Quaternionic Approach of Canal Surfaces Constructed by Some New Ideas

Abstract

A canal surface is a surface constructed as the envelope of a family of spheres with the parametric (non constant radii) radii r(s) and a space curve $${\alpha (s)}$$ called its center. If the radii of the generating spheres are constant r then the canal surface is called tubular surface such as right circular cylinder, torus, right circular cone, surface of revolution. The quaternions are number systems generalized by the complex numbers. Irish mathematician Hamilton (Lond Edinb Dublin Philos Mag J Sci 25(3):489–495, 1844) discovered them in 1843 and applied to mechanics in three-dimensional space. Many laws of curves and surfaces in differential geometry used quaternions such as quaternionic helices (Coken and Tuna in Appl Math Comput 155:373–389, 2004; Gok et al. in Appl Clifford Algebras 21:707–719, 2011; Kahraman et al. in Appl Math Comput 218:6391–6400, 2012) and canal surfaces (Aslan and Yayli in Adv Appl Clifford Algebras, 2015; Babaarslan and Yayli in ISRN Geom 2012:Article ID 126358, p 8, 2012). This paper has two purposes. The first purpose is to form canal surfaces whose centers are spherical indicatrices of a spatial curve with a new idea in terms of alternative moving frame $${\{N,C,W\}}$$ The second purpose is to obtain canal surfaces by using the quaternion product and matrix representation. Moreover, we give some related examples with their figures.