Binary Subdivision Schemes for Functions over Irregular Knot Sequences

For a wide class of stationary subdivision methods, we derive necessary and suucient conditions for these schemes to produce C k continuous limit curves. These stationary schemes include those arising from midpoint subdivision of irregularly-spaced knot sequences. We also describe a matrix method for computing various derivative schemes associated with such stationary schemes. x1. Introduction Subdivision is a powerful tool for creating and representing complex curved shapes. Consider the process due to Chaikin 2] depicted in gure 1. A polygon is transformed into a new polygon with twice as many segments. For this particular transformation, the vertices of the new polygon are placed 1 4 and 3 4 of the way between the old vertices. Applying this process repeatedly yields a polygon with a great number of segments that closely approximate a smooth curve. What is this smooth curve? Riesenfeld 9] shows that the curve is a uniform quadratic B-spline whose control points are the original polygon. Fig. 1. A subdivision method.