An Algorithm For Computing Integral Values With NURBS Curve

Non-uniform Rational B-spline(NURBS) is becoming an international standard,used for data transformation, the geometric design of the product and geometric modeling. It also sit in computer aided geometric design of the top spot in the 10 frontier research topics. Its importance and value in the international graphics community is self-evident, should be called the second to none. Compared to the algorithm of polynomial spline, although NURBS algorithm is similar to it in many ways, the rational representation method of NURBS curve is very complex,therefore the calculation about differential and integral is very complicated, even it’s difficult to solve the concrete results. This leads to in the process of solving NURBS curve integral values, we will face an important research topic, that’s how to solve out the accurate results, and how to use some specific algorithm to calculate the integral results that we cannot solve.In this thesis, the curvilinear integral quantitative calculation method of the research mainly includes two parts, the area of the surrounding area by axes and NURBS curve, and the arc length of NURBS curve(1)For the area of the surrounding area by axes and NURBS curve, to low(e.g. 2) rational Bezier curves and NURBS curve integral, whether the exact solution can be obtained by the special calculation method directly. Namely according to a point on the curve or some geometric properties of the curve, a formula is deduced by calculation, and then directly calculated the integral results which need to get by the formula. For this problem, even just quadratic curve, if using the conventional formula of integral calculation, we need to integrate the quadratic rational function after derivation, due to the integrated function is complicated, so we need so much advanced integral techniques to solve out the results, and if the function cannot be integrated, so we can’t work out the final result. And based on the matrix representation of NURBS cubic curve, precise calculation is given directly under NURBS cubic curve integral formula in comparison is more intuitive and easy to understand and easy to apply.And giving a concrete example, and apply the precise calculation formula of NURBS curve area, to calculate the data of Bonjean’s curve in ship statics. For instance, water surface area, static moment, floatation coordinate, waterlinecoefficient, vertical prismatic coefficient.(2)For the arc length of NURBS curve, to low(e.g 3) NURBS curve, we can through the node insert algorithm, converts cubic NURBS curves to Bezier curve. So as long as we calculate the arc length of the rational Bezier curve, then we can calculate the arc length of NURBS curve by accumulative sum. For the high order NURBS curve, whether there is similar to rational Bezier curve approximation curve, and the curve of the integral result is easy to be obtained. In this thesis, using the ideas of curve segment, to generate a smooth curve through cutting control polygon approximation NURBS curve. With discrete algorithm applied to the latter the correlation integral is transformed into a summation of polynomial integral, thus it can calculate the arc length of curve.