中文摘要：

几何迭代法,又称渐进迭代逼近(progressive-iterative approximation,PIA),是一种具有明显几何意义的迭代方法.它通过不断调整曲线曲面的控制顶点,生成的极限曲线曲面插值(逼近)给定的数据点集.文中从理论和应用2个方面对几何迭代法进行了综述.在理论方面,介绍了插值型几何迭代法的迭代格式、收敛性证明、局部性质、加速方法,以及逼近型几何迭代法的迭代格式和收敛性证明等.进而,展示了几何迭代法在几个方面的成功应用,包括自适应数据拟合、大规模数据拟合、对称曲面拟合,以及插值给定位置、切矢量和曲率矢量的曲线迭代生成,有质量保证的四边网格和六面体网格生成,三变量B-spline体的生成等.

英文摘要：

Geometric iterative method, also called progressive-iterative approximation (PIA), is an iterative me-thod with clear geometric meaning. Just by adjusting the control points of curves or surfaces iteratively, the limit curve or surface will interpolate (approximate) the given data point set. In this paper, we introduce the geometric iterative method in two aspects, i.e., theory and application. In theory, we present the iterative formats of the interpolatory and approximating geometric iteration methods, respectively, show their convergence and local property, and develop the accelerating techniques. Moreover, some successful applications of the geometric itera-tive methods are demonstrated, including adaptive data fitting, large scale data fitting, symmetric surface fitting, generation of the curve interpolating given positions, tangent vectors, and curvature vectors, generation of the quadrilateral and hexahedral mesh with guaranteed quality, and generation of the trivariate B-spline solid, etc.