Evaluating map projection efficiency for large-area representation: a comparative analysis using tissot’s indicatrix

Abstract

The Earth's curved surface cannot be represented on a plane without distortion, as established by Gauss's Theorema Egregium. Because a sphere possesses constant positive Gaussian curvature while a plane has zero curvature, no length-preserving mapping between them can exist, and every large-area map must accept some geometric compromise. For maps spanning continents or the entire globe, these distortions grow large enough to influence spatial perception and quantitative analysis, yet projection selection in practice often relies on convention rather than objective evaluation. This study evaluates eight widely used projections-Mercator, Plate Carrée, Lambert cylindrical equal-area, Gall–Peters, sinusoidal, Mollweide, Robinson, and Winkel Tripel-using a unified framework based on Tissot's indicatrix. Area, angular, and shape distortions were computed across a global grid and condensed into a single area-weighted Airy–Kavrayskiy efficiency index. Validation against closed-form analytical solutions achieved a maximum discrepancy of 1.3 × 10⁻⁶. Results confirm the fundamental trade-off between conformal and equal-area projections. Compromise projections performed best overall, with Winkel Tripel and Robinson achieving efficiency indices of 0.269 and 0.292 respectively, stable across varying latitude truncation caps. Notably, the Plate Carrée outperformed all equal-area projections despite preserving no single geometric property exactly. These results provide a transparent and reproducible basis for projection selection in large-area mapping.