The Optimal Rational Parametric Plane Algebraic Curve

Parametric form and implicit form are two common representations of curves andsurfaces.These two representations show their advantages and disadvantages in differentapplications,it is relatively easy to do inside-outside tests for implicit representations,while itis convenient to render and manipulate parametric curves and surfaces,and most of thegeometric modeling systems choose the parametric form.Hence the transformations betweenthe two representations become important problems.In this paper, has done some work on rational parameterization of plane algebraic curve.Rational parameterization of algebraic curves in the mathematical sense has been resolved,but in engineering applications, technical staff are often concerned about the optimalparameterization of the curve segment specified on the algebraic curve. Given the currentparametric criteria and the optimal parameterization algorithm based on the standard, is stillnot very satisfactory, so still has some theoretical and practical significance for the optimalrational parameterization of curve segment specified on a plane algebraic curve.Closest to the arc length parameterization as the standard, a more general evalution isproposed for rational parametric effect on a plane algebraic curve, and a rational parametricoptimal or approximation formula of quadratic algebraic curve on any given curve segment isconstructed, with a strong self-adaptive. For the rational parameterization of the section ofcircular, symmetrical nature of the part of elliptical or hyperbolic, also gives a theorem andthe proof.