Quaternionic approach of slant ruled surfaces. (English) Zbl 1492.53008
Summary: In this study, we show that the quaternion product of quaternionic operator whose scalar part is a real parameter and vector part is a curve in \(\mathbb{R}^3\) and a spherical striction curve represents a slant ruled surface in \(\mathbb{R}^3\) if the vector part of the quaternionic operator is perpendicular to the position vector of the spherical striction curve. In \(\mathbb{R}^3\), exploitting this operator, we define the slant ruled surface corresponding to the natural lift curve on the subset of the tangent bundle of unit 2-sphere, \(T\bar{M}.\) Then, we classify \(\vec{\bar{q}}\)-, \(\vec{\bar{h}}\)- and \(\vec{\bar{a}}\)- slant ruled surfaces. Furthermore, these surfaces can also be expressed with 2- parameter homothetic motions. Finally, we give the geometric interpretations of this operator with some examples.
MSC:
| 53A05 | Surfaces in Euclidean and related spaces |
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