Abstract
Two extensions of the quadratic nonuniform B-spline curve with local shape parameter series, called the W3D3C1P2 spline curve and the W3D4C2P1 spline curve, are introduced in the paper. The new extensions not only inherit most excellent properties of the quadratic nonuniform B-spline curve but also can move locally toward or against the fixed control polygon by varying the shape parameter series. They are C1 and C2 continuous separately. Furthermore, the W3D3C1P2 spline curve includes the quadratic nonuniform B-spline curve as a special case. Two applications, the interpolation of the position and the corresponding tangent direction and the interpolation of a line segment, are discussed without solving a system of linear functions. Several numerical examples indicated that the new extensions are valid and can easily be applied.
Highlights
Introduction e nonuniform polynomialB-spline curve has been popularly applied in computer-aided geometric design (CAGD) [1]
Before we apply the two extensions of the quadratic nonuniform B-spline curve to interpolation, we present the position and the corresponding tangent direction of them at u ui+1
U (u3, u3, u3, u4, · · ·, un, un+1, un+1, un+1) can be used to interpolate the first control point P1 and the last control point Pn. e corresponding tangent direction at P1 or Pn is parallel to P1P2 or Pn−1Pn, respectively
Summary
Given a nonuniform knot sequence ui+i ∞−∞, let · · · < u0 < u1 < · · · < un−1 < un < · · ·, where we refer to U E piecewise linear transformation function t(u) can transform the point in the interval [ui, ui+1) to the one in the unit interval [0, 1) as follows:. With the help of the linear transformation function t(u), two kinds of piecewise polynomial basis functions with local shape parameter series are defined as follows. +iN ∞−I∞,i(aun)d+i ∞β−∞i+i ∞w−∞ith two and local shape parameter the W3D4C2P1 basis functions NII,i(u)+i ∞−∞ with one local shape parameter series ci+i ∞−∞ are defined to be the following functions: Ne,i(u). Ne0,i(t(u)), Ne1,i(t(u)), Ne2,i(t(u)), 0, u ∉ ui, u ∈ ui, ui+1, u ∈ ui+1, ui+2, u ∈ ui+2, ui+3, ui+3,
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