Abstract
We present a general formula to generate the family of odd-point ternary approximating subdivision schemes with a shape parameter for describing curves. The influence of parameter to the limit curves and the sufficient conditions of the continuities from C0 to C5 of 3- and 5-point schemes are discussed. Our family of 3-point and 5-point ternary schemes has higher order of derivative continuity than the family of 3-point and 5-point schemes presented by [Jian-ao Lian, On a-ary subdivision for curve design: II. 3-point and 5-point interpolatory schemes, Applications and Applied Mathematics: An International Journal, 3(2), 2008, 176-187]. Moreover, a 3-point ternary cubic B-spline is special case of our family of 3-point ternary scheme. The visual quality of schemes with examples is also demonstrated.
Highlights
Subdivision schemes are important and powerful tools for generation of smooth curves and surfaces from a set of control points by means of iterative refinement
We present a general formula to generate the family of odd-point ternary approximating subdivision schemes with a shape parameter for describing curves
If the limit curve/surface approximate the initial control polygon and that after subdivision, the newly generated control points are not in the limit curve/ surface, the scheme is said to be approximating. It is called interpolating if after subdivision, the control points of the original control polygon and the new generated control points are interpolated on the limit curve/surface
Summary
Subdivision schemes are important and powerful tools for generation of smooth curves and surfaces from a set of control points by means of iterative refinement. Khan and Mustafa [3] offered ternary six-point interpolating subdivision scheme. Lian [15] has introduced a -ary 3-point and 5-point interpolating schemes for arbitrary odd integer a 3. Lian [16] offered 2m -point and (2m 1) -point interpolating a -ary schemes for curve design. Intervals are too narrow to provide freedom for curve designing This motivates us to present the family of odd-point ternary schemes with high smoothness and more degree of freedom for curve designing.
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