text: The local variational theory of hyperbolic LCSs builds on their original definition as strongest repelling or repelling material surfaces in the flow over the time interval . At an initial point  , let 
  denote a unit normal to an initial material surface  (cf. Fig. 6). By the invariance of material lines, the tangent space  is mapped into the tangent space of 
  by the linearized flow map . At the same time, the image of the normal  normal under  generally does not remain normal to .
  Therefore, in addition to a normal component of length , the advected normal also develops a tangential component of length  (cf. Fig. 7).
  If , then the evolving material surface  strictly repels nearby trajectories by the end of the time interval . Similarly, 
  signals that  strictly attracts nearby trajectories along its normal directions. A repelling (attracting) LCS over the interval  can be defined as a material surface  whose net repulsion  is pointwise maximal (minimal) with respect to perturbations
  of the initial normal vector field . As earlier, we refer to repelling and attracting LCSs collectively as hyperbolic LCSs.
  Solving these local extremum principles for hyperbolic LCSs in two and three dimensions yields unit normal vector fields to which hyperbolic LCSs should everywhere be tangent. The existence of such normal surfaces also requires a Frobenius-type integrability condition in the three-dimensional case. All these results can be summarized as follows:
  Repelling LCSs are obtained as most repelling shrink lines, starting from local maxima of . Attracting LCSs are obtained as most attracting stretch lines, starting from local minima of . These starting points serve are initial positions of exceptional saddle-type trajectories in the flow. An example of the local variational computation of a repelling LCS is shown in FIg. 8. The computational algorithm is available in LCS Tool.
  In 3D flows, instead of solving the Frobenius PDE (see table above) for hyperbolic LCSs, an easier approach is to construct intersections of hyperbolic LCSs with select 2D planes, and fit a surface numerically to a large number of such intersection curves. Let us denote the unit normal of a 2D plane  by . The intersection curve of a 2D repelling LCS surface with the plane  is normal to both  and to the unit normal  of the LCS. As a consequence, an intersection curve  satisfies the ODE
  whose trajectories we refer to as reduced shrink lines. (Strictly speaking, this equation is not an ordinary differential equation, given that its right-hand side is not a vector field, but a direction field, which is generally not globally orientable). Intersections of hyperbolic LCSs with  are fastest contracting reduced shrink lines. Determining such shrink lines in a smooth family of nearby  planes, then fitting a surface to the curve family so obtained yields a numerical approximation of a 2D repelling LCS.