The Dual Expression of Parallel Equidistant Ruled Surfaces in Euclidean 3-Space

Abstract

In this study, we examine the dual expression of Valeontis’ concept of parallel p-equidistant ruled surfaces well known in Euclidean 3-space, according to the Study mapping. Furthermore, we show that the dual part of the dual angle on the unit dual sphere corresponds to the p-distance. We call these ruled surfaces we obtained “dual parallel equidistant ruled surfaces” and we briefly denote them with “DPERS”. Furthermore, we find the Blaschke vectors, the Blaschke invariants and the striction curves of these DPERS and we give the relationships between these elements. Moreover, we show the relationships between the Darboux screws, the instantaneous screw axes, the instantaneous dual Pfaff vectors and dual Steiner rotation vectors of these surfaces. Finally, we give an example, which we reinforce this article, and we explain all of these features with the figures on the example. Furthermore, we see that the corresponding dual curves on the dual unit sphere to these DPERS are such that one of them is symmetric with respect to the imaginary symmetry axis of the other.

The Invariants of Dual Parallel Equidistant Ruled Surfaces

“…See [41] for the real and dual Equations (2) and (3). The dual pitch length, the dual pitch angle and the drall (parameter of distribution) of the closed ruled surface → M(s) are, respectively [22],…”

“…The definition and properties of dual parallel equidistant ruled surfaces (DPERS) are also explained in [41]. In this section, the relations that will be used in the continuation of the study will be given.…”

“…where − → a i (s) and − → b i (s) are the real and dual components of the vectors − → A i (s) and − → B i (s), [41]. Additionally, φ * (s), z(s), r(s) are the perpendicular projection distances on the unit vectors − →…”

“…In Ref. [41], we defined parallel equidistant ruled surfaces (DPERS) and compared some of their geometric properties. In addition, we have shown that one of these surfaces is symmetrical to the other.…”

Notes on Curves at a Constant Distance from the Edge of Regression on a Curve in Galilean 3-Space

“…The ruled surfaces in G 3 are three types. Definitions of the ruled surfaces of type A, B, C and current studies can be viewed in [6,15,[17][18][19][20]. Our goal is to define the ruled surfaces using r λ as the base curve and r λ as the director curve, and to see if they can be developed.…”