Survey on Geometric Iterative Methods with Applications

Geometric iterative method, also called progressive-iterative approximation (PIA), is an iterative method 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 iterative 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.