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Airborne LiDAR (Light Detection and Ranging) data is widely applied in building reconstruction, with studies reporting success in typical buildings. However, the reconstruction of curved buildings remains an open research problem. To this end, we propose a new framework for curved building reconstruction via assembling and deforming geometric primitives. The input LiDAR point cloud are first converted into contours where individual buildings are identified. After recognizing geometric units (primitives) from building contours, we get initial models by matching basic geometric primitives to these primitives. To polish assembly models, we employ a warping field for model refinements. Specifically, an embedded deformation (ED) graph is constructed via downsampling the initial model. Then, the point-to-model displacements are minimized by adjusting node parameters in the ED graph based on our objective function. The presented framework is validated on several highly curved buildings collected by various LiDAR in different cities. The experimental results, as well as accuracy comparison, demonstrate the advantage and effectiveness of our method. {The new insight attributes to an efficient reconstruction manner.} Moreover, we prove that the primitive-based framework significantly reduces the data storage to 10-20 percent of classical mesh models.