Keywords: iterative methods, semilocal convergence, third order methods, geometric problems.

One of the most usual ways to approximate nonlinear equations is the use of iterative methods. An iterative method starts from one or several initial approximations to the solution and, using an iterative function, computes a new (possibly better) approximation to the solution.

In computer aided geometric design, iterative methods, like Newton's method, are often used for solving geometric problems, such as finding the intersection between two curves or two surfaces. The geometric problem is posed as a nonlinear systems of equations, and the iterative method is applied to solve it. In this work we use this particular problem to illustrate the different convergence conditions that can be obtained, depending on the representation (explicit, implicit or parametric) of the considered curves or surfaces.

Using geometric approximations of the curves or surfaces, we explore the use of different type of approximations (based on lines, circles, parabolas and generalized hyperbolas) performing a comparison of the different strategies. We are interested in both theoretical and numerical aspects. We present the order of convergence of the different approximations and derive sufficient hypotheses that ensure their convergence to the solution of the problem. Finally, we review other geometric problems, like computing the distance between a point and a curve, computing the distance between two curves or surfaces, computing a common tangent to two curves, where this strategy can be considered.

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