Exchange variations of generalized dual parallel curves and surfaces

Parallel curves (or offset curves) and parallel surfaces (or offset surfaces) have a big importance for CAD/CAM, robotics, cam design and many industrial applications, especially for mathematical modelling of cutting paths milling machines. Any vector space has a corresponding dual vector space that consists of all linear functions on vector space. Dual spaces are used in mathematics such as describing measures, distributions, and Hilbert spaces. Consequently, the dual space is an important concept in functional analysis. This paper proposes a novel definition of generalized and standard dual parallel curves and surfaces. Additionally, we give some properties of generalized dual parallel curves and surfaces using this novel definition. We also express the variation of the generalized dual parallel curves, the first and second variation of area change of the standard dual parallel surfaces and the first variation of area change of the generalized dual parallel surfaces.