Null-frame embankment-type ruled surfaces in Minkowski 3space via the adapted null frame
Keywords:
Adapted null frame; Minkowski 3-space; null curves; embankment surfaces; lightlike ruled surfaces; Gaussian curvature; developabilityAbstract
This paper develops a new geometric framework for constructing embankment-type ruled surfaces along null (lightlike) spine curves in the Lorentzian setting of Minkowski 3-space E31. The Adapted NullFrame Embankment Method (ANFEM) introduces a specially constructed null triad {L,N,S} that replaces the classical Frenet and orthogonal modified frames in the degenerate causal regime, where standard orthonormality conditions break down. Using this frame, three families of embankment-type ruled surfaces are defined as envelopes of one-parameter families of Lorentzian cone structures aligned with null tangent directions: the NullFrame Embankment Surface (ANFEM-ES), the Null-Frame Embankment-Like Surface (ANFEM-ELS), and the Null-Frame Tubembankment-Like Surface (ANFEM-TLS). Explicit parametric forms are derived for each family, together with the corresponding coefficients of the first and second fundamental forms. Closed analytical expressions for the Gaussian curvature K and mean curvature H are obtained, from which precise differential conditions for developability and minimality are established. A comparative simulation study demonstrates that the ANFEM surfaces exhibit curvature behavior that is structurally distinct from the spacelike orthogonal modified frame (OMF) surfaces of the base construction, with the null geometry producing parabolic ruling patterns rather than hyperbolic cross-sections. Two illustrative examples employing helical and polynomial spine curves confirm the regularity of the construction and the non-degeneracy of the resulting curvature fields. The proposed framework extends the scope of Lorentzian surface theory to the lightlike regime, with prospective applications in relativistic wave-front modeling, null geodesic surface design, and computer-aided kinematic simulation along lightlike world lines.
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