An Analytically Derived Gauss–Legendre Quadrature for Axis-Aligned Ellipse–Ellipse Intersection

Accurate and efficient evaluation of the intersection area between two axis-aligned ellipses is essential in applications where the coordinate system or underlying geometry naturally imposes alignment. However, most existing numerical integration techniques are designed for arbitrarily oriented ellipses, and their generality typically requires adaptive refinement or solving higher-degree algebraic intersection formulations, leading to greater computational cost than necessary in the axis-aligned case. This study introduces two analytically derived, fixed-cost Gauss–Legendre quadrature formulations for computing the intersection area in the axis-aligned configuration. The first is a sine-mapped Gauss–Legendre quadrature, which applies a trigonometric transformation to improve conditioning near endpoint singularities while retaining constant-time evaluation. The second is an enhanced two-panel affine-normalized formulation, which splits the intersection domain into two sub-intervals to increase local accuracy while maintaining a fixed computational cost. Both methods are benchmarked against adaptive Simpson integration, polygonal discretization, and Monte Carlo sampling over 10,000 randomly generated ellipse pairs. The two-panel formulation achieves a mean relative error of 0.003% with runtimes more than twenty times faster than the adaptive reference and remains consistently more efficient than the polygonal and Monte Carlo approaches while exhibiting comparable or superior numerical behavior across all tested regimes.