NURBS , Spline Surfaces and Blossoming in Two - dimensions

In this lecture we generalize the polar form and blossoming and introduce NURBS(non-uniform rational B-spline), then we extend the discussion to 2D about surfaces. 1 NURBS NURBS is an industry standard tool for the representation and design of geometry. It is the non-uniform and rational extension of B-splines. We will introduce these properties one by one using blossoming. A cubic polar form is a multi-affine 1 and symmetric function f (u 1 , u 2 , u 3) such that: • multi-affine: f (α 1 t 1 + α 2 t 2 , u 2 , u 3) = α 1 f (t 1 , u 2 , u 3) + α 2 f (t 2 , u 2 , u 3), ifα 1 + α 2 = 1 • symmetric: f (u 1 , u 2 , u 3) does not depend on arguments' order. As described in last lecture, there exists a unique polar form f (u 1 , u 2 , u 3) for every polynomial function f (t) such that f (t) = f (t, t, t). Blossoming is a way to compute points on the curve or new control points by linear recursive interpolation of control points. Blossoming for Bezier curves and B-splines is described in lecture 2. Here we describe how non-uniform B-splines can be constructed by blossoming. For uniform B-splines, control point P i corresponds to f (i, i + 1, i + 2). So it is determined by 3 consecutive uniform-spaced parameter values (also called knots). However, we can use the multi-affine properties to build the blossom pyramid even if the initial control points are