Least eccentric ellipses for geometric Hermite interpolation

We present a rational Bézier solution to the geometric Hermite interpolation problem. Given two points and respective unit tangent vectors, we provide an interpolant that can reproduce a circle if possible. When the tangents permit an ellipse, we produce one that deviates least from a circle. We cast the problem as a theorem and provide its proof, and a method for determining the weights of the control points of a rational curve. Our approach targets ellipses, but we also present a cubic interpolant that can find curves with inflection points and space curves when an ellipse cannot satisfy the tangent constraints.     

空间曲线几何Hermite插值的B样条方法

在给定的GC2插值条件,利用de Boor的构造平面曲线的GC2-Hermite插值方法,构造了一条具有两个自由度的三次B样条插值曲线,并证明插值曲线是局部存在的且具有4阶精度.     

三次Hermite插值曲线的能量优化

在给定插值点的位置矢量及切矢量的情况下,通过在两相邻节点引入两个新的节点,提出了一类保持[C1]连续的三次Hermite插值曲线的构造方法,分别通过基于曲率、挠率的能量函数对其进行优化,给出了能量最小化的参数取值公式。讨论了参数对曲线形状的影响,实例表明了方法的有效性。     

用细分螺线插值容许G2Hermite 数据

为使几何细分方法生成的平面螺线段插值平面容许G2Hermite 数据,基于 平面双圆弧插值理论提出了该方法首末端点处新的细分规则。理论分析表明,修改后的细分 方法所得极限曲线是曲率单调、不变号的螺线段,且插值首末端点处的点、切向、曲率。数 值算例表明,修改后的细分方法收敛速度较快,极限曲线具有较好的形状。     

Hermite interpolation with simple planar PH curves by speed reparametrization

We introduce a new method of solving C¹ Hermite interpolation problems, which makes it possible to use a wider range of PH curves with potentially better shapes. By characterizing PH curves by roots of their hodographs in the complex representation, we introduce PH curves of type K(t−c)²ⁿ⁺¹+d. Next, we introduce a speed reparametrization. Finally, we show that, for C¹ Hermite data, we can use PH curves of type K(t−c)²ⁿ⁺¹+d or strongly regular PH quintics satisfying the G¹ reduction of C¹ data, and use these curves to solve the original C¹ Hermite interpolation problem.     

空间曲线的高阶几何Hermite插值

在给定空间曲线两个端点的位置、切方向、曲率向量和挠率的情况下 ,用参数化五次B啨ｚｉｅｒ曲线来对这条空间曲线进行几何Hermite插值 ,证明了插值问题局部可解 ,解有两个自由度 ,还给出了一种确定这两个自由度的方法 ;并证明了在适当的假设下插值有 6阶逼近度     

On Hermite interpolation with B-splines

Polar forms are used to find the B-spline control points for Hermite interpolation.     

Geometric Hermite interpolation for space curves

This paper considers the geometric Hermite interpolation for spacial curves by parametric quartic Bézier curve. In additon to position and tangent direction, the curvature vector is prescribed at each knot. We prove that under appropriate assumptions the interpolant exists locally with one degree of freedom. Moreover, we prove the interpolant is 6th order accurate.     

High accuracy spline interpolation for 5-axis machining

This article presents a new algorithm for simultaneous 5-axis spline interpolation. The algorithm basically merges two concepts: (1) the interpolation of the toolpath with Pythagorean Hodograph (PH) curves and (2) the analytic solution of the inverse kinematic problem using the template equation method. The first method allows one to obtain a analytic relation between the arclength and the parameter of the toolpath curve. This one enables one to control the velocity of the tool on the workpiece. The second method allows one to determine the analytic solution of the parameterized inverse kinematic problem that permits us to introduce arbitrary number of geometric parameters. A natural selection of the possible parameters can be the parameters of tool geometry and the workpiece placement. This way, the off-line generated inverse solutions—that transform the cutter contact curve into axis values—can be online compensated, as soon as the exact parameter values are become known. Based on these two approaches, a robust and fast method for the simultaneous 5-axis spline interpolation is developed. The result of this new algorithm is time-dependent axis splines which represent the given toolpath with high accuracy.