkness and other conditions of reduced visibility. A white reflex reflector provides an observation angle of 0.33\({}^{\circ}\) (EU ECE regulations), not less than 1680 millicandela/lux at a light entrance angle of 0\({}^{\circ}\), not less than 1120 millicandela/lux of light at 1\({}^{\circ}\) up and 10\({}^{\circ}\) down and not less than 560 millicandela/lux including the entrance angle at 20\({}^{\circ}\) left and 20\({}^{\circ}\) right. [5]. The corner-cube retro-reflector (CCR) is based on groups of three perpendicular planes, as shown in Figure 2. Conventionally, the dihedral angle between any pair of reflecting faces is made to be almost exactly 90\({}^{\circ}\)[1], so that the reflected beam is exactly antiparallel to the incident beam [13]. If the angles differ from 90\({}^{\circ}\) by an amount, the reflected beam will be converged or diverged in multiple beams to achieve the required application [1]. Through an array of pins, corner-cube retro-reflectors can be produced as shown in Figure 2. The orientation of each face is given by the normal unit \(\hat{n_{1}}\), \(\hat{n_{2}}\), and \(\hat{n_{3}}\) for each face. The reflection from each face reverses the component of the light's velocity vector that is normal to the face. Let \(\overset{\rightarrow}{V}\) and \(\overset{\rightarrow}{V}\) be the directions of a ray before and after reflection, respectively, with the vector \(\overset{\rightarrow}{V}=\overset{\rightarrow}{V}-2(\overset{ \rightarrow}{V}\cdot\hat{n})\)\(\hat{n}\), where \(\hat{n}\) is normal to the face. Applying the above formula three times yields the direction of  
Figure 1: Retro-reflected light by (**a**) the Economic Commission for Europe (ECE) regular reflex reflector, and (**b**) the ECE SuperPin reflex reflector. the reflected beam for a particular order of reflection. Formulas for the direction of the reflected rays after the three reflections are given by [PERSON]'s formula:  
\[\overrightarrow{t}=\overrightarrow{q}+2\overrightarrow{\alpha}(\alpha \overrightarrow{\alpha}-\beta\overrightarrow{b}+\gamma\overrightarrow{c}) \tag{1}\]  
where \(\overrightarrow{t}\) is the final direction; \(\overrightarrow{q}\) is the original direction: \(\alpha\), \(\beta\), \(\gamma\) are the small angles by which the angles between the three mirrors exceed right angles and \(\overrightarrow{a}\), \(\overrightarrow{b}\), and \(\overrightarrow{c}\) are normal to the three mirrors taken in order in a right-hand sense. Equation (1) is valid at the first-order when the mirrors are nearly mutually perpendicular. The angle \(\alpha\) is the angle between the faces whose normals are \(\overrightarrow{b}\), and \(\overrightarrow{c}\). The normals may be strictly perpendicular; that is, they do not need to include the small deviations caused by the dihedral-angle offsets. The directions of the reflected rays were computed by applying the law of reflection three times. The unit normals to the faces can be computed as follows and as shown in Figure 3. Let the normals to the faces without dihedral-angle offsets be the unit vectors \(\hat{t}\), \(\hat{t}\), and \(\hat{k}\) along the three coordinates x, y, and z, respectively.