  Subdivision is a well-known and established method for generating smooth
curves and surfaces from discrete data by repeated refinements. The typical
input for such a process is a mesh of vertices. In this work we propose to
refine 2D data consisting of vertices of a polygon and a normal at each vertex.
Our core refinement procedure is based on a circle average, which is a new
non-linear weighted average of two points and their corresponding normals. The
ability to locally approximate curves by the circle average is demonstrated.
With this ability, the circle average is a candidate for modifying linear
subdivision schemes refining points, to schemes refining point-normal pairs.
This is done by replacing the weighted binary arithmetic means in a linear
subdivision scheme, expressed in terms of repeated binary averages, by circle
averages with the same weights. Here we investigate the modified
Lane-Riesenfeld algorithm and the 4-point scheme. For the case that the initial
data consists of a control polygon only, a naive method for choosing initial
normals is proposed. An example demonstrates the superiority of the above two
modified schemes, with the naive choice of initial normals over the
corresponding linear schemes, when applied to a control polygon with edges of
significantly different lengths.
