Study On Approximating Rational Bézier Curves Using Polynomial Bézier Curves

Rational Bézier curves have been widely used for geometric modeling in computeraided geometric design(CAGD). It realizes the uniform representation of conic sectionand polynomial parametric curves. Rational Bézier curves not only enjoy all theproperties that their nonrational counterparts possess, but also can be modified theirshapes by changing their weights without moving their control poits. However,differential and integral operations of rational curves are very complicated, even cannotbe calculated. And it is not easy to obtain geometric information of rational curves, forexample, curvature, torsion, and so on. Therefore, approximation of rational curves bypolynomial curves is an important and fundamental task in CAD/CAM. Usingpolynomial curves to approximate rational Bézier curve is more and more important, tofind a good approximation method, achieve a better approximation effect, solve theproblem of rational curve is of great importance and value.It discusses the polynomial approximation problem of rational Bézier curve, bymeans of the criterion of L2as metric standard, consider ’removing’ the numerator of therational Bézier curve expression, do deformation of the approximation formulas. Thismethod avoids the problem of rational Bézier function integral, reduces the difficulty ofthe operations. Through the corresponding numerical examples, we can know: It has agood approximation effect without the limit of endpoints; interpolation endpoints, agood approximation effect can be achieved by increasing order of polynomial curve; asthe endpoints keeping geometric continuity, the approximation effect of approximationcurve with the same degree is generally better than that of keeping the same orderparametric continuity at two endpoints.