In computer-aided geometric design (CAGD), the Bernstein basis was extended to the B-spline basis through knot vectors and recursive construction, shifting from a global polynomial form to a locally supported piecewise representation. The Ball basis, composed of quadratic and cubic polynomials, is similar to the Bernstein basis of degree 3. This paper proposes a generalization for a spline-type and creates a piecewise polynomial extension, called the Ball-Spline basis, consisting of symmetric polynomial segments of degrees 2, 3, 3, and 2, arranged in a symmetric structure with the highest continuity order—C1 continuity between the quadratic and cubic segments, and C2 continuity between the two cubic segments. The cubic basis is constructed by multi-order spline technology and generated to higher degrees by an integral method. Compared with the B-spline basis, the proposed Ball-Spline basis shares its fundamental properties, such as positivity, normality, and local support, and generates design curves with fewer control points under the same approximation accuracy in certain examples. Thecurves generated by the Ball-Spline basis functions exhibit numerical stability under knot perturbations and admit interpretable geometric and physical properties.
Wang et al. (Sun,) studied this question.