In this study, we define a variational field for constructing a family of Frenet curves of the length l lying on a connected oriented hypersurface and calculate the length of the variational curves due to the ED-frame field in Euclidean 4-space. And then, we derive the intrinsic equations for the variational curves and also obtain boundary conditions for this type of curves due to the ED-frame field in Euclidean 4-space.
We know that Bonnet surfaces are the surfaces which can admit at least one non-trivial isometry that preserves the principal curvatures in the Euclidean three-dimensional space. In this study, firstly, we have examined the required conditions for the canal surfaces, which are called the special swept surfaces, to be Bonnet surfaces. After that, we have defined the Bonnet canal surfaces in the Euclidean three-dimensional space and have obtained some special results for the Bonnet canal surfaces. We have studied the Bonnet canal surfaces, which will be formed, when the curve generating the canal surface is a special curve. In addition, we have given an example of the Bonnet canal surface by considering the conditions.
The objective of this study is to define an ovaloid surface on the convex closed space-like surfaces of constant breadth when principal curvatures of these surfaces are continuous, non-vanishing functions, and to obtain some special geometrical properties of this ovaloid surface by using the radius of curvature, diameter of the surface in [Formula: see text].
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