Complex Geodesic Curvature for Real-Time LiDAR Feature Classification

Conventional point-based algorithms largely ignore the strict sequential order of rotating multi-beam Light Detection and Ranging (LiDARs). We exploit this intrinsic structure by modeling each scan ring as a discrete curve on a conical manifold, encoded as a one-dimensional complex-valued signal. This representation preserves full 3D information while enabling efficient, principled computation of geodesic curvature via discrete difference calculus. Augmented with local smoothness and range-gradient features, this curvature forms the basis of an unsupervised classification pipeline that uses adaptive ring-wise thresholds to label points as planar, edge, or corner features. A scan-topology graph then aggregates these 1D labels into coherent 3D primitives, all in linear time. Experiments on synthetic and large-scale urban datasets confirm the method's theoretical accuracy and practical utility. Our C++ implementation processes at over 80 Frames Per Second (FPS) on a single Central Processing Unit (CPU) core, significantly outperformings traditional Principal Component Analysis (PCA)-based and clustering methods in geometric accuracy, label purity, and temporal stability. By uniting first-principles differential geometry with real-time performance, our framework provides a transparent, parameter-light alternative to learning-based pipelines, offering robust landmarks for Simultaneous Localization and Mapping (SLAM) and consistent semantics for dynamic scene understanding.