In this work, we derive the Euler-Lagrange equation for an elastic line which is lying on a pseudo-hypersurface in pseudo-Euclidean spaces E-v(n). Following this, we check the solutions which depend on the boundary conditions whether they are geodesic on a pseudo-hypersurface or not. The relaxed elastic line on a pseudo-hyperplane, a pseudo-hypersphere, and pseudo-hyperbolic space is a geodesic. However, the relaxed elastic line on a pseudo-hypercylinder, is a space-like geodesic. (c) 2005 Elsevier Inc. All rights reserved.