Advanced Graphics Study Guide

Part 3: Splines
A: Bezier curves
B: B-splines
C: NURBS
...back to part 2 | on to part 4...

3B) B-splines

Why B-splines?

• minimise the order of the polynomial pieces (order k)
• maximise the continuity between pieces (continuity C(k-2))
• minimise the number of control points controlling a piece (k points)
• have positive basis functions
• have basis functions which partition unity, implying that each piece lies inside its control points' convex hull
• are invarient with respect to affine transforms

Exercises How many control points are required for a quartic Bezier and how many for a quartic B-spline? Why are cubics the default for B-spline use? Explain the difference between Uniform, Open Uniform, and Non-Uniform knot vectors. What are the advantages of each type? [2000/9/4] (b) A non-rational B-spline has knot vector [1,2,4,7,8,10,12]. Derive the first of the third order (second degree) basis functions, N1,3(t), and graph it. If this knot vector were used to draw a third order B-spline, how many control points would be required? [7 marks] [2001/8/4] (a) For a given order, k, there is only one basis function for uniform B-splines. Every control point uses a shifted version of that one basis function. How many different basis functions are there for open-uniform B-splines of order k with n + 1 control points, where n >= 2k - 3? [6 marks] (b) Explain what is different in the cases where n < 2k - 3 compared with the cases where n >= 2k - 3. [3 marks] (c) Sketch the different basis functions for k = 2 and k = 3 (when n >= 2k - 3). [4 marks] (d) Show that the open-uniform B-spline with k = 3 and knot vector [ 0 0 0 1 1 1 ] is equivalent to the quadratic Bezier curve. [7 marks] [2002/7/9] (d) Derive the formula of and sketch a graph of N3,3(t), the third of the quadratic B-spline basis functions, for the knot vector [ 0 0 0 1 3 3 4 5 5 5 ]. [6 marks]

Part 3: Splines
A: Bezier curves
B: B-splines
C: NURBS
...back to part 2 | on to part 4...