A basic goal of this paper is to investigate the tubular surface constructed by the spherical indicatrices of any spatial curve in the Euclidean $3-$ space. This kind of tubular surface is designed for the alternative moving frame $\{N,C,W\}$ in conjunction with finding a relationship between the tubular surfaces and their special curves, such as geodesic curves, asymptotic curves, and minimal curves. The minimal curve $\gamma $ on a surface is defined by the property that its fundamental coefficients satisfy Eq. (3.7) along the curve $\gamma $. At the end of this article, we exemplify these curves on the tubular surfaces with their figures using the program Mathematica.