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Abstract: This paper presents a practical polyline approach for approximating the Hausdorff distance between planar free-form curves. After the input curves are approximated with polylines using the recursively splitting method, the precise Hausdorff distance between polylines is computed as the approximation of the Hausdorff distance between free-form curves, and the error of the approximation is controllable. The computation of the Hausdorff distance between polylines is based on an incremental algorithm that computes the directed Hausdorff distance from a line segment to a polyline. Furthermore, not every segment on polylines contributes to the final Hausdorff distance. Based on the bound properties of the Hausdorff distance and the continuity of polylines, two pruning strategies are applied in order to prune useless segments. The R-Tree structure is employed as well to accelerate the pruning process. We experimented on Bezier curves, B-Spline curves and NURBS curves respectively with our algorithm, and there are 95% segments pruned on approximating polylines in average. Two comparisons are also presented: One is with an algorithm computing the directed Hausdorff distance on polylines by building Voronoi diagram of segments. The other comparison is with equation solving and pruning methods for computing the Hausdorff distance between free-form curves. [Copyright &y& Elsevier]

Abstract: This paper proposes an algorithm for calculating the orthogonal projection of parametric curves onto -spline surfaces. It consists of a second order tracing method with which we construct a polyline to approximate the pre-image curve of the orthogonal projection curve in the parametric domain of the base surface. The final 3D approximate curve is obtained by mapping the approximate polyline onto the base surface. The Hausdorff distance between the exact orthogonal projection curve and the approximate curve is controlled under the user-specified distance tolerance. And the continuity of the approximate curve is , where is the user-specified angle tolerance. Experiments demonstrate that our algorithm is faster than the existing first order algorithms. [Copyright &y& Elsevier]

A method for generating an interpolation closed G1 arc spline on a given closed point set is presented. For the odd case, i.e. when the number of the given points is odd, this paper disproves the traditional opinion that there is only one closed G1 arc spline interpolating the given points. In fact, the number of the resultant closed G1 arc splines fulfilling the interpolation condition for the odd case is exactly two. We provide an evaluation method based on the arc length as well such that the choice between those two arc splines is made automatically. For the even case, i.e. when the number of the given points is even, the points are automatically moved based on weight functions such that the interpolation condition for generating closed G1 arc splines is satisfied, and that the adjustment is small. And then, the G1 arc spline is constructed such that the radii of the arcs in the spline are close to each other. Examples are given to illustrate the method. [Copyright &y& Elsevier]

Abstract: This paper proposes a geometric iteration algorithm for computing point projection and inversion on planar parametric curves based on local biarc approximation. The iteration begins with initial estimation of the projection of the prescribed test point. For each iteration, we construct a biarc that locally approximates a segment on the original curve starting from the current projective point. Then we compute the projective point for the next iteration, as well as the parameter corresponding to it, by projecting the test point onto this biarc. The iterative process terminates when the projective point satisfies the required precision. Examples demonstrate that our algorithm converges faster and is less dependent on the choice of the initial value compared to the traditional geometric iteration algorithms based on single-point approximation. [Copyright &y& Elsevier]

Abstract: Curves on surfaces play an important role in computer aided geometric design. In this paper, we present a parabola approximation method based on the cubic reparameterization of rational Bézier surfaces, which generates G1 continuous approximate curves lying completely on the surfaces by using iso-parameter curves of the reparameterized surfaces. The Hausdorff distance between the approximate curve and the exact curve is controlled under the user-specified tolerance. Examples are given to show the performance of our algorithm. [Copyright &y& Elsevier]