In the treatment for different vision impairments, such as refractive errors, strabismus, and cataract, the angles α or angle κ can both influence the course and outcome [11, 14, 15].

The angle κ, also known as angle λ, is defined as the angle between the pupillary axis (the line that passes perpendicularly through the center of the pupil and the center of curvature of the cornea) and the visual axis (the line that connects the fixation point with the fovea) [16]. Angle κ was predominantly situated towards the nasal side in myopic subjects aged 16–51 years in Thailand (97%) [17], whereas the finding in this study was diametrically opposite (89.37%). Besides the difference in ages (7.24 ± 2.08 vs. 31.66 ± 7.77 years) between the two studies, the refraction status may be one of the important factors (0.05 ± 1.98 D vs. -4.9 ± 2.29 D). Previous studies have shown that angle κ is correlated with horizontal coma, and there may be a mechanism in the posterior surface of the cornea or the lens to balance the horizontal coma on the anterior surface of the cornea [18, 19]. The angle κ situated towards the temporal side in contrast to the norm during emmetropization may affect the strabismus surgery design [11]. A large angle κ may correspond to a smaller radius of scleral curvature (steeper sclera) in the design of scleral lenses; however, the effect of the quadrant distribution of angle κ on the radius of scleral curvature has not been discussed [10]. Our study revealed that angle κ in children is predominantly distributed inferotemporally, followed by superotemporally. This may influence the design of corneal contact lenses, such as OK lenses, and strabismus surgery. The angle κ-corrected corneal topography can help in the distinguishment of pseudo-deviation after orthokeratology treatment, and the accurate measurement of angle κ is crucial in determination of the actual strabismus angle. The details require elucidation in future studies on specific populations.

Publication concluded angle κ correlates with the axial length (Pearson’s r = − 0.813, p < 0.001) and spherical equivalent correlation (SER) (Pearson’s r = 0.685, p = 0.003) [10]. However, no correlations between these two parameters and angle κ were found in this study. There were three possible reasons for the differences when compared with previous differences: first, we included children aged 3–15 years, while the other related study included adults aged 24–54 years, and angle κ may vary between children and adults; second, a larger sample size was included in this study (219 vs. 24), which reduces the appearance of false positive results; and third, a mixed-effects model was used in this study to conduct the analysis, which controls for the interactions between hierarchical clustered factors.

It was found that male corneas with flatter corneas (larger CRs) and larger WTW corresponded to larger angles κ. We used GEE to eliminate the hierarchical clustered factors. Gender may be an independent factor influencing angles κ, and comparisons based on WTWs need to be conducted under gender grouping for better comparison of angles κ. A previous study on the 14–81 year-old population found no significant gender differences [16]. Therefore, age may be the main influencing factor. The development of the axial length and corneal curvature differs between males and females [20]. Thus, the findings of this study, which focused on children, differed from the findings of previous studies conducted on adults.

The interocular comparison showed that the differences in interocular WTWs affected the interocular angles κ. Comparing the fellow eyes, we eliminated the possible influencing factors in angle κ between individuals. Differences in interocular WTWs may affect the visual pathways of children, resulting in differences in angle κ. Therefore, in the treatment of congenital cataracts, strabismus, or refractive errors, the effect of differences should also be taken into account on treatment strategies and outcomes between the fellow eyes in pediatric patient.

Previous studies of angle α predominantly focused on the cataract population because it is a more stable pre- and post-cataract surgery measurement compared to angle κ [15, 21]. This study showed that flatter corneas with smaller WTWs had larger angles α, which is similar to the findings on angles κ and consistent with the relationship between angles α and WTW reported in the cataract population [22]. We also found that angle α predominantly oriented towards the nasal side of the optic axis; angle α in the cataract population was mainly situated towards the temporal side of the visual axis (nasal side of the optic axis), and Angle α in the cataract population was shown to decrease non-linearly and shift towards the nasal side of the visual axis (the temporal side of the optic axis) as the axial length increased [15]. The mean axial length in this study was 23.24 ± 1.14 mm; thus, angle α predominantly distributed on the nasal side of the optic axis. Angle α in both the populations showed similar characteristics. Currently, studies on the role of angle α in the treatment of congenital cataracts, strabismus, and refractive errors are scarce, which needs to be addressed with further studies.

The current study has the following limitation. Firstly, we did not standardize for different levels of accommodation in subjects, as previous studies have shown that angles κ are not significantly different between subjects with different levels of accommodation [23]. Secondly, the effect of conditioning on angle κ may be different in children, rendering the other limitation of this study.