We propose a novel paradigm for cone-beam computed tomography (CBCT) reconstruction from ultra-sparse X-ray projections, by introducing a framework that generates auxiliary X-ray projections under controlled geometric parameters. This innovation overcomes the limitations of conventional methods that are constrained to producing fixed-angle projections. Our approach is organized into three key modules: the XGen, X-Correction, and CT-Correction module. Through the XGen module, we generate projections based on any given geometric parameters to supplement the geometric information in the projection domain. The X-Correction module then introduces geometric corrections to harmonize the generated projections. Finally, through the CT-Correction module, the reconstructed image undergoes refining, thereby enhancing the image quality within the image domain. We have validated our model on several datasets including a large-scale publicly available lung CT dataset (LIDC-IDRI with 1018 patients); an extensive abdominal CT dataset (AbdomenCT-1K, with a selected 1k patients); and our proprietary pelvic CT dataset, collated from a hospital (445 patients). Real walnut projection data were also incorporated for genuine projection validation. Compared to the traditional projection generation methods and the state-of-the-art ultra-sparse reconstruction techniques on 2-view and 10-view tasks, our method has demonstrated consistently superior performance across various tasks.