  This paper proposes to generalize linear subdivision schemes to nonlinear
subdivision schemes for curve and surface modeling by refining vertex positions
together with refinement of unit control normals at the vertices. For each
round of subdivision, new control normals are obtained by projections of
linearly subdivided normals onto unit circle or sphere while new vertex
positions are obtained by updating linearly subdivided vertices along the
directions of the newly subdivided normals. Particularly, the new position of
each linearly subdivided vertex is computed by weighted averages of end points
of circular or helical arcs that interpolate the positions and normals at the
old vertices at one ends and the newly subdivided normal at the other ends.
  The main features of the proposed subdivision schemes are three folds:
  (1) The point-normal (PN) subdivision schemes can reproduce circles, circular
cylinders and spheres using control points and control normals;
  (2) PN subdivision schemes generalized from convergent linear subdivision
schemes converge and can have the same smoothness orders as the linear schemes;
  (3) PN $C^2$ subdivision schemes generalizing linear subdivision schemes that
generate $C^2$ subdivision surfaces with flat extraordinary points can generate
visually $C^2$ subdivision surfaces with non-flat extraordinary points.
  Experimental examples have been given to show the effectiveness of the
proposed techniques for curve and surface modeling.
