$C^{\ast }$-partner curves with modified adapted frame and their applications

In this study, the curve theory, which occupies a very important and wide place in differential geometry, has been studied. One of the most important known methods used to analyze a curve in differential geometry is the Frenet frame, which is a moving frame that provides a coordinate system at each point of the curve. However, the Frenet frame of any curve cannot be constructed at some points. In such cases, it is useful to define an alternative frame. In this study, instead of the Frenet frame that characterizes a regular curve in Euclidean space $E^3$, we have defined a different and new frame on the curve. Since this new frame is defined with the aid of the Darboux vector, it is very compatible compared to many alternative frames in application. Therefore, we have named this new frame the "modified adapted frame" denoted by $\{N^{\ast },C^{\ast },W^{\ast }\}$. Then, we have given some characterizations of this new frame. In addition to that, we have defined $N^{\ast }$-slant helices and $C^{\ast }$-slant helices according to $\{N^{\ast },C^{\ast },W^{\ast }\}$. Moreover, we have studied $C^{\ast }$-partner curves using this modified adapted frame. Consequently, by investigating applications, we have established the relationship between $C^{\ast }$-partner curves and helices, slant helices.