## 3 Results and Discussion
### Function Simulation Results  
Theoretically, the chroma (C) of the blue sapphires had the greatest correlation with **b** due to their blue tone. The correlations of **C** and **b**, **L** and **b**, and **L** and **C** of the sapphires were fitted with functions (Figure 4). As shown in Figure 4, the chroma (**C**) and **b** values of the two sapphire cut shapes exhibited a linear negative correlation, meaning that the chroma (**C**) was linearly positively correlated with the absolute value of **b**. The relationship between the chroma (**C**) and **b** values of the oval-shaped sapphires conformed to the formula \(\text{C}_{\text{LBR}}=-1.07\times\text{b}_{\text{LBR}}+0.02\), \(\text{R}^{2}=0.99\) (Figure 4a), while that of the pear-shaped sapphires conformed to the formula \(\text{C}_{\text{LBP}}=-1.09\times\text{b}_{\text{LBP}}-0.92\), \(\text{R}^{2}=0.99\) (Figure 4b). In the CIE 1976, \(\text{L}^{*}\) a* b* is the uniform color space, and the negative direction of the b* axis represents variations toward blue. The  
Figure 4: Function fitting of the sapphire color parameters. (**a**) Linear fitting between **b** and **C** for oval-shaped sapphires; (**b**) linear fitting between **b** and **C** for pear-shaped sapphires; (**c**) function fitting between **L** and **b** for oval-shaped sapphires; (**d**) function fitting between **L** and **b** for pear-shaped sapphires; (**e**) function fitting between **L** and **C** for oval-shaped sapphires; and (**f**) function fitting between **L** and **C** for pear-shaped sapphires. smaller the b value, the larger the absolute value of **b** in the negative direction of the b* axis, and the greater the chroma (**C**). For both the cut shapes of the sapphires, the lightness (**L**) showed a quadratic correlation with **b** and **C** (Figure 4c-f). As the **L** increased, the **b** value first decreased and then increased, meaning the absolute value of **b** first increased and then decreased (Figure 4c,d). Similarly, as **L** increased, the **C** value first increased and then decreased (Figure 4e,f). This rule can be best illustrated by the following extreme example. The color was black, with its chroma of 0, when the lightness was at its lowest. And the color became visible and the chroma increased, until it reached its extremum, as the lightness increased. The lightness continued to increase until 100, the color became white, and the chroma (**C**) returned to 0. The function fitting results were as follows:  
The relationship between the **b** values and the lightness of the oval-shaped sapphires conformed to the formula \(\mathrm{b_{LBR}}=0.02\times\mathrm{L_{LBR}}^{2}-2.22\times\mathrm{L_{LBR}}-4.14\), R\({}^{2}=0.43\) (Figure 4c). When \(\mathrm{L_{LBR}}=55.5\), the value of \(\mathrm{b_{LBR}}\) was at its minimum (\(-65.745\)), with an \(\mathrm{L_{LBR}}\) of 55.5. This meant that the blue perception of the sapphires increased with an increase in lightness when the lightness of the oval-shaped sapphires was less than 55.5. When the lightness of the oval-shaped sapphires exceeded 55.5, the blue perception of the sapphires decreased as the lightness increased. The relationship between the **C** values and lightness of the oval-shaped sapphires conformed to the formula \(\mathrm{C_{LBR}}=-0.02\times\mathrm{L_{LBR}}^{2}+2.27\times\mathrm{L_{LBR}}+5.97\), R\({}^{2}=0.39\) (Figure 4e). The \(\mathrm{C_{LBR}}\) was at its maximum (70.38) with an \(\mathrm{L_{LBR}}\) of 56.75. This meant that the chroma of the sapphires increased with an increase in the lightness when the lightness of the oval-shaped sapphires was less than 56.75. When the lightness of the oval-shaped sapphires exceeded 56.75, the chroma of the sapphires decreased as the lightness increased. The relationship between the **b** values and lightness of the pear-shaped sapphires conformed to the formula \(\mathrm{b_{LBP}}=0.03\times\mathrm{L_{LBP}}^{2}-2.80\times\mathrm{L_{LBP}}-3.29\) (R\({}^{2}=0.54\)) (Figure 4d).