Fitting point clouds with normal vectors by geometric algorithm

插值是b样条曲线曲面应用中最典型的问题之一。在本研究中，我们不仅对插值一组点感兴趣，而且对b样条曲线和曲面在这些点上给出的法向量感兴趣。点法向拟合的应用包括通过增加法向控制来增强自由曲线设计，以及精确逼近偏移曲线/曲面。我们的方法依赖于一个非常简单的几何算法，该算法基于局部点-曲线/曲面距离计算，法向量的正交性条件的施加以及偏移过程，以全局方式迭代更新输入控制点。通过实例验证了所提算法的有效性。

Interpolation is one of the most typical problems that arise in the application of B-spline curves and surfaces. In this research we are interested in not only interpolating a set of points, but also normal vectors given at these points by the B-spline curves and surfaces. Applications of point-normal fitting include enhancement in the free-form curve design by adding the normal control, and accurate approximation of offset curves/surfaces. Our method relies on a surprisingly simple geometric algorithm which iteratively updates the input control points in a global manner based on local point-curve/surface distance computations, imposition of orthogonality conditions of the normal vector, and offsetting procedures. Examples are provided to demonstrate the effectiveness of the proposed algorithms.