IDENTITY OF THE TUBE

The presence of a right tubular surface in the Euclidean vector space R3 is sufficient to establish a straightforward metric relation, valid for various tube profiles. First, an implicit Cartesian equation of a prismatic tube with a parallelogram cross-section is defined and proven (Theorem 1) using the concept of the signed volume of a tetrahedron. The main metric relation is then established for right tubular surfaces (Theorem 2): it aggregates the quadratures of the traces left by a right tube intersecting the coordinate planes of an orthonormal Cartesian reference frame. Furthermore, a second tubular identity (Theorem 3) along with a two-dimensional consequence (Theorem 4) are presented.Keywords : volume signé du tétraèdre, tube prismatique, identité du tube, équation du parallélogramme, équation du tube, théorème de Green-Riemann.MSC2020 : 51N20, 68W30, 26A06.