EJMT Abstract

Canal surfaces are a special occurrence of envelopes of parameterized families of surfaces in 3D space, namely envelopes of families of spheres centered on a given space curve. When the radius of the sphere is constant, they are called tubular surfaces, or simply tubes. In every country, numerous examples can be found all around (water pipes, canals for traffic of merchandises, etc. All these can incite to study such objects or models in mathematics. Dynamic Geometry Systems are an efficient tool to model such concrete objects, as they enable both a geometric construction and an exploration of the corresponding algebraic data (equations, parametric presentation). In order to have a more powerful tool for the algebraic computations, a Computer Algebra System can be used, whence a new need to communicate between DGS and CAS. In the past, a concrete construction of such canal surfaces was viewed as an artistic creation, whose abstract basis was quite complicated to grasp. For the beginner, understanding the switch from a concrete occurrence towards abstraction (equations, etc.) was a non-trivial issue. Neither was the reverse direction. We start from a geometric description of a spinal curve and of spheres centered on it, from there we derive equations whose solutions describe the desired surface. Nowadays, these paths can be completed by a new step, from an abstract presentation with equations towards a concrete realization using 3D printing. This approach, from real world to mathematical presentation, and then back to concretization, is typical of STEAM education.