Construction of Canal Surfaces With Variable Generatrix And Parallelism Plane Based on Plane’s Equiaffine Transformations

A kinematic approach to the construction of surfaces with equiaffine invariant for a volume of a compartment bounded by a surface, and a square of cross sections, parallel to a parallelism plane on the basis of the plane’s equiaffine transformations has been proposed in this paper. In particular, in the paper have been considered parametric equations of canal surfaces with a parallelism plane xOy, which sections have been transformed by elliptic, parabolic and hyperbolic rotations. A curve-prototype lies in the section z = 0, and a curve-image lies in the section z = h after transformation with rotations set parameters. The parametric equations have been obtained in a general form with randomness of the choice both a curve-prototype for canal surface construction, and functions defining a rate of rotation parameters change. Some general properties of the obtained surfaces have been considered, as well as some special cases of surfaces are considered, and trajectories of generating lines have been defined for them. In the case of parameter’s linear variation along the z-axis the elliptical rotation’s surfaces represent spiral and straight helical surfaces with constant and variable step however the generatrix trajectory envelops elliptical cone and cylinder. In all cases, the generatrices of the elliptical rotation’s surfaces are space curves. In the case of all parameters’ linear variation the parabolic rotation’s surfaces represent ruled surfaces. One particular case of parabolic rotation’s surfaces is the surface with two parallelism planes: xOy for equiaffine changing curve-section and xOz for generatrix. In the cases related to non-linear variation of only one parameter, generatrices for surfaces of parabolic and hyperbolic rotations are plane curves. Considered surfaces require further investigation with a view to their possible use in mechanisms and structures.