Automated Exploration of Envelopes and Offsets with Networking of Technologies

We explore envelopes and offsets of plane curves using automated methods, based on the usage of Computer Algebra Systems and Dynamical Geometry Systems. Work is performed using both parametric presentations and implicit equations. Implicitization involves well-known algorithms from the theory of Gröbner bases. In particular, irreducibility of the involved polynomials, whence irreducibility of the obtained curves is checked using the latest developments of the software tool GeoGebra Discovery. We conclude with comments on the 4C’s of twenty-first century Education, applied to mathematics and show how networking between technologies provides a suitable environment to apply and develop a 5th C, namely curiosity.