Nonlinear Subdivision Schemes with Free Parameters Based on Circle Average

To make the subdivision curves more controllable,a class of nonlinear subdivision schemes with free parameters by circle average was proposed.First,a new nonlinear weighted average of two points and their corresponding normals was introduced,which was called circle average.Next,we presented a new nonlinear 4-point interpolatory scheme and a new 3-point approximating scheme with a free parameter,respectively.This was done by replacing the weighted binary arithmetic means in a linear scheme,expressed in terms of repeated binary average,with circle average.Finally,the convergence,circle-preserving and C1 continuity of the new subdivision schemes were analyzed.Some numerical examples show that the limit curve of the nonlinear 4-point interpolatory scheme is continuous without self-intersection when applied to a control polygon with edges of significantly different lengths.At the same time,the selection of parameters and initial normal vectors can effectively control the shape of the limit curve.It can be known from diagram of the change of curvature that the limit curves generated by the new subdivision schemes are more smoother than linear 4-point interpolatory scheme.