6533b7dcfe1ef96bd1272724

RESEARCH PRODUCT

Courbe d'une fraction rationnelle et courbes de Bézier à points massiques

subject

description

Modelling polynomial curves or arcs with Bezier curves can be seen as a basis conversion not so easy for the rational curves. The classical representation of Rational curves based on controlled points with non negative weights as in NURBS does not cover all rational curves. This can be fixed by using the rational Bezier representation by mass points that are weighted points with negative or null weights. The curve of any rational function includes arcs denoted as connex components. These curves and their asymptotic lines are here modelled by the use of mass control points. The asymptotic lines are described by a point that are one weighted point or a vector. An algorithm proposes to represent any arc of a rational function f = P Q where P, Q are polynomials. The algorihtm starts by the rational Bezier representation by mass points with the parameter running on [0, 1], isolates any connex component of the curve by obtaining the zeroes of the denominator. The real line isthus split in subintervals. The next step consists in a homographic parameter change that transforms the [0, 1] interval into any previous interval. The parameter change does not modify the degree curve and points out the mass control points Bézier representation including the asymptotic line.