A \(C^2\) four-point subdivision scheme with fourth order accuracy and its extensions. (English) Zbl 1080.65526

Dyn, Nira; Floater, Michael S.; Hormann, Kai

Dæhlen, Morten (ed.) et al., Mathematical methods for curves and surfaces: Tromsø 2004. Sixth international conference on mathematical methods for curves and surfaces, celebrating the 60th birthday of Tom Lyche, Tromsø, Norway, July 1–6, 2004. Brentwood, TN: Nashboro Press (ISBN 0-9728482-4-X/hbk). Modern Methods in Mathematics, 145-156 (2005).

Summary: We present a new four-point subdivision scheme that generates \(C^2\) curves. It reproduces cubic polynomials, has a basic limit function with support \([-4, 3]\), and is close to being interpolatory. The refinement rule is based on local cubic interpolation, followed by evaluation at \(1/4\) and \(3/4\) of the refined interval. We investigate the approximation properties of this four-point scheme and extend it to a new family of \(2n\)-point schemes. The performance of the new schemes is demonstrated by several examples.
For the entire collection see [Zbl 1065.65003].

MSC:

65D18

Numerical aspects of computer graphics, image analysis, and computational geometry

Keywords:

four-point subdivision scheme; curves; refinement; cubic interpolation