Evaluation Of Line Profile And Spatial Straightness Error Based On Hausdorff Distance

As a similarity measure, Hausdorff distance is widely used in various engineering application areas, like image matching, pattern recognition, computer vision and fingerprint verification. In the mechanical field, it has been applied in CAD/CAM, haptic rendering simulation, robot path planning, curve/surface approximation. As a result of the development of technology, the research of Hausdorff distance in terms of discrete geometric objects have been used in related engineering fields. However, the research of Hausdorff distance in terms of continuous geometric objects is less, except for few research of the computation of Hausdorff distance between freefrom curves/surfaces. The computation of Hausdorff distance between two freefrom curves and its application in geometric error evaluation are mainly discussed in the paper. This paper proceeds as follow.First, an global algorithm based on subdivision techniques for computing the minimum distance between a point and a parameter curve/surface is proposed. The algorithm transforms the computation of the minimum distance between a point and a Bezier curve/surface into the geometrically intuitive a curve/surface tangent to the parameter axis/plane, by building the corresponding mathematical models, with the subdivision searching strategy. This algorithm has strong stability, as well as has not to find all roots without the initial value. The efficiency has been improved by combined with Newton method with the algorithm.Second, the conditions for the computations of the events where the Hausdorff distance can take place and the corresponding constraint equations have been discussed. A new method containing two steps for the computation of the approximate and precise Hauddorff distance between two freefrom curves is generated. The first step of the method is calculating the approximate solutions by sampling one curve; the second step is establishing a corresponding local optimization model according to the shape and position of the sub-curves and finding the precise Hausdorff distance. This method turns the computation of Hausdorff distance into the computation of the minimum distance between a point and curve/surface, which improves the efficiency and stability of computation. The correctness of the method has been verified by the numerical examples.Finally, the evaluation methods of the line profile error and spatial straightness error based on the minimum directed Hausdorff distance are established under the condition of the Hausdorff distance is applied in geometric error evaluation. The results of the proposed methods meet the minimum zone condition of those errors. The mathematical programming model and its linearization method for the minimum directed Hausdorff distance are presented. The error of the linearized model is analyzed during the processing, and a MATLAB function to solve the nonlinear model and the corresponding criterion for minimum zone condition is provided. Last, by comparing the numerical examples with the exiting results respectively, the evaluation accuracy, stability and efficiency of this method are verified.