## 2 Literature Review
### Boundary Construction and Linear Interpolation  
Accurately defining fixation boundaries is crucial for understanding users' visual behavior in eye-tracking technology. From a cognitive standpoint, fixation boundaries can be interpreted as externalized representations of perceptual grouping, reflecting how viewers segment scenes into meaningful spatial units. As [PERSON] [68] suggests, the human visual system uses boundary cues to organize visual input into coherent structures, which facilitates higher-level interpretation and learning. Ensuring boundary continuity and precision is, therefore, essential, not only for computational stability, but also for preserving the integrity of spatial perception and attentional organization. While traditional fixation boundary methods rely on discrete sampling points to represent fixation locations, they often fail to capture spatial smoothness and local continuity within fixation distributions, which may affect the geometric stability of boundary construction [69]. Regarding boundary construction, [PERSON] and [PERSON] [70] first proposed a linear interpolation-based boundary construction method, which connects adjacent sampling points to form enclosed fixation regions. While this method is computationally simple, it overlooks the curvature characteristics of eye movement trajectories, potentially leading to inaccurate boundary fitting. To improve boundary-fitting accuracy, [PERSON] et al. [71] introduced a polynomial interpolation method using quadratic polynomials to describe trajectories between sampling points more precisely, thereby significantly enhancing boundary construction accuracy. However, polynomial interpolation may lead to overfitting issues, especially when data points are unevenly distributed. To address this, [PERSON] et al. [72] proposed a technique that combines linear interpolation with local optimization to mitigate the problem of overly large boundaries caused by traditional convex hull algorithms. In high-precision ranges (\(-0.5^{\circ}\) to \(0.5^{\circ}\)), this method achieved an accuracy rate of 60-99%. However, it may still produce discontinuities in boundary fitting when handling rapid eye movements. To tackle the issues of boundary continuity and accuracy, [PERSON] et al. [73] introduced piecewise linear interpolation, applying local smoothing constraints to optimize boundary shapes. [PERSON] et al. [74] further extended this approach to adaptive interpolation techniques, dynamically adjusting interpolation parameters based on sampling point density to accommodate eye movement trajectories at varying speeds. Additionally, [PERSON] and [PERSON] [75] proposed a real-time algorithm based on local linear interpolation, significantly improving processing efficiency through parallel computation. However, [PERSON] [76] pointed out that balancing real-time performance and accuracy remains a challenge, particularly in high-frequency eye movement data processing, where boundary update latency is an unresolved issue. It is worth noting that most of the above methods rely on a single interpolation function for boundary calculations. Recent studies by [PERSON] et al. [77] and [PERSON] and [PERSON] [78] have proposed multi-function or hybrid interpolation strategies to enhance computational accuracy and robustness in boundary construction. In this study, we propose a novel approach using two complementary functions (LimitX and LimitY) to calculate specific positions on the boundary. This method enhances continuity and numerical robustness in cases with irregular fixation distributions, aligning with recent advances in hybrid interpolation strategies for boundary modeling [79]. Although most modern eye-tracking systems include algorithms to handle common artifacts such as blinks, traditional modeling approaches often do not explicitly account for residual calibration errors or transient drifts, which may still impact boundary accuracy [80; 81]. [PERSON] et al.