Geometric properties and offset surface construction for slant timelike-ruled surfaces

This work investigates slant timelike-ruled surfaces and their evolute offsets in Minkowski 3-space $\mathbb{E}_{1}^{3}$. Using the symmetry of evolute curves, we derive a parametric formulation for skew timelike-ruled surfaces and establish conditions ensuring the coaxial alignment of the central normal with the ruling direction of the corresponding offset surface. The geometric properties are examined through the Blaschke and Darboux frames, leading to curvature characteristics and fundamental invariants. Special cases, such as timelike-developable and timelike-binormal surfaces, are analyzed with illustrative examples. These findings contribute to a deeper understanding of the differential geometry of timelike-ruled surfaces and their evolute offsets in Lorentzian space.