Advanced Graphics
Examples class 1
 [2.3/4] (a) Describe the situations in which it is sensible
to use a wingededged data structure to represent a polygon mesh and,
conversely, the situations in which a wingededged
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,
Cncontinuity mean. [2 marks]
 [4.4/2] Derive the constraints on control point positions which
ensure that two quartic Bézier curves join with (a)
C0continuity, (b) C1continuity, and (c)
C2continuity. [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 nonnegative
 are zero only at the ends of the
parameter interval
 are nonzero 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 Bspline use?
[2 marks]
 [5.4/5]
 For a given order, k, there is only one basis
function for uniform Bsplines. Every control point uses a shifted version of
that one basis function. How many different basis functions are there for
openuniform Bsplines 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
openuniform Bspline 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.
