Thread-parallelisation and single-instruction multiple data (SIMD) ”vectorisation” of software components in HEP computing has become a necessity to fully benefit from current and future computing hardware. In this context, the Geant-Vector/GPU simulation project aims to re-engineer current software for the simulation of the passage of particles through detectors in order to increase the overall event throughput. As one of the core modules in this area, the geometry library plays a central role and vectorising its algorithms will be one of the cornerstones towards achieving good CPU performance. Here, we report on the progress made in vectorising the shape primitives, as well as in applying new C++ template based optimisations of existing code available in the Geant4, ROOT or USolids geometry libraries. We will focus on a presentation of our software development approach that aims to provide optimised code for all use cases of the library (e.g., single particle and many-particle APIs) and to support different architectures (CPU and GPU) while keeping the code base small, manageable and maintainable. We report on a generic and templated C++ geometry library as a continuation of the AIDA USolids project. The experience gained with these developments will be beneficial to other parts of the simulation software, such as for the optimisation of the physics library, and possibly to other parts of the experiment software stack, such as reconstruction and analysis.