Rational curves and splines are one of the building blocks of computer graphics and geometric modeling. Although a rational curve is more flexible than its polynomial counterpart, many properties of polynomial curves are not applicable to it. For this reason, it is very useful to know if a curve presented as a rational space curve has a polynomial parameterization. In the paper, an algorithm is presented that decides whether a polynomial parameterization exists, and computes the parameterization.
In algebraic geometry, it is known that a rational algebraic curve is polynomially parameterizable iff it has one place at infinity. This criterion has been used in earlier methods to test the polynomial parameterizability of algebraic plane curves. The resulting algorithm is complicated, and it also requires that the parametric curves be implicitized. This causes problems for rational space curves. The paper gives a simple condition that is both necessary and sufficient for the polynomial parameterizability of rational space curves. The calculation of the polynomial parameterization is simple, and involves only a rational reparameterization of the curve.
rational curves, polynomial curves, reparameterization, algorithms, algebraic geometry
Rational curves are a central tool in graphics and modeling. The rational formulation has the ability to represent conics (in fact all genus 0 algebraic curves) 12 as well as free-form (controlled) curves ' . The coordinates for each point on the curve can be expressed as x(t) y(t) z(t) a,b] where x(t), y(t), z(t) and w(t) are polynomials in R[t], the ring of all polynomials in t whose coefficients are real numbers. A curve given in this representation is said to be polynomial if w(t)= 1; otherwise, it is a rational curve.
A rational curve is represented in its homogeneous form, Q(t)= (x(t), y(t), z(t), w(t)), which is equivalent to the projection of a polynomial curve in a