Abstract
In this paper, we analyzed the problem of constructing a family of surfaces from a given spacelike (or timelike) geodesic curve. Using the Frenet trihedron frame of the curve in Minkowski space, we express the family of surfaces as a linear combination of the components of this frame, and derive the necessary and sufficient conditions for the coefficients to satisfy both the geodesic and the isoparametric requirements. Finally, examples are given to show the family of surfaces with common geodesic.