In digital dentistry, it is increasingly common to consider both teeth and soft tissue when designing treatments or visualising patient models. However, scans of teeth and tissue are usually acquired separately, and few studies have explored the joint geometry of the facial surface and dentition, despite its many potential clinical applications. This work explores how linear and multi-linear shape models of lips and teeth can bridge the gap between disparate 3D scanning modalities. We construct a multi-linear shape model from 284 CT scans. The model disentangles soft-tissue thickness from the underlying dental-arch shape. We use a novel ray-casting approach for determining tissue thickness, and obtain the dental arch shapes from segmented tooth scans in an automated manner. We demonstrate how the model can jointly fit to tooth-arch shapes and face-scans from lip scans, thereby enabling the alignment of separately acquired intra-oral and facial scans. The ray-casting approach outperforms previous work on modelling soft-tissue thickness, as it represents the lip region more accurately. The model is successfully used to estimate the tooth arch shape, and this in turn leads to a good alignment of intra-oral scans within the lip shape. Dento-facial models have many uses in the world of digital dentistry. By modelling the dependence between lip soft-tissue and the shape of the arch, we demonstrate the inference of the tooth arch shape for unseen lips. Further work can expand the model to include appearance and expression, and enable yet more applications.