B-spline Curve Interpolation Research Articles

An accurate NURBS curve interpolation algorithm with short spline interpolation capacity

In this paper, a novel and accurate real-time non-uniform rational B-spline curve interpolation algorithm is proposed. This algorithm not only considers chord errors, feedrate fluctuations, jerk-limited, and acceleration/deceleration (Acc/Dec) capabilities of the machine, but also optimizes the look-ahead process. In the meanwhile, it improves machining efficiency by adding the circular buffer and pre-interpolation (non-off-line) and enhances the real-time performance by removing the time-consuming calculation from the interrupt service routine. Furthermore, the proposed interpolation algorithm can interpolate both the long spline and the short spline with uniform method. The advantages of the proposed method were confirmed by the simulation results.

Uniform B-Spline Curve Interpolation with Prescribed Tangent and Curvature Vectors

This paper presents a geometric algorithm for the generation of uniform cubic B-spline curves interpolating a sequence of data points under tangent and curvature vectors constraints. To satisfy these constraints, knot insertion is used to generate additional control points which are progressively repositioned using corresponding geometric rules. Compared to existing schemes, our approach is capable of handling plane as well as space curves, has local control, and avoids the solution of the typical linear system. The effectiveness of the proposed algorithm is illustrated through several comparative examples. Applications of the method in NC machining and shape design are also outlined.

Research on CNC Key Technology for High Speed Milling

At present, high speed machining (HSM) features in high efficiency, high precision, high flexibility and high quality. HSM technology involves many factors, including cutting mechanisms, machining performance, tool path and other aspects. The key techniques on HSM such as linear interpolation, circular interpolation, cubic B-spline curve interpolation, non-uniform rational B-spline (NURBS) curve interpolation and their respective characteristics are paid more attention and expatiated.

Interpolation Algorithm on Numerical Control Technology

High speed machining (HSM) features in high efficiency, high precision, high flexibility and high quality. The key techniques on HSM such as linear interpolation, circular interpolation, cubic B-spline curve interpolation, non-uniform rational B-spline (NURBS) curve interpolation and their respective characteristics are paid more attention and expatiated. Circle arcs are approximated by using intersecting polygons instead of general inscribed polygons. NURBS interpolation algorithms can reduce feedrate fluctuation and improve machining accuracy.

Simultaneous perimeter measurement for 3D object with a binocular stereo vision measurement system

A simultaneous measurement scheme for multiple three-dimensional (3D) objects’ surface boundary perimeters is proposed. This scheme consists of three steps. First, a binocular stereo vision measurement system with two CCD cameras is devised to obtain the two images of the detected objects’ 3D surface boundaries. Second, two geodesic active contours are applied to converge to the objects’ contour edges simultaneously in the two CCD images to perform the stereo matching. Finally, the multiple spatial contours are reconstructed using the cubic B-spline curve interpolation. The true contour length of every spatial contour is computed as the true boundary perimeter of every 3D object. An experiment on the bent surface's perimeter measurement for the four 3D objects indicates that this scheme's measurement repetition error decreases to 0.7 mm.

Point-tangent/point-normal B-spline curve interpolation by geometric algorithms

We introduce a novel method to interpolate a set of data points as well as unit tangent vectors or unit normal vectors at the data points by means of a B-spline curve interpolation technique using geometric algorithms. The advantages of our algorithm are that it has a compact representation, it does not require the magnitudes of the tangent vectors or normal vectors, and it has C 2 continuity. We compare our method with the conventional curve interpolation methods, namely, the standard point interpolation method, the method introduced by Piegl and Tiller, which interpolates points as well as the first derivatives at every point, and the piecewise cubic Hermite interpolation method. Examples are provided to demonstrate the effectiveness of the proposed algorithms.

Adaptive parametric interpolation scheme with limited acceleration and jerk values for NC machining

Parametric interpolation has many advantages over the traditional linear or circular interpolation in computer numerical control (CNC) machining. The existing work in this regard is reported to have achieved constant feedrate, confined chord error and limited acceleration/deceleration in one interpolator. However, the excessive jerk still exists due to abrupt change in acceleration profile, which will cause shock to the machine as well as deteriorate the surface accuracy. In this paper, an adaptive interpolation scheme incorporating machine’s dynamics capability consideration is proposed and illustrated in details. In the proposed algorithm, the commanded feedrate is maintained at most of the time and adaptively reduced in large curvature areas to meet the demand of the machining accuracy requirement, while at the same time, the acceleration and jerk values are limited within the machine’s capabilities during the whole interpolation process. It ensures a high machining accuracy, eliminates the phenomenon of overshoot/undershoot and reduces mechanical shock to the machine tools. The real-time performance of this interpolator is also measured to demonstrate its practical application. Two non-uniform rational B-spline (NURBS) curve interpolation experiments are provided to verify the feasibility and advantages of the proposed scheme.

Surface reconstruction based on point cloud from laser scanning system

The characteristics of the point cloud obtained with laser scanning system are discussed in the paper, and a new algorithm of surface reconstruction based on point cloud is put forward. Based on the theory of B-spline curve and surface interpolation, the freeform curve and surface reconstruction method - we called the surface as loft surface - are introduced in detail. This method of surface reconstruction is adequately utilized the characteristics of the point cloud from laser scanning system. The experimental results testify that the method is effectual and feasible.

