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It is well-known that the standard level set advection equation does not preserve the signed distance property, which is a desirable property for the level set function representing a moving interface. Therefore, reinitialization or redistancing methods are frequently applied to restore the signed distance property while keeping the zero-contour fixed. As an alternative approach to these methods, we introduce a modified level set advection equation that intrinsically preserves the norm of the gradient at the interface, i.e. the local signed distance property. Mathematically, this is achieved by introducing a carefully chosen source term being proportional to the local rate of interfacial area generation. The introduction of the source term turns the problem into a non-linear one. However, we show that by discretizing the source term explicitly in time, it is sufficient to solve a linear equation in each time step. Notably, without further adjustment, the method works in the case of a moving contact line. This is a major advantage since redistancing is known to be an issue when contact lines are involved. We provide a first implementation of the method in a simple first-order upwind scheme in both two and three spatial dimensions.