Evolution of Space Curves Using Type-3 Bishop Frame

Document Type : Research Articles

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Department of Mathematics Faculty of Science, Assiut University, Assiut, Egypt

Abstract

In this paper, Firstly, we introduce a new version of Bishop frame using a common vector field as normal vector field of a regular curve and call this frame as type-3 Bishop frame. Geometric visualization for both Frenet frame and type-3 Bishop frame are plotted. Secondly, time evolution equation for type-3 Bishop frame and type three bishop curvatures are obtained and we plotted the evolving curve via the numerical integration of type-3 Bishop Serret Frenet equations. Finally, surfaces generated by evolving a regular space curve in $\mathbb{R}^3$ as it evolves over time with arbitrary velocities attached to type-3 Bishop frame are obtained. We plotted these surfaces via numerical integration of Gauss-Weingarten equations.

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