Nurbs Curves and Surfaces Tutorial

Every curve or surface can be defined by a set of parametric functions. For instance, (x,y,z) coordinates of the points of the curve can be given by:) (), (), (t Z z t Y y t X x = = = t being the parameter and X, Y, Z being polynomial functions in t. If X, Y and Z are 1 st degree polynomials, a line segment will be defined. In that case, two knowns only (i.e. two points or a point and a slope) will be sufficient to define this curve. If X, Y, Z are 2 nd degree polynomials, a parabola segment will be defined and 3 knowns will be necessary to describe it (i.e. 3 points or two points and a tangent). For higher degree polynomials, describing the curve will involve more knowns. This number of knows is what we call the order of the curve, and is always the degree of the polynomial plus 1. Most of the time, cubic polynomials 1 are used to represent curves. Indeed, more knows are needed for higher degree polynomials, what makes modelling difficult to handle. On the other hand, lower degree polynomials describe too restrictive curves, being either lines or parabolas, which are always planar curves. Various approaches have been imagined by mathematicians, for instance, Bézier curves, Hermite curves, Catmull-Rom splines and B-Splines. More information on those curves can be found in [CGPP].