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Illustration of OSOP: The red solid curve is the given curve, while the curves in blue and in green are the extended segment of the given curve and the resulting interpolation curve with its control polygon in green, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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Citations
... The parametric and geometric continuity constraints between any two curves and surfaces are derived in [20,21]. Chen and Ma [22], present a local piecewise geometric interpolation method for the B-spline interpolation problem with tangent directional constraints. The method is based on an unclamping technique and knot extension. ...
In this article, an algorithm to obtain a smooth path (free from obstacles) that can be optimized by shortest path distance, bending energy and curvature variation energy will be presented. Previously, scholars used various tools to generate smooth path such as Clothoid, Log-Aesthetic curves (LACs), and Bézier curves. The limited number of solutions from the aforementioned curves become one of the drawback to generate smooth path planning. Therefore, providing a number of solutions that can generate smooth path planning becomes the objective of this study. In this paper to generate a smooth path, five templates of spiral transition curves having three different shape parameters with monotone curvature (either increase or decrease) by cubic GHT-Bézier curves are proposed. Moreover, few examples of path planning technique via cubic GHT-Bézier spiral curve to show the flexibility of smooth path by minimization of the shortest path (minimum arc length) L, bending energy E and curvature variation energy V are presented. The superiority of cubic GHT-Bézier spiral path smoothing techniques as compared to Clothoid and LACs is also demonstrated.
... A vital problem is the projection of a point onto a surface or parametric curve in order to get the foot point and calculate the corresponding parameter values, which is widely applied in the field of Geometric Modeling [1][2][3][4][5][6][7], Computer Graphics and Computer Vision [5][6][7], Computer Aided Geometric Design [8][9][10][11][12], geometric design [13][14][15], geometry sculpt [16], scientific research and engineering applications [17] and computer animation [18][19][20]. ...
... Piegl et al. [5] present a method to project a point onto NURBS surfaces with three stpes: decompose a NURBS surface into quadrilaterals, project the test point onto the nearest quadrilateral, and finally get the parameter from the nearest quadrilateral. Mortenson [21] converted the projection problem into the Symmetry 2017, 9,210 2 of 13 problem of obtaining a polynomial's root, and then adopted the Newton-Raphson approach to get the root. ...
Regarding the point projection and inversion problem, a classical algorithm for orthogonal projection onto curves and surfaces has been presented by Hu and Wallner (2005). The objective of this paper is to give a convergence analysis of the projection algorithm. On the point projection problem, we give a formal proof that it is second order convergent and independent of the initial value to project a point onto a planar parameter curve. Meantime, for the point inversion problem, we then give a formal proof that it is third order convergent and independent of the initial value.
Curve interpolation with B-spline is widely used in various areas. This problem is classic and recently raised in application scenario with new requirements such as path planning following the tangential vector field under certified error in CNC machining. This paper proposes an algorithm framework to solve Hausdorff distance certified cubic B-spline interpolation problem with or without tangential direction constraints. The algorithm has two stages: The first stage is to find the initial cubic B-spine fitting curve which satisfies the Hausdorff distance constraint; the second stage is to set up and solve the optimization models with certain constraints. Especially, the sufficient conditions of the global Hausdorff distance control for any error bound are discussed, which can be expressed as a series of linear and quadratic constraints. A simple numerical algorithm to compute the Hausdorff distance between a polyline and its B-spline interpolation curve is proposed to reduce our computation. Experimental results are presented to show the advantages of the proposed algorithms.
Curve extension is a useful tool in the computer‐aided design (CAD) community. A given B‐spline curve usually needs to be extended by another curve to reach one or more target points. In this work, we aim to enlarge the representation domain of the extending part to achieve the optimal extending result in the global solution space. Inspired by this, we have made three contributions. First, we use piecewise polynomial, that is, a nonuniform B‐spline, instead of one polynomial segment to extend the original curve. Compared to one polynomial segment, curves consisting of piecewise polynomial have stronger modeling ability and therefore expand the solution space of the problem. For extension to multiple target points, we are the first to directly give extension results based on all target points rather than extending to every target step by step. Third, we exploit the matrix representation of B‐splines to obtain an explicit solution for this extension problem. The detailed formula derivations and experimental results are provided to show the validity and effectiveness of our method.
B-spline curve extension is an important operation in computer aided design systems. In this paper, we present a new extension algorithm for B-spline curves. The algorithm uses curve unclamping to generate a uniform B-spline curve segment from the original curve and gradually extends the segment to pass through every target point. Algorithms of uniform B-spline curves are used such that our algorithm has a low time cost and can easily handle arbitrary-order derivative constraints at the target points. Generalization for non-uniform rational B-spline curve extension is also discussed, and examples show the efficiency of our method. © 2015 Science China Press and Springer-Verlag Berlin Heidelberg
