Computing knots by quadratic and cubic polynomial curves
- PDF / 969,580 Bytes
- 14 Pages / 612 x 808 pts Page_size
- 80 Downloads / 197 Views
Computing knots by quadratic and cubic polynomial curves Fan Zhang1,2 ( ), Jinjiang Li1,2 , Peiqiang Liu1,2 , and Hui Fan1,2 c The Author(s) 2020.
aided design, engineering, scientific computing, and computer graphics is the construction of curves and surfaces with high precision and smoothness. They require different attributes for different applications [1–6]. To meet these requirements, good interpolation techniques and parameterization methods are needed. For scientific computation and engineering application, constructing curves and surfaces with high polynomial accuracy is desirable. This paper focuses on how to determine the parameter values, or knots, for a given set of points with high precision.
Abstract A new method is presented to determine parameter values (knot) for data points for curve and surface generation. With four adjacent data points, a quadratic polynomial curve can be determined uniquely if the four points form a convex polygon. When the four data points do not form a convex polygon, a cubic polynomial curve with one degree of freedom is used to interpolate the four points, so that the interpolant has better shape, approximating the polygon formed by the four data points. The degree of freedom is determined by minimizing the cubic coefficient of the cubic polynomial curve. The advantages of the new method are, firstly, the knots computed have quadratic polynomial precision, i.e., if the data points are sampled from a quadratic polynomial curve, and the knots are used to construct a quadratic polynomial, it reproduces the original quadratic curve. Secondly, the new method is affine invariant, which is significant, as most parameterization methods do not have this property. Thirdly, it computes knots using a local method. Experiments show that curves constructed using knots computed by the new method have better interpolation precision than for existing methods.
1.2
Previous work
For a given set of data points to be interpolated, Pi = (xi , yi ), i = 1, . . . , n, the aim of parameterization is to assign a parameter or knot value ti , t1 < ti < tn , for each Pi . The interpolated curve can be seen as the path of a particle through space, while the parameter t can be regarded as time, so the parameterization gives the location of the particle at each moment of time. For the same set of data, even with the same interpolation methods, Keywords knot; interpolation; polynomial curve; constructing curves with different parameterizations affine invariant will result in a different interpolant. The choice of parameterization method will have a noticeable effect on the interpolated curve. Uniform parameterization 1 Introduction is only suitable for cases when the intervals between 1.1 Problem consecutive data points are equal. In applications, A fundamental problem in the fields of computer- three non-uniform parameterization strategies are widely used: the chord length method [7], Foley’s 1 School of Computer Science and Technology, Shandong method [8], and the centripetal method [9]. The chord Technology and
Data Loading...
