A generalized curve subdivision scheme of arbitrary order with a tension parameter

This article presents a generalized subdivision scheme of arbitrary order with a tension parameter for curve design. The scheme is built upon refinement of a family of generalized B-splines that unify classic B-splines with algebraic-trigonometric B-splines and algebraic-hyperbolic B-splines. The scheme of order k   produces Ck−2Ck−2-continuous limit curves representing such splines. Many known subdivisions are special cases of the proposed subdivision scheme. By assigning an appropriate initial tension parameter, many analytic curves commonly used in engineering applications, such as Lissajous curves, conics, trigonometric function curves, hyperbolic function curves, catenary curves and helixes, etc., can also be exactly defined under the generalized subdivision scheme. Numerous examples are also provided to illustrate how the initial tension parameter and the control points are assigned for reproducing such analytic curves.

Research highlights
► The limit curve can be adjusted by changing a tension parameter.
► The scheme of order k   produces a family of uniform generalized B-spline curves with Ck−2Ck−2 continuity.
► Many known subdivision schemes built-upon uniform B-spline curves, uniform algbraic-trigonometric (UAT) B-spline curves and uniform algebraic-hyperbolic (UAH) B-spline curves are properly unified under the proposed generalized subdivision scheme.
► Analytic curves, such as Lissajous curves, conics, helixes, etc., can also be produced as special cases.
► The scheme can also generate special circular splines.