Special Space Curves Characterized by $\det(α^{(3)}, α^{(4)}, α^{(5)}) = 0$

Abstract

By using the facts that the condition$\det(α^{(1)}, α^{(2)}, α^{(3)}) = 0$ characterizes a plane curve and the condition $\det(α^{(2)}, α^{(3)}, α^{(4)}) = 0$ characterizes a curve of constant slope, we present special space curves characterized by the condition $\det(α^{(3)}, α^{(4)}, α^{(5)}) = 0$, in different approaches. It is shown that the space curve is Salkowski if and only if $\det(α^{(3)}, α^{(4)}, α^{(5)}) = 0$. The approach used in our investigation can be useful in understanding the role of the curves characterized by $\det(α^{(3)}, α^{(4)}, α^{(5)}) = 0$ in differential geometry.