This research aims to study Darboux sweeping surface in isotropic space I1 3 . We went through the geometric characteristics of sweeping surfaces in I1 3 . The first and second fundamental forms of the sweeping surface were evaluated. Furthermore, we investigate the mean and Gaussian curvature of the sweeping surface. We also show that the parametric curves on these surfaces are non-geodesic and non-asymptotic. Then, we derive the necessary and sufficient conditions for the sweeping surface to become a developable sweeping surface, minimal sweeping surface, and Weingarten surface. Finally, an example to illustrate the application of the results is introduced.

This research aims to study the Sweeping surface which is generated by the motion of the straight line (the profile curve) while this movement of the plane in the space is in the same direction as the normal to a cubic Bezier curve (spine curve). In geometrical modeling, sweeping is an essential and useful tool and has some applications, especially in geometric design. The idea depends on choosing a geometrical object which is the straight line, that is called the generator, and sweeping it along a cubic Bezier curve (spine curve), which is called trajectory, along the Cubic Bezier curve (spine curve) in an isotropic space has produced an Isotropic Bezier Sweeping Surfaces (IBSS). This study discusses Isotropic Bezier Sweeping Surfaces (IBSS) with the Bishop frame. We studied a special case of a surface sweep, which is the cylindrical surface resulting from a path curve that is a straight line. We have calculated the 1st fundamental and 2nd fundamental forms for this surface. The parametric description of the Weingarten Isotropic Bezier Sweeping Surfaces (IBSS) is also calculated in terms of Gaussian and mean curvatures. Mathematica 3D visualizations were used to create these curvatures. Finally, we characterized new associated surfaces according to the Bishop frame on (IBSS), such as studying minimal and developable isotropic Bezier sweeping surfaces (IBSS).