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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.

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# Computing knots by quadratic and cubic polynomial curves

Show Author's information Fan Zhang1,2( )Jinjiang Li1,2Peiqiang Liu1,2Hui Fan1,2
School of Computer Science and Technology, Shandong Technology and Business University, Yantai 264005, China
Co-Innovation Center of Shandong Colleges andUniversities: Future Intelligent Computing, Yantai 264005, China

## 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.

Keywords: knot, interpolation, polynomial curve, affine invariant

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Publication history
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## Publication history

Accepted: 17 June 2020
Published: 17 October 2020
Issue date: December 2020

## Acknowledgements

This work was supported in part by the following: National Natural Science Foundation of China under Grant Nos. 61602277 and 61772319, Natural Science Foundation of Shandong Province under Grant Nos. ZR2016FQ12 and ZR2018BF009, Key Research and Development Program of Yantai City under Grant No. 2017ZH065, CERNET Innovation Project under Grant No. NGII20161204, and Science and Technology Innovation Program for Distributed Young Talents of Shandong Province Higher Education Institutions under Grant No. 2019KJN042.