Hermite interpolation by piecewise rational surface

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Abstract

In this paper a bivariate rational Hermite interpolation is developed to create a space surface using both function values and the first-order partial derivatives of the function being interpolated as the interpolation data. The interpolation function has a simple and explicit rational mathematical representation with parameters. There are two schemes to construct the interpolation functions, an example shows that both of them approximate the function being interpolated very well. The basis of this interpolation is derived and the interpolating function can be C1 in all the interpolation region when the parameters satisfy a simple condition. When a patch of the interpolating surface does not satisfy the need of design, say it is too high or too low at a point and its neighbourhood, the values of the interpolating function in these points need to be decreased or increased under the condition that the interpolating data are not changed. For this, sufficient conditions are derived, and it is shown that this can be done just by selecting suitable parameters in the interpolation.

Introduction

Spline interpolation is a useful and powerful tool in computer-aided geometric design. Mathematicians have studied many kinds of spline interpolation methods to meet the needs of the ever-increasing model complexity and incorporate manufacturing requirements, such as polynomial spline, triangular spline, β-spline, Box spline, vertex spline and others [1], [2], [3], [4], [5], [6], [7], [8]. On the other hand, one of the disadvantages of the spline method is its uniqueness; it is not possible to do local modifications while leaving invariant interpolating data. In recent years, the univariate rational spline interpolation with parameters has been constructed [9], [10], [11], [12], [13], [14], [15]. Those kinds of interpolation spline not only have simple mathematical representation, but they can be used for the modification of local curves by selecting suitable parameters under the condition that the interpolating data are not changed. In this case, the uniqueness of the interpolating curves for the given interpolating data becomes the uniqueness of the interpolating curves for the given interpolating data and the parameters. Motivated by the univariate rational spline interpolation, the bivariate rational interpolation spline with parameters, based on the function values and the partial derivative values, has been studied in [16], but those interpolation surfaces are not smooth at the joints of the two patches. To improve this, this paper will create a kind of bivariate rational Hermite interpolation which will be C1 in the whole interpolation region.

Similar as in the univariate interpolation case, since there are parameters in the interpolation function, the interpolating surface varies as the parameters change. So, the variation of the parameters makes the modification of the interpolating surface possible under the condition that the interpolation data are not changed. If a patch of the interpolating surface is too high or too low at a point and its neighbourhood, by adjusting the parameters, the surface can be constrained to be “down” or “up”, so the shape of the interpolating surface can be modified to the wanted shape. It is called the shape control method by parameters. The problem is how to select the suitable parameters? There will be an answer in this paper.

The paper is arranged as follows. In Section 2, the new bivariate rational spline based on function values and the first-order partial derivatives with parameters is constructed, it is called the bivariate rational Hermite interpolation. Section 3 deals with the smoothness of the interpolating surfaces. When the knots are equally spaced in the x-direction and some of the parameters satisfy a simple condition, the interpolating function is C1 in the interpolating region. Section 4 is about the basis of this bivariate interpolation. The shape control method will be given in Section 5. Section 6 is about numerical examples.

Section snippets

Interpolation

Let Ω : [a, b; c, d] be the plane region, and {(xi,yj,fi,j,fi,jx,fi,jy),i=1,2,,n;j=1,2,,m} be a given set of data points, where a = x1 < x2 <  < xn = b and c = y1 < y2 <  < ym = d are the knot sequences, and fi,j,fi,jx,fi,jy represent f(xi,yj),f(xi,yj)x,f(xi,yj)y, respectively. Let hi = xi+1  xi, lj = yj+1  yj, and for any point (x, y)  [xi, xi+1; yj, yj+1] in the (x, y)-plane, let θ=x-xihi and η=y-yjlj. First, for each y = yj, j = 1, 2,  , m, construct the x-direction interpolating curve Pi,j(x) in [xi,xi+1] [14]; this

The smoothing conditions

This section deals with the smoothing conditions of the interpolating function Pi,j(x, y) defined by (2). The rational interpolating function Pi,j(x) defined by (1) has continuous first-order derivative when x  [x1, xn], so it is easy to see that the bivariate interpolating function Pi,j(x, y) defined by (2) has continuous first-order partial derivatives Pi,j(x,y)y and Pi,j(x,y)x in the interpolating region [x1, xn; y1, ym] except Pi,j(x,y)x for every y  [yj, yj+1], j = 1, 2,  , m  1, at the points (xi

Basis of the interpolation

For the interpolation defined by (1), it is easy to see that Pi,j(x) can be rewritten asPi,j(x)=ω0,0(θ)fi,j+ω1,0(θ)fi+1,j+ω0,1(θ)hifi,jx+ω1,1(θ)hifi+1,jx,whereω0,0(θ)=(1-θ)2((1+θ)αi,j+θ)(1-θ)αi,j+θ,ω1,0(θ)=θ2((1-θ)αi,j+(2-θ))(1-θ)αi,j+θ,ω0,1(θ)=θ(1-θ)2αi,j(1-θ)αi,j+θ,ω1,1(θ)=-θ2(1-θ)(1-θ)αi,j+θ,and they satisfyωi,j(k)(θ)=1,θ=i,andk=j,0,otherwise.The set {ωi,j(θ), i, j = 0, 1} are called the basis of the interpolation (1). It is obvious that when αi,j = 1, the terms {ωi,j(θ), i, j = 0, 1} are the

Shape control of the interpolating surface

This section deals with a local control of the interpolating surface. Let Ω be the interpolation region, [xi, xi+1; yj, yj+1]  Ω, and let Pi,j(x, y) be the interpolating function defined by (2). Iffi,jy>0,fi+1,jy>0,andfi,j+1y<0,fi+1,j+1y<0,the interpolating surface on part of the region [xi, xi+1; yj, yj+1] may protrude at some points, for example, say at the point Pi,j(x, y), where xi < x < xi+1, yj < y < yj+1. How the suitable parameters αi,j,αi,j+1 and αi,j can be selected that the value of Pi,j

Numerical example

Example 2

Assume the function being interpolated, f(x, y), is defined on [1, 2; 1, 2], and the data are given in the following Table 2.

For this interpolation, the parameters α1,1,α1,2 and β1,1 can be any positive real numbers. For instance, let α1,1=α1,2=1.1, β1,1 = 0.9, and let P(x, y) be the interpolating function defined by (2) in [1, 2; 1, 2] with the data and parameters above. Fig. 1 is the graph of P(x, y), and some values of P(x, y) are given in Table 3.

From Table 3, some values of P(x, y) are greater than

Conclusions

  • There are two schemes of the interpolation for the given interpolation data, one is interpolating from the x-direction first, another is from the y-direction first. The interpolation uniqueness is held for both of them, and both of the two interpolation functions approximate the function being interpolated very well as shown in Example 1. Thus, the interpolation process can begin from either of the two variables.

  • The interpolating functions have explicit expressions with parameters αi,j and βi,j

Acknowledgement

The supports of the National Nature Science Foundation of China and the Nature Science Foundation of Shandong Province of China are gratefully acknowledged.

References (16)

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