Answers to each part of this question should be brief: a sentence or two, or a quick sketch. (1 mark for each part)
Explain how the implicit equation for a two-dimensional ellipse can be constructed from the implicit straight-lines that form its major and minor axes. (4 marks) Why is this construction advantageous when it comes to translating or rotating the ellipse? (2 marks)
A prolate spheroid (or rugby ball, if you prefer...)
needs to be modelled in a geometric modeller, as it is the shape of the
cavity in a machine being designed to mold gel vitamin capsules.
The major axis of the spheroid is 12mm long, and its minor axis is
8mm long. The major axis lies along the vector 
, and the
centre of the spheroid is the origin of coordinates. Using the method
that you described in answer to the first part of this question, construct
an implicit equation for the spheroid in three dimensions. Don't bother
to multiply the equation out. (8 marks)
A section through the machine is being pictured by a ray-tracing program. Where (if it does at all) does the ray x = 10 - 10t, y = 10 - 10t, z = t strike the spheroid? (6 marks)
The object shown in Figure Q3.a has nine faces (as numbered; faces 1, 2, and 3 being hidden from view). Using the conventions that a concave edge has the value 1 and a convex edge has the value 2, construct the face adjacency graph (8 marks) and the face adjacency matrix (4 marks) for the object.
Figure Q3.b shows a pattern in the form of a face adjacency graph of a feature that is to be searched for in the object. What sort of shape is the feature? (Make a sketch - 5 marks.) Ring (ideally using a different colour than the one you're writing everything else with) one occurence of the feature in your face adjacency graph for the original object. (3 marks)
A parametric quadratic patch:
is to be cut with a ball-nosed cutter of radius 0.5. Where is the centre of the cutter when it is cutting the patch at s=0, t=0 and also when it is at s=0, t=1? (Assume that the vector-product of gradients in the s and t directions in that order points from solid to air.) (12 marks)
If the patch were being cut, what calculations would need to be done to ensure that the tool was not gouging parts of the surface that should have been preserved. (No need for formulae - 8 marks)
Describe, using the shape shown in Figure Q5.a as an example, how a two-dimensional straight-edged drawing can be decomposed into convex regions. (8 marks)
How does this help with the problem of user-input to geometric modellers? (3 marks)
Could the method be extended to cover circular arcs such as those in Figure Q5.b? (6 marks) What problems would you anticipate with allowing curved edges in general? (3 marks)
© Adrian Bowyer 1996