AN EXAMINATION OF THE CONDITION UNDER WHICH A CONCHOIDAL SURFACE IS A BONNET SURFACE IN THE EUCLIDEAN 3-SPACE

In this study, we examine the condition of the conchoidal surface to be a Bonnet surface in Euclidean 3-space. Especially, we consider the Bonnet conchoidal surfaces which admit an infnite number of isometries. In addition, we study the necessary conditions which have to be fulflled by the surface of revolution with the rotating curve <em>c</em>(<em>t</em>) and its conchoid curve <em>c_d</em>(<em>t</em>) to be the Bonnet surface in Euclidean 3-space.

Dual curves associated with the Bonnet ruled surfaces

An interest problem arises to determine the surfaces in the Euclidean three space, which admit at least one nontrivial isometry that preserves the principal curvatures. This leads to a class of surface known as a Bonnet surface. The intention of this study is to examine a Bonnet ruled surface in dual space and to calculate the dual geodesic trihedron of the dual curve associated with the Bonnet ruled surface and derivative equations of this trihedron by the dual geodesic curvature. Also, we find that the dual curvature, the dual torsion for the dual curves associated with the Bonnet ruled surface which are different from any dual curves. Moreover, some examples are obtained about the Bonnet ruled surface.

On Backlund transformations of surfaces by extended Harry-Dym flow

The present paper deals with the introduction of Backlund transformations by extended Harry-Dym flow and with the aid of the extended version of the Riccati mapping method is obtained new solutions. Then, we give the Backlund transformation of the Schrodinger flow and obtain its the Bonnet surface. In finally, results obtained with the mathematical model are evaluated by applying to mathematica.

Timelike Tangent Developable Surfaces and Bonnet Surfaces

A criterion was given for a timelike surface to be a Bonnet surface in 3-dimensional Minkowski space by Chen and Li, 1999. In this study, we obtain a necessary and sufficient condition for a timelike tangent developable surface to be a timelike Bonnet surface by the aid of this criterion. This is examined under the condition of the curvature and torsion of the base curve of the timelike developable surface being nonconstant. Moreover, we investigate the nontrivial isometry preserving the mean curvature for a timelike flat helicoidal surface by considering the curvature and torsion of the base curve of the timelike developable surface as being constant.