Abstract
We apply the Discrete Fourier Transform to the construction of B-Spline curves to gain more insight into their structure. As a B-Spline curve is determined by its control polygon, this analysis is intimately linked to the Fourier analysis of the control polygon. To do this we apply Fast Fourier transform (FFT) algorithm to the structure of B-Spline curve and its rational form. We get inner structure of original B-Spline curve in the transform domain again in the form of B-Spline curve, having control polygon as regular or star polygon. Using the technique mentioned in the paper we get the same curve without change of shape in the transformed case of polygon points. We also extend the idea for the interval form of B-Spline Curve.
Highlights
This paper is concerned with analysis of B-Spline curve using Fast Fourier Transform
As a B-Spline curve is determined by its control polygon, by transforming the points of control polygon, we transform the B-Spline curve from one domain to another using Discrete Fourier Transform (DFT) through algorithm Fast Fourier Transform (FFT)[1,2,5]
We show that the obtained base B-Spline curve have very nice properties with control polygon points as powers of twiddle factor of DFT[3]
Summary
This paper is concerned with analysis of B-Spline curve using Fast Fourier Transform. One classical and mathematically very interesting method is to use trigonometric functions This is the basis for the theory of Fourier analysis. Our main aim in this paper is to analyze the B-Spline curve to study the inner structure of curve using Fourier Transform. 3 we analyze the B-Spline curve and its rational form using DFT. We show that the obtained base B-Spline curve have very nice properties with control polygon points as powers of twiddle factor of DFT[3]. The case for, open B-Spline curve having order equal to number of polygon points gives the inner structure of Bézier curve as mentioned by Paul Barry [3]. In the last section we extend the idea for the interval form of B-Spline curve where the control polygon is defined by rectangular or circular interval to capture the uncertainty characteristic of design
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