Conformal Sampling Method Research Based On Curvature Analysis Of Cubic B-spline Curve

The attention of this paper is sampling data in3D reconstruction in the background ofreverse engineering. In reconstruction process that used contours to reconstruction objects, thecontours that have smooth and symmetrical points are an important guarantee for a smoothsurface that has be reconstructed and high-quality grid. Sampling of data points determinesthe merits of the modeling reconstruction. However, the data on the original contours can notmeet these conditions. Generally speaking, the original data are sampled. And the existingsampling methods can be done under specific situations, and they are difficult to express thesufficient conditions of sampling standard into formula or theorem. The main contents of theproblem can be summarized as follows: firstly, the geometric model cannot be effectivelyreconstructed or the results of reconstruction are distorted because of unnecessary samplingdense. Secondly, when the sampling data points are too dense, the reconstruction process willincrease the computational overhead, extend computing time and cost more memory space.So the attention of this paper is to research a sampling algorithm that has less computation,less storage and general application on the base of these problems.The research used a series of contours as a research object to study a conformal samplingmethod. We will fit curves to these points before sampling data in order to get the contoursthat contain smooth and symmetrical points. The contents of this research contains: fittingoriginal data into cubic B-spline curves; extracting the characteristic points of curves;dividing cubic B-spline curves on the characteristic points; Calculating minimum curvatureradius of every curve; Sampling data. The main contributions of this research include:(1) Cubic B-spline curve fitting. The methods proposed by LesA Piegl and Wayne Tillerare adopt to fit curves to the original points. The input of the algorithm is original points in3Dspace, but the output is cubic B-spline curves.(2) Extracting characteristic points of curves. The characteristic points were defined asthe non-singular inflection points and singular points in this paper. According to thedefinitions of inflection points in the mathematics and CAGD theory, we deduced formulas that can compute the characteristic points of cubic Bezier curves on the basis of consideringthe curvature and expression of cubic Bezier curves and proposed an algorithm that canextract characteristic points of curves. However, because the multiple effects from degree,control polygon and knot vector, it is difficult to analyze the shape characteristics of B-splinecurve, and we try to convert cubic B-spline curves into cubic Bezier curves in order to reducecomputation. So we proposed an algorithm that can divide cubic B-spline curve into severalcubic Bezier curves. Theoretical analysis and experimental results indicated that the algorithmhas reasonability and feasibility.(3) Curve subdivision. We use the existing algorithm to them into two segments of cubicBezier curves at the characteristic points in order to guarantee monotonicity of every curve.(4) Calculating the minimum curvature radius of curves. It is an important part in thesampling method. We must calculate the maximum curvature of every cubic Bezier curve,which requests us to research an algorithm that can obtain the maximum curvature rapidly andaccurately. We summarized the judging conditions of curvature changes after analyzing thechange of curvature. The maximum curvature can be calculated by these judging conditions.(5) Sampling data. We proposed a sampling method based on curvature analysis of cubicBezier curve, and the method includes detailed sampling standards and algorithm steps.