Rational Pentagonal Bézier Representation Of Circular Arcs And Blowup Sampling Of Algebraic Curves

In this paper, we introduce our research work in two aspects. 1.Rational pentagagonal Bezier representation of circular arcsBezier curve plays an important role in the fields of computer aided geometry design (CAGD) and computer aided modeling, A Bezier curve is determined only by control points ,so it's conveninent and intuitive for designing.But there is a drawback that classical conic curves including circular arcs which frequently occur in CAGD/CAM can't be exactly represented with Bezier curves.Then exact rational parametric forms of these classical curves are of values.The paper[l][2]gave the necessary and sufficient condition for ratioal cubic and quartic Bezier curves representing circular arcs.They also pointed out that rational cubic Bezier curves can only represent circular arcs whose circular angles are less than 240° and rational quartic Bezier curves can represent abitrary circular arcs whose circular angles are less 2π except for a whole circule .In this paper ,we present the necessary and sufficient condition for rational pentagagonal Bezier representation of circular arcs. Furthmore we prove that rationl pentagagonal Bezier curves can represet abitrary circular arcs including the whole circle.We also give some figures to illustrate our conclusions.2.Blow sampling method of algebraic curvesAlgebraic curves exactly describe corresponding curve objects by equations. These objects often have comprehensive topologcal structure and plenty geometric shapes. While it is difficult to draw these curves if they are not rationl curves.So the visualization of algebraic curves must proceed sampling to obtain enough sampling points to display the topological structures and geometric shapes of the algebraic curves.The existed sampling methods mainly include four kinds:ray-casting method,continuation,enumeration and particle method. Stochastic differential equation sampling method SSM in[6]belongs to particle method ,which has three char-acters:fast samplng speed,uniformly in probability distributed sampling points and high accurate sampling results.We advance SSM to make it more suitable for sam-pling algebraic curves with multi-flexes and disconnected branches.This is dynamic fission sampling method (DFS). For nonsingular curves,we can get high accurate sampling results with DFS method.While for those curves with many singular points or a large multiplicity singular point,there will be bad sampling errors near singular points. Just for this ,we introduce the idea of blowup to deal with such curves before sampling them.We decomposed all the singular points through continously blowing up algebraic curves at singular points.Then we succeed to work out the difficult sampling at singular points in essence.