Conversion and evaluation for two types of parametric surfaces constructed by NTP bases

In this paper, a new generalized Ball basis, normalized totally positive (NTP) basis given by Delgado and Peña, is investigated. The conversion formulae between the basis and the Bernstein basis are derived. We also prove that these formulae not only are valuable for studying the geometric properties, such as subdivision, of the curves and surfaces constructed by this generalized Ball basis, but also can improve the computational speed of the Bézier curves and surfaces. After the Bézier surface (curve) is converted into the generalized Ball surface (curve), the time complexity for evaluation can be reduced from cubic to quadratic, of the degree of the surface (curve). However, the intrinsic property, such as shape-preserving property, is not changed. So, the generalized Ball surface and curve have a great future in application of geometric design.