Applied Mathematics & Information Sciences

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Geometry and kinematics have been intimately connected in their historical evolution and, although it is currently less fashionable, the further development of such connections is crucial to many computer-aided design and manufacturing. In this paper, the evolution of the translation surfaces and their generating curves in E3 are investigated. Integrability conditions of the Gauss-Weingarten equations are obtained. Kinematics of moving frame fields associated to these surfaces are described. The evolution equations of the Christoffel symbols, the second fundamental quantities and Gauss-Codazzi equations for the motion are established. Thus, the evolution equations of the curvatures in terms of their intrinsic geometric formulas are derived. Two examples of translation surfaces and their motions are considered and plotted.