Approximate Implicitization Of Parametric Curves And Efficient Approximation Of A Planar Algebraic Curve

Due to its nice properties in certain applications, implicit equations are widely used in CAGD, Computer Graphics, Inverse Engineering, and have been studied by many re-searchers in recent years. In this thesis, we have proposed algorithms for approximate im-plicitization of parametric curves and efficient approximation of planar curves. The thesis consists of two parts:implicitization of parametric curves and efficient approximation of a planar algebraic curve.In part one, we give a monomial order; then we project the new vector from interpola-tion to the orthogonal complement space of the vectors spanned successively, If the length of projection is less than a given threshold, we say the corresponding polynomial vanishing at the given point set approximately. And put the polynomial to be the approximate implicit equation of the parametric curve. A further algorithm is obtained of interpolating normal direction based on the first algorithm. Especially for the case of planar parametric curves, nontrivial examples have been presented to prove the effectiveness of the method.In part two, we talk about problem of efficient approximation of a planar algebraic curve. We assume that the topology of the curve has been computed. Therefore, we are under the premise of that we have known topology of algebraic curves, then consider how to efficiently approximate the regular curve segments. First, we trace a distance along the pos-itive tangent direction of a given point at the curve, and then we use the alternating Newton iteration method to get another point on the curve. Repeating this process, we can achieve the purpose of approximation of the curve. We have proved the convergence and accuracy of the method.