Lizheng Lu

An explicit method for G 3 merging of two Bézier curves

This paper presents an explicit method for the G 3 merging problem of two Bezier curves. The main idea is to express the L 2 distance as a quadratic function of some parameters provided by G 3 continuity conditions. An efficient non-iterative algorithm is proposed to obtain the optimal merged curve when the L 2 distance is minimized. The uniqueness of the global minimum is also proven. This method can be applied to two adjacent curves with different degrees and has the ability to obtain satisfactory merging results by using curves of lower degree. The efficiency and accuracy of the proposed explicit method are illustrated through several comparative examples.

Gram matrix of Bernstein basis

This note presents explicit expressions for the inverses of the Gram matrix of the Bernstein basis and its principal submatrices, by taking the advantages of the transformations between the Bernstein basis and the constrained dual Bernstein basis. Using the symmetry property, fast calculation of these matrices and their inverses is achieved. Significant improvements are obtained for applications including polynomial approximation of functions and degree reduction of Bezier curves.

Planar quintic G2 Hermite interpolation with minimum strain energy

In this paper, we study planar quintic G^2 Hermite interpolation with minimum strain energy. To match arbitrary G^2 Hermite data, a quintic curve is expressed in terms of four free parameters that encode the local reparameterization at the endpoints and are available for further optimization. We express the approximate strain energy as a quartic function in four parameters, whose minimum can be found by solving an optimization problem of two parameters relating to the magnitudes of endpoint tangent vectors. A feasible region is used while searching the optimal values of these two parameters such that the interpolating curve can preserve tangent directions and avoid singularities at the endpoints. We then solve this constrained minimization problem via the proximal gradient method. Several comparative examples are provided to demonstrate the effectiveness of the proposed method and applications to shape design are also shown.

Explicit algorithms for multiwise merging of Bézier curves

This paper presents a novel scheme, called C r , s multiwise merging, for merging multiple segments of Bezier curves using a single Bezier curve. It is considered as an extension of the existing pairwise merging, to avoid the limitations caused by recursively applying pairwise merging to the multiple case. An explicit algorithm is developed to obtain the merged curve, which preserves C r and C s continuity at the endpoints and is optimal in the sense that the L 2 or l 2 distance is minimized. As an application we develop explicit algorithms for G 1 multiwise merging, always producing better results than C 1 multiwise merging.

A note on quintic polynomial approximation of generalized Cornu spiral segments

In two recent papers Cripps et al. (2010) [3], and Cross and Cripps (2012) [2], quintic polynomial approximations of the generalized Cornu spirals have been studied by considering G^2 continuity and G^3 continuity at the end points respectively. The quintic curve is constructed so that the maximum curvature error is within the specified tolerance. In this paper, we provide corrections to the typing errors in Cross and Cripps (2012) [2], and propose a simpler and more efficient method for the G^2-constrained quintic polynomial approximation where the G^2 conditions are always satisfied by four free variables. Also, we introduce a new error measure of the maximum curvature error, which can greatly reduce the computational time when looking for the solution. Numerical experiments demonstrate the effectiveness of the new measure.

High-quality point sampling for B-spline fitting of parametric curves with feature recognition

Abstract Determining a sequence of points from parametric curves is essential for the construction of least-squares B-spline fitting curves in geometric modeling and related applications. Uniform sampling methods always determine points uniformly in arc length, curvature or a weighted form, but feature points intuitively indicating the curve profile are generally ignored. In this paper, we focus on high-quality point sampling so as to better capture the original curve shape. Firstly, all the feature points (inflection points and extreme curvature points) on the curve are recognized by a parabolic interpolation method. Then more auxiliary points are adaptively added according to the characteristic function defined as a weighted sum of arc length and total curvature characteristics. By adjusting the weight in the characteristic function, it provides more flexibility and controllability to obtain auxiliary points. Finally, we convert the original curve into a B-spline curve by interpolating these feature points and auxiliary points using the progressive–iterative approximation method. Numerical examples demonstrate the effectiveness and high quality of our proposed method.

Note on multi-degree reduction of Bézier curves via modified Jacobi-Bernstein basis transformation

Recently, Bhrawy etźal. (2016) proposed a novel method for C b - 1 , a - 1 -constrained degree reduction of Bezier curves by using the basis transformations between the modified Jacobi polynomials (MJPs) and Bernstein polynomials. In this note, we provide corrections to the formulas wrongly presented in their article. Then we propose G b - 1 , a - 1 -constrained degree reduction of Bezier curves using MJPs, which can always produce more satisfactory results.

An iterative algorithm for G 2 multiwise merging of Bézier curves

This paper presents an iterative algorithm for G 2 multiwise merging of Bezier curves. By using the G 2 constraint, the L 2 distance is represented after simplification as a quartic polynomial in two parameters relating to the magnitudes of end tangents of the merged curve. These two parameters are restricted in a feasible region, in order for the merged curve to preserve the specified directions of end tangents. Then G 2 multiwise merging is formulated as a constrained minimization problem, and the classic projected Newton method is applied to find the minimizer. Some extensions of multiwise merging using G 3 constraints, other energy functionals and curve representations are also outlined. Several comparative examples are provided to demonstrate the effectiveness of the proposed method.

Quintic polynomial approximation of log-aesthetic curves by curvature deviation

Log-aesthetic curves (LACs), possessing monotone curvature and including many classical curves, have been widely used to describe fair shapes in geometric modeling. However, they are generally represented in non-polynomial form and are thus not compatible with current CAD systems. In this paper we present quintic polynomial approximation of LAC segments. For a given LAC segment, a quintic G 2 interpolating Bezier curve is obtained by minimizing a curvature-based error metric, with the advantage of being more likely to preserve the monotone curvature property. Numerical experiments demonstrate that our method can usually generate better results than the previous methods in terms of the deviation in positions and curvatures.