A Geometric Gaussian Mixture Representation of Plane Curves

Researchers introduce a Gaussian Mixture Model (GMM) framework that represents plane curves as probabilistic geometric primitives, encoding both tangential and normal uncertainty. This mathematical approach enables uncertainty-aware geometric modeling applicable to CAD, robotics, and digital twin applications.

This research presents a theoretical advancement in geometric representation by bridging deterministic curve modeling with probabilistic methods. The core innovation converts polygonal curve approximations into Gaussian mixture components, where each segment captures uncertainty through both tangential and normal distributions. This framework addresses a fundamental gap in traditional CAD and geometric modeling systems, which typically treat curves as idealized one-dimensional entities without accounting for manufacturing tolerances, measurement errors, or inherent geometric uncertainty.

The mathematical elegance of matching first and second central moments creates analytically tractable models that preserve local geometric properties while extending beyond classical formulations. The framework's versatility—handling smooth, closed, open, non-regular, and self-intersecting curves—demonstrates its broad applicability across diverse geometric scenarios. This generality contrasts with specialized approaches that handle only specific curve classes.

The practical implications span multiple high-value domains. In CAD systems, uncertainty quantification enables tighter tolerance specifications and risk assessment during design phases. For robotics and probabilistic trajectory planning, the framework provides principled methods to model obstacles and paths with confidence bounds, improving collision avoidance and motion safety. Digital twin applications benefit from incorporating real-world measurement uncertainty into virtual representations, enhancing simulation accuracy.

The experimental validation on canonical curves suggests the GMM representation captures essential geometric characteristics—local tangents, normals, and arc lengths—translating to accurate global shape preservation. Future work likely focuses on computational efficiency for high-dimensional applications and integration with existing CAD software architectures. This research contributes to the growing intersection of probabilistic modeling and computational geometry, with immediate relevance for precision engineering and autonomous systems.

• →Gaussian Mixture Models provide analytically tractable probabilistic representation of plane curves with uncertainty encoding
• →Framework handles diverse curve types including smooth, closed, self-intersecting, and non-regular curves
• →Applicable to uncertainty-aware CAD systems, robotics obstacle modeling, and digital twin applications
• →Local geometric properties (tangent, normal, arc length) are preserved while extending beyond deterministic formulations
• →Adaptive discretization and variable normal uncertainty parameters enable flexible geometric modeling