Topology reasoning jointly detects 3D lanes and traffic elements from multi-view images and infers their structural connectivity. Current methods model lanes as discrete polylines, lacking smoothness, analytical tangent directions, and global spatial support for attention, while providing sparse topology supervision. We propose TopoCurve, a geometry-driven architecture for 3D topology reasoning grounded in a structured parametric lane representation. Lanes are modeled as endpoint-fixed cubic Bézier curves, enabling continuous geometry with exact endpoints and analytically defined directionality. We exploit this shared curve geometry across the entire pipeline. Endpoint distance and tangent alignment are encoded with multi-scale Fourier features and injected into the topology head. Sampled curve points serve as geometry-aligned references for deformable cross-attention spanning the full lane. Parallel curve-anchored attention branches provide diverse predictions for one-to-many topology supervision. These components form a tightly coupled cascade where representation enables geometric reasoning, guides feature aggregation, and supports denser supervision. TopoCurve achieves 50.6 OLS on the OpenLane-V2 benchmark without any post-processing, establishing a new state-of-the-art among end-to-end camera-only methods, and outperforms all existing approaches on endpoint detection (56.8 vs. 52.6 on DET_p).

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