## 3 Methodology
### Determining the Head/Face Required Dimensions
This was considered through:  
\[a_{3}\approx\tan^{-1}\frac{B_{9}-B_{2}}{B_{10}} \tag{5}\]  
\[a_{2}\approx 2B_{2} \tag{6}\]  
The stability of the glasses also requires a stable triangular relationship between the frame and the bridge of the nose, and this was considered through:  
\[a_{11}\approx\cos^{-1}\frac{B_{11}^{2}+B_{13}^{2}-B_{12}^{2}}{2B_{11}B_{13}} \tag{7}\]  
Optical centre and interpupillary distance of the lens is also an important functional consideration. The position of the lens needs to be determined according to the position of the pupil, and the inner and outer corners of the orbit. The eyelid, and the cheekbones are used to determine the size of the lens. \[a_{1}\approx B_{3} \tag{8}\]  
\[a_{4}\leq B_{14} \tag{9}\]  
In Formulas (1)-(9), points \(a_{1}\)-\(a_{11}\) correspond to eyewear size parameters as shown in detail in Table 1 and Figure 4. Dimensions are represented by \(B_{1}\)-\(B_{14}\) points as shown in Figure 5b. The feature points with serial numbers 1-23 are represented in the coordinate system as \((x_{n},y_{n},z_{n})\), \(n=1,2\cdot\cdot 23\). Then  
\[ \begin{array}{c}B_{1}=\left|x_{7}-x_{4}\right|,\ B_{2}=\left|x_{5}-x_{4} \right|,\ B_{3}=\left|x_{8}-x_{9}\right|,\ B_{4}=\left|x_{5}-x_{9}\right|,\\ B_{5}=\left|y_{7}-y_{8}\right|,\ B_{6}=\left|y_{7}-y_{22}\right|,\ B_{7}= \left|y_{22}-y_{23}\right|,\ B_{8}=\left|x_{22}-x_{23}\right|,\\ B_{9}=\left|x_{13}-x_{12}\right|,\ B_{10}=\left|x_{4}-z_{12}\right|,\\ B_{11}=\left|\sqrt{\left(z_{4}-z_{22}\right)^{2}+\left(y_{4}-y_{22}\right)^{2 }}\right|;B_{12}=\left|\sqrt{\left(z_{22}-z_{10}\right)^{2}+\left(y_{22}-y_{1 0}\right)^{2}}\right|,\\ B_{13}=\left|\sqrt{\left(z_{4}-z_{5}\right)^{2}+\left(y_{4}-y_{5}\right)^{2 }}\right|,\ B_{14}=\left|z_{3}-z_{11}\right|.\end{array}  \tag{10}\]  
To correlate the aforementioned points to human head measurements orthogonal pictures were used. Before taking the pictures, small location circles were attached to the face. These markers (Figure 5a) were used to calibrate the feature points. Additionally, auxiliary calibration points were positioned at the intersection of horizontal and vertical lines. Pictures were taken with the head parallel to the Frankfurt plane, with the body upright and the shoulders flat. Auxiliary calibration points were used to ensure head posture was kept in position in comparison to the horizontal and vertical axes. Pictures were post-processed (see Figure 5a), to sharpen the edges of the image. The edge contour of the head was then extracted, and the photos were gridded according to the calibrated feature points. Then the coordinate origin was determined, and the plane coordinate system was established. According to the plane coordinate system, the dimension of each feature point in the photo was obtained. The outline of each marker was traced, and its real dimension was obtained using:  
\[H_{0}=H_{p}\times\frac{\frac{\sum_{i=1}^{n}D_{i}}{n}}{D_{0}} \tag{11}\]where \(H_{0}\) is the real size measurement, \(D_{0}\) is the measurement of the small circles, and \(n\) is the number of small circles measured and the picture. The eyewear feature points and their correlation to the human head are given in Table 1.  
The correlated points obtained from the analyzed eyewear and the relevant human head anatomy (Figure 5b) were used to accurately create a parametric 3D model of a human head. Considering the human head morphology, several planes were used to divide the 3D model (see Figure 6) in order to transform it into a 3D coordinate system that was used as a foundation for the parametric modelling. This coordinate system was composed of 18 planes at points 2-7, 10-15, 17-20 and 22; and another one at the midpoint between 3 and 4. Due to the high level of detail around the eye, this area was divided in more detailed sections to facilitate the accuracy of the measurements. The 3D coordinate system has a vertical plane passing through point 4, a horizontal plane passing through point 6 and another plane passing through point 22.