## Examples class 1

• [2.3/4] (a) Describe the situations in which it is sensible to use a winged-edged data structure to represent a polygon mesh and, conversely, the situations in which a winged-edged data structure is not a sensible option for representing a polygon mesh. (b) What is the minimum information which is required to successfully draw a polygon mesh using Gouraud shading? [4 marks]
• [4.4/1] Explain what C0-, C1-, C2-, Cn-continuity mean. [2 marks]
• [4.4/2] Derive the constraints on control point positions which ensure that two quartic Bézier curves join with (a) C0-continuity, (b) C1-continuity, and (c) C2-continuity. [6 marks]
• [not in the study guide] Bézier basis functions satisfy the following list of properties. Explain the significance of each property for the designer of a Bézier curve. [5 marks]
• sum to one (form a `partition of unity')
• are non-negative
• are zero only at the ends of the parameter interval
• are non-zero at the ends of the parameter interval only for one basis function
• have a symmetric counterpart which is the result of reflecting the basis function through the central parameter value
• [5.4/2] Why are cubics the default for B-spline use? [2 marks]
• [5.4/5]
• 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]
• Explain what is different in the cases where n < 2k - 3 compared with the cases where n >= 2k - 3. [3 marks]
• Sketch the different basis functions for k = 2 and k = 3 (when n >= 2k - 3). [4 marks]
• Show that the open-uniform B-spline with k = 3 and knot vector [0, 0, 0, 1, 1, 1] is equivalent to the quadratic Bézier curve. [7 marks]
• [5.7/3] What are the advantages of NURBS over Bézier curves and surfaces? (i.e. why have NURBS, in general, replaced Bézier curves and surfaces in CAD?) [4 marks]
• [not in the study guide] For each of the properties of Bézier basis functions, above: (a) which properties apply to NURBS in general? (b) which apply to NURBS only with a certain type of knot vector? For each case where this is true, what type of knot vector is required? [7 marks]

This exercise set is marked out of 50. This should take 90 minutes in an examination.