  Under mean radius of curvature flow, a closed convex surface in Euclidean
space is known to expand exponentially to infinity. In the 3-dimensional case
we prove that the oriented normals to the flowing surface converge to the
oriented normals of a round sphere whose centre is determined by the initial
surface.
  To prove this we show that the oriented normal lines, considered as a surface
in the space of all oriented lines, evolve by a parabolic flow which preserves
the Lagrangian condition. Moreover, this flow converges to a holomorphic
Lagrangian section, which form the set of oriented lines through a point.
  The coordinates of this centre point are projections of the support function
into the first non-zero eigenspace of the spherical Laplacian and are given by
explicit integrals of initial surface data.
