The study of parametrized families of curves and surfaces is a classical topic of great importance in applied science and engineering. It suffers from a lack of rigorous theory and of theorems. The usage of technology such as Computer Algebra Systems (CAS) may give this mathematical domain a new role in STEM education.
Two central features of technology are used here: the graphical register of the CAS, and the algebraic algorithms provide automated proof of the results.
Their respective roles are different when working in 2D and in 3D. In this last case, the joint influence of visualization problems and the non-availability of certain tools such as a slider push the central aspects of the study towards automated proofs.