Parametric Curve Intersection via Monotonic Segmentation

The patent divides curves into monotonic segments based on parameter ranges where the curves are monotonic in Cartesian space. This segmentation allows the use of simple bisection methods on each segment while avoiding the reliability issues of Newton-Raphson near cusps and tangent points, as each segment guarantees a single intersection that can be found reliably.

Instead of using derivative-based methods (Newton-Raphson) that fail when derivatives are small, the patent inverts the approach by using parameter-based monotonicity analysis and bisection methods that work reliably regardless of derivative magnitude. This inversion transforms the problem from finding roots of distance functions to finding parameter values where curve positions coincide.

By segmenting curves into monotonic regions first, the patent eliminates the need for nested Newton-Raphson loops. Each monotonic segment can be processed independently with a single bisection loop, removing the computational overhead of nested iterations while maintaining intersection detection capability.

The patent performs partial action by dividing the curve into segments and applying intersection detection to each segment separately. This approach finds all intersections through multiple simple iterations rather than using one complex nested iteration, improving computational efficiency.

The patent merges the intersection detection logic into a unified framework based on monotonic segmentation. By combining curve splitting, monotonicity analysis, and intersection finding into a cohesive algorithm, the patent reduces code dispersion and improves maintainability while preserving detection reliability.

The monotonic segmentation approach serves multiple functions: it enables reliable intersection detection, handles tangent and cusp cases, and provides a unified code structure. This multi-functionality reduces the need for separate specialized routines dispersed throughout the codebase.