B-Spline Curve Fitting with Normal Constrains in Computer Aided Geometric Designed
We present an algorithm for fitting data points with normal constrains in Computer Aided Geometric Designed. Different from previous methods which interpolate the normal vectors accurately, the purpose of our method is to approximate these normal vectors.
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College of Mathematics and Computer Science, Jiangxi Science and Technology Normal University, Nanchang, Jiangxi, China 2 JiangXi Normal University Science and Technology College, Nanchang 330000, Jiangxi, China [email protected]
Abstract. We present an algorithm for fitting data points with normal constrains in Computer Aided Geometric Designed. Different from previous methods which interpolate the normal vectors accurately, the purpose of our method is to approximate these normal vectors. We make the normal vectors on the fitting curve are close to the normal constrains with their corresponding parameters. According to approximate the normal vectors constrains, we can control the shape of fitting curve. Our purpose is to minimize the square sum of inner products of normal vectors on fitting curve and normal constrain vectors. In interpolation, we need to minimize the square sum of the inner products and interpolate these data points, so we use Lagrange multiplier method to solve the control points. In approximation, the inner product terms is made as the fair terms, we minimize the weighted sum of the distance term and fair term to get the control points of the fitting curve. Finally, an example of this algorithm is demonstrated. Keywords: B-spline curve fitting Local fitting Whole fitting Parameterization Computer Aided Geometric Designed
1 Introduction B-spline curve fitting for ordered data points is the core problem in reverse engineering. Interpolation and approximation are two basic types of fitting technique. Interpolation means that a curve passing through these ordered points, and approximation means that a curve closing these data points. The B-spline curve interpolation to data points is very well-considered in CAGD, but some problems such as parameterization, shape controlling have not been well solved. B-spline curve interpolates data points as well as some constrains to preserve the shape [1–3]. Shape preserving fitting techniques with B-spline curves/surfaces are explained in Hoschek and Lasser [3], Piegl and Tiller [1], Hoschekand Jüttler [2]. B-spline curve not only interpolating data points but also the tangent vectors or normal vectors are used in many case to control the shape of fitting curve, then this method is a research hotspot in reverse engineering. Piegl and Tiller [1] have first proposed a method to interpolate data points with these the first order derivatives. Each data point corresponds to an equation, meanwhile © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 M. Atiquzzaman et al. (Eds.): BDCPS 2020, AISC 1303, pp. 1282–1289, 2021. https://doi.org/10.1007/978-981-33-4572-0_184
B-Spline Curve Fitting with Normal Constrains
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each derivatives corresponds to an equation. There are 2(n + 1) equations to solve this point-tangent interpolation, and we will obtain 2(n + 1) control points of the interpolating curve. An explosion of the control points is a waste of time and space. Shiochi [5, 6] have also proposed a
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