Aortic shape analysis plays a key role in cardiovascular diagnostics, treatment planning, and understanding disease progression. We present a robust, fully automated pipeline for aortic shape analysis from cardiac MRI, combining deep learning and statistical techniques across segmentation, 3D surface reconstruction, and mesh registration. We benchmark leading segmentation models—including nn-UNet, TotalSegmentator, and MedSAM2—highlighting the effectiveness of domain-specific training and transfer learning on a curated dataset. Following segmentation, we reconstruct high-quality 3D meshes and introduce a DL-based mesh registration method that directly optimises vertex displacements. This approach significantly outperforms classical rigid and non-rigid methods in geometric accuracy and anatomical consistency. Using the registered meshes, we perform statistical shape analysis on a cohort of 599 healthy subjects. Principal Component Analysis reveals dominant modes of aortic shape variation, capturing both global morphology and local structural differences under rigid and similarity transformations. Our findings demonstrate the advantages of integrating traditional geometry processing with learning-based models for anatomically precise and scalable aortic analysis. This work lays the groundwork for future studies into pathological shape deviations and supports the development of personalised diagnostics in cardiovascular medicine.