# HCL Control Strategy for an Adaptive Roadway Lighting Distribution
## 2 Materials and Methods
### Freeform Surface  
Freeform surface generally refers to an irregular or non-symmetric smooth surface, which varies from that of an ellipse, a parabola or conic curve formed with a specific shape. Freeform surface was mainly used in the design of the exterior structure of car bodies or ships in the early days. The application of optical design is mainly based on the relationship between light distribution and target illuminance, which is conformed to Snell's law, Etendue conservation and the Edge-ray principle, etc. When a derivative formula of the curved surface is established then we can obtain the control points of the component structure and model the freeform surface. In this paper, the distribution of the LED light and the required light type of the observation surface are analyzed using zonal flux; the incident light source is divided into grids and correspond to the outgoing position of the target plane. Figure 5 is a display of the light energy mapping of the freeform surface.  
When solving or optimizing freeform surfaces, a large number of complicated mathematical calculations of differential equations are commonly used. When it comes to basic geometric optics, the intersection of the linear equations, based on the incident light vector and the tangent vector, is the control point for the curve. The calculation process is relatively simple and could apply to most of the geometric curved surfaces. However, when the number of control points is insufficient, for instance, if we calculate at intervals of 5\({}^{\circ}\) or 10\({}^{\circ}\), it may lead to deviation in the tangent vector [17]. Therefore, when designing, a large number of control points should be used to create a freeform surface structure to reduce errors although it will also increase the computing time during modeling. In addition, when it comes to different lens applications or output angle beam pattern, the reasonable number  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Zonal Angle \(\theta\)** & **Average Intensity** & **Zonal Coefficient** & **Zonal Flux** \\ (\({}^{\circ}\)) & **(cd)** & **2\(\pi\)**(\(cos\theta_{1}\)**\(-cos\theta_{2}\))** & **(lm)** \\ \hline
0–10 & 31.6 & 0.095 & 3 \\ \hline
10–20 & 30.6 & 0.283 & 8.7 \\ \hline
20–30 & 28.7 & 0.463 & 13.3 \\ \hline
30–40 & 26.0 & 0.628 & 16.3 \\ \hline
40–50 & 22.4 & 0.774 & 17.4 \\ \hline
50–60 & 18.2 & 0.897 & 16.3 \\ \hline
60–70 & 13.4 & 0.993 & 13.3 \\ \hline
70–80 & 8.2 & 1.058 & 8.7 \\ \hline
80–90 & 2.8 & 1.091 & 3 \\ \hline \multicolumn{3}{c}{Total zonal flux (lm)} & \multicolumn{2}{c}{100} \\ \hline \hline \end{tabular}
\end{table}
Table 2: Zonal flux analysis for Lambertian LED (100 lm).  
Figure 5: Freeform surface light energy mapping diagram [36].  
of control points demanded will primarily be based on related designing experience. While the goniometer measures luminaire distribution at intervals of 2.5\({}^{\circ}\), the optical lens design in this paper calculates at intervals of 1\({}^{\circ}\) when analyzing, its deviation is less than 1% when compared with using intervals of 0.1\({}^{\circ}\) in horizontal plane analysis. Moreover, when calculated at intervals of 1\({}^{\circ}\), the software efficiency is quite good and the control points of the freeform surface are very accurate.