Abstract
In this paper, we study to express the theory of curves including a wide section of Euclidean geometry in terms of dual vector calculus which has an important place in the three -dimensional dual space $\mathbb{D}^{3}$. In other words, we study $DAW(k)$-type curves $\left( 1\leq k\leq 3\right)$ by using Bishop frame defined as alternatively of these curves and give some of their properties in $\mathbb{D}^{3}$. Moreover, we define the notion of evolutes of dual spherical curves for ruled surfaces. Finally, we give some examples to illustrate our findings.
Highlights
We study to express the theory of curves including a wide section of Euclidean geometry in terms of dual vector calculus which has an important place in the three -dimensional dual space D3
We define the notion of evolutes of dual spherical curves for ruled surfaces
The analytical tools in the study of 3-dimensional kinematics and differential geometry of ruled surfaces are based on dual vector calculus
Summary
The analytical tools in the study of 3-dimensional kinematics and differential geometry of ruled surfaces are based on dual vector calculus. A differentiable curve on dual unit sphere in D3 represents a ruled surface in E3 [2] Ruled surfaces are those surfaces which are generated by moving a straight line continuously in the space [3]. In [8], the authors studied DAW (k)− type curves on the dual unit sphere using Frenet frame. We investigate the DAW (k)−type curves and give the curvature conditions of these curves using a Bishop frame which has many properties that make it ideal for mathematical research It has applications in the area of biology and computer graphics, for example it may be possible to compute information about the shape of sequences of DNA using a curve defined by Bishop frame. It may provide a new way to control virtual cameras in computer animations
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