The approach of the paper is on spacelike circular surfaces in the Minkowski 3-space. A spacelike circular surface is a one-parameter family of Lorentzian circles with a fixed radius regarding a non-null curve, which acts as the spine curve, and it has symmetrical properties. In the study, we have parametrized spacelike circular surfaces and have provided their geometric and singularity properties such as Gaussian and mean curvatures, comparing them with those of ruled surfaces and the classification of singularities. Furthermore, the conditions for spacelike roller coaster surfaces to be flat or minimal surfaces are obtained. Meanwhile, we support the results of the approach with some examples.

The main result of this paper is constructing a surfaces family with the similarity of Bertrand curves in Euclidean 3–space ${\mathbb{E}}^3$. Then, by utilizing the Serret–Frenet frame, we conclude the sufficient and necessary conditions of surfaces family interpolating Bertrand curves as common asymptotic curves. Consequently, the expansion to the ruled surfaces family is also depicted. As implementations of our main results, we demonstrate some examples to confirm the method.