A new approach for lofted B-spline surface interpolation to serial contours

This paper addresses the problem of B-spline surface interpolation to serial contours, where the number of points varies from contour to contour. A traditional approach to the problem creates a set of B-spline curves via B-spline curve interpolation to each contour, makes them compatible via degree elevation and knot insertion and performs B-spline surface lofting to get a B-spline surface that interpolates them. The approach tends to result in an astonishing number of control points in the lofted B-spline surface. This situation arises mainly from the inevitable process of progressively merging different knot vectors to make the B-spline curves compatible. This paper presents a new approach for fixing this problem. The approach includes a novel process of obtaining a set of compatible B-spline curves from the given contours. The process is based on universal parameterisation [1, 2], allowing the knots to be selected freely but leading to a more stable linear system for B-spline curve interpolation. Since the number of control points in each compatible B-spline curve is equal to the highest number of contour points, the proposed approach can realise efficient data reduction and provide a compact representation of a lofted B-spline surface while keeping the desired surface shape. Some experimental results demonstrate its usefulness and quality.

Lofted B-spline surface interpolation by linearly constrained energy minimization

This paper proposes a new approach for lofted B-spline surface interpolation to serial contours, where the number of points varies from contour to contour. The approach first finds a common knot vector consisting of fewer knots that contain enough degrees of freedom to guarantee the existence of a B-spline curve interpolating each contour. Then, it computes from the contours a set of compatible B-spline curves defined on the knot vector by adopting B-spline curve interpolation based on linearly constrained energy minimization. Finally, it generates a B-spline surface interpolating the curves via B-spline surface lofting. As the energy functional is quadratic, the energy minimization problem leads to that of solving a linear system. The proposed approach is efficient in computation and can realize more efficient data reduction than previous approaches while providing visually pleasing B-spline surfaces. Moreover, the approach works well on measured data with noise. Some experimental results demonstrate its usefulness and quality.

B-스플라인 곡면기법을 이용한 압출금형 곡면의 표현방법에 관한 연구

To construct the extrusion die surface, a B-Spline surface scheme based on the cubic B-Spline curve interpolation method is proposed in the present work. The inlet and outlet profiles are described with B-Spline curves by using the centripetal method for uniform parameterization. The interior control points of surface are generated using the derivative characteristics of B-Spline curve. A complete B-Spline surface is constructed by using appropriate coordinate transformation and knot deletion. In the present study, a quantitative measure for the control of surface is suggested by introducing the tangential vector and inclination angles at the inlet and outlet sections. To verify the validity of the proposed method, automatic surface generation is carried out for the various types of extrusion die surface.

Choosing nodes and knots in closed B-spline curve interpolation to point data

Node and knot placement for closed B-spline curve interpolation to point data heavily depends on whether the degree of B-splines is odd or not. For odd degree B-splines, the natural method of setting knots to coincide with nodes (i.e. parameters) works very well and provides the good quality. However, when the degree is even, the usual methods including the natural method can have problems and result in the bad quality. This paper presents a method, called the shifting method, which works well for even degree B-spline interpolation. It has nearly the same properties as the natural method does for odd degree B-splines. It is simple and provides the good quality of a resultant curve.

Quadratic B-spline curve interpolation

Traditional approach in performing even-degree B-spline curve/surface interpolation would generate undesired results. In this paper, we show that the problem is with the selection of interpolation parameter values, not with even-degree B-spline curves and surfaces themselves. We prove this by providing a new approach to perform quadratic B-spline curve interpolation. This approach generates quadratic B-spline curves whose quality is comparable to that of cubic interpolating B-spline curves. This makes quadratic B-spline curves better choices than cubic B-spline curves in some applications in graphics and geometric modeling, since it is cheaper to render/subdivide a quadratic curve and it is easier to find the intersection of two quadratic curves.

A Study on Five Flank Machining of Centrifugal Compressor Impellers

Referring to machining technologies used for turbomachinery components, generally there are two cutting strategies: point cutting and flank cutting. Based on considerations of the cost, efficiency, and surface roughness, flank cutting by using a five-axis machining tool is a promising way to machine blade surfaces of turbomachinery components constructed by ruled surfaces. In this article, a flank cutting technology for centrifugal compressor impellers is developed by using the B-spline curve interpolation, ruled surface construction, and coordinate transformations. Also, an impeller with 12 blades is actually machined to verify the proposed approach by using a five-axis machining center.

A universal parametrization in B-spline curve and surface interpolation

We propose here a new universal parametrization for B-spline interpolation. The new parametrization is based on the values t i where B-splines N i, k ( t) are maximum in case of order k. The resulting interpolation curve X( t) is transformation invariant and more natural looking, in general, than those obtained by other methods. Using a fixed knot vector, t i 's are independent of interpolating points { P i }, and hence the computation of X( t) can be done more efficiently. In addition, the new method works well in any order k.

Neural network methods for NURBS curve and surface interpolation

New algorithms based on artificial neural network models are presented for cubic NURBS curve and surface interpolation. When all the knot spans are identical, the NURBS curve interpolation procedure degenerates into that of uniform rational B-spline curves. If all the weights of data points are identical, then the NURBS curve interpolation procedure degenerates into the integral B-spline curve interpolation.

Nonlinear shape restoration of distorted images with coons transformation

Image shape restoration based on mathematical transformation is a successful approach to nonlinear distortions in computer vision, robot vision and pattern recognition. The key of this process is to find the distortion function and its inverse function. Usually, the distortion function is unknown or unclear. Even in the case of that when the function is known, it remains difficult to compute or estimate the parameters necessary for the restoration. To overcome this problem, in this paper, Coons transformation utilizing the boundary functions for the distorted images have been used to approximate the exact distortion function. The boundary functions are calculated using B-spline curve interpolation, which is coincided with the necessary condition of major elements that constitute a Coons transformation. The performance of the proposed method to nonlinearly distorted images has been demonstrated in experiments with interesting results.