The local variational theory of elliptic LCSs targets material surfaces that locally maximize material shear over the finite time interval  of interest. This means that at initial point each point  of an elliptic LCS , the tangent space  is the plane along which the local Lagrangian shear  is maximal (cf. Fig 7).
Introducing the two-dimensional shear vector field
and the three-dimensional shear normal vector field
the criteria for two- and three-dimensional elliptic LCSs can be summarized as follows:
For 3D flows, as in the case of hyperbolic LCSs, solving the Frobenius PDE can be avoided. Instead, one can construct intersections of a tubular elliptic LCS with select 2D planes, and fit a surface numerically to a large number of these intersection curves. As for hyperbolic LCSs above, let us denote the unit normal of a 2D plane  by . Again, the intersection curves of elliptic LCSs with the plane  are normal to both  and to the unit normal  of the LCS. As a consequence, an intersection curve  satisfies the reduced shear ODE
whose trajectories we refer to as reduced shear lines. (Strictly speaking, the reduced shear ODE 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 tubular elliptic LCSs with  are limit cycles of the reduced shear ODE. Determining such limit cycles in a smooth family of nearby  planes, then fitting a surface to the limit cycle family yields a numerical approximation for 2D shear surface. A three-dimensional example of this local variational computation of an elliptic LCS is shown in Fig. 11.