Smooth Curve Interpolation Research Articles

Shape Preserving Data Interpolation Using Rational Cubic Ball Functions

A smooth curve interpolation scheme for positive, monotone, and convex data is developed. This scheme uses rational cubic Ball representation with four shape parameters in its description. Conditions of two shape parameters are derived in such a way that they preserve the shape of the data, whereas the other two parameters remain free to enable the user to modify the shape of the curve. The degree of smoothness isC1. The outputs from a number of numerical experiments are presented.

Catmull-Rom curve fitting and interpolation equations

Computer graphics and animation experts have been using the Catmull-Rom smooth curve interpolation equations since 1974, but the vector and matrix equations can be derived and simplified using basic algebra, resulting in a simple set of linear equations with constant coefficients. A variety of uses of Catmull-Rom interpolation are demonstrated, including data point interpolation, image surround and key frame animation.

Data visualization using rational spline interpolation

A smooth curve interpolation scheme for positive, monotonic, and convex data has been developed. This scheme uses piecewise rational cubic functions. The two families of parameters, in the description of the rational interpolant, have been constrained to preserve the shape of the data. The rational spline scheme has a unique representation. The degree of smoothness attained is C 1 .

A rational cubic spline for the visualization of monotonic data: an alternate approach

A smooth curve interpolation scheme for monotonic data was developed by Sarfraz (Comput. Graph. 24(4) (2000) 509). This scheme is reviewed and a new alternate approach, which was indicated in Sarfraz (Comput. Graph. 26(1) (2002) 193), has been introduced with all details and solid proofs here. In theory, the new scheme uses the same piecewise rational cubic functions as in Sarfraz (2000), but it utilizes a rational quadratic in practice. The two families of parameters, in the description of the rational interpolant, have been constrained to preserve the monotone shape of the data. The new rational spline scheme, like the old one in Sarfraz, 2000, has a unique representation. The degree of smoothness attained is C 2 and the method of computation is robust. Both, old as well as new shape preserving methods, seem to be equally competent. This is shown by a comparative analysis. There is a trade off between the two methods as far as computational aspects are concerned. However, the new method is superior to the old method as far as accuracy is concerned. This fact has been proved by making an error analysis over the actual and computed curves.

Visualization of shaped data by a rational cubic spline interpolation

A smooth curve interpolation scheme for positive and monotonic data has been developed. This scheme uses piecewise rational cubic functions. The two families of parameters, in the description of the rational interpolant, have been constrained to preserve the shape of the data. The rational spline scheme has a unique representation. In addition to preserve the shape of positive and/or monotonic data sets, it also possesses extra features to modify the shape of the design curve as and when desired. The degree of smoothness attained is C 1.

A rational cubic spline for the visualization of monotonic data

A smooth curve interpolation scheme for monotonic data has been developed. This scheme uses piecewise rational cubic functions. The two families of parameters, in the description of the rational interpolant, have been constrained to preserve the shape of the data. The rational spline scheme has a unique representation. The degree of smoothness attained is C 2 which is more powerful than a previous C 1 method.

Smooth curve interpolation with generalized conics

Efficient algorithms for shape preserving approximation to curves and surfaces are very important in shape design and modelling in CAD/CAM systems. In this paper, a local algorithm using piecewise generalized conic segments is proposed for shape preserving curve interpolation. It is proved that there exists a smooth piecewise generalized conic curve which not only interpolates the data points, but also preserves the convexity of the data. Furthermore, if the data is strictly convex, then the interpolant could be a locally adjustable GC 2 curve provided the curvatures at the data points are well determined. It is also shown that the best approximation order is O ( h 6 ). An efficient algorithm for the simultaneous computation of points on the curve is derived so that the curve can be easily computed and displayed. The numerical complexity of the algorithm for computing N points on the curve is about 2 N multiplications and N additions. Finally, some numerical examples with graphs are provided and comparisons with both quadratic and cubic spline interpolants are also given.

Smooth curve interpolation

In a recent paper [1] a method was presented for smooth curve interpolation by means of minimising the energy of an idealised elastic beam constrained to pass through given data points. This method of obtaining the curve required an enormous amount of computation, since it involved the repeated solution of a fourth order differential equation and the minimisation of a function of many variables, the number of variables being equal to the number of data points. In this paper a much simpler method is presented for finding the minimum energy curve in which the minimisation procedure is eliminated. An ALGOL procedure to perform the curve fit is supplied.

Smooth-curve interpolation: A generalized spline-fit procedure

A method is presented for finding the smoothest curve through a set of data points. “Smoothest” refers to the equilibrium, or minimum-energy configuration of an ideal elastic beam constrained to pass through the data points. The formulation of the smoothest curve is seen to involve a multivariable boundary-value minimization problem which makes use of a numerical solution of the beam non-linear differential equation. The method is shown to offer considerable improvement over the spline technique for smooth-curve interpolation.