Depth completion is a vital task for autonomous driving, as it involves reconstructing the precise 3D geometry of a scene from sparse and noisy depth measurements. However, most existing methods either rely only on 2D depth representations or directly incorporate raw 3D point clouds for compensation, which are still insufficient to capture the fine-grained 3D geometry of the scene. To address this challenge, we introduce Tri-Perspective View Decomposition (TPVD), a novel framework that can explicitly model 3D geometry. In particular, (1) TPVD ingeniously decomposes the original point cloud into three 2D views, one of which corresponds to the sparse depth input. (2) We design TPV Fusion to update the 2D TPV features through recurrent 2D-3D-2D aggregation, where a Distance-Aware Spherical Convolution (DASC) is applied. (3) By adaptively choosing TPV affinitive neighbors, the newly proposed Geometric Spatial Propagation Network (GSPN) further improves the geometric consistency. As a result, our TPVD outperforms existing methods on KITTI, NYUv2, and SUN RGBD benchmarks. Furthermore, we build a novel depth completion dataset named TOFDC, which is acquired by the time-of-flight (TOF) sensor and the color camera on smartphones. Project page.