Final-year Geometric Modelling Course

1996 Exam

1

1. How does the depth buffer make pictures of three-dimensional objects?

2. Show how a face `model' can represent a non-solid object.

3. What is the difference between a face `model' and a boundary-representation (B-rep) model?

4. The extended Euler-Poincaré formula for checking topological consistency in a B-rep modeller is . What do the variables mean?

5. Draw a shape for which the simple Euler-Poincaré formula (F+V-E = 2) is not valid, and show that it isn't.

6. Some real objects have curved surfaces. Some simple geometric modellers have to approximate curved surfaces with small facets. Give an example of a use of a model where facets would be inconvenient.

7. Give an example of a use of a faceted model where the faceting wouldn't matter.

8. What sort of engineering components would a Duct-type modeller be good at modelling?

9. What can NURBS patches do that B-spline patches can't?

10. How does a NURBS differ from a B-spline mathematically?

11. In a design, it is important that a simple-shaped moving part does not hit a panel that is represented by a NURBS patch. What easy test would ensure that no collision occurred?

12. How can the difference operator be removed from set-theoretic (CSG) models?

13. What is the implicit inequality that defines a sphere of radius 4 centred at the point (5,3,9)? (Don't multiply out the brackets.)

14. Why is interval arithmetic useful with solids represented by implicit inequalities?

15. In feature recognition, what is destructive solid geometry?

16. What is the backwards-growing feature recognition technique?

17. Why is it easier, in general, to make a picture of a boundary-representation geometric model than of a set-theoretic one?

18. What are attributes in a geometric modeller?

19. A T-slot-cutter (long thin cylindrical shaft, short fat cutting cylinder at the bottom) moves in a horizontal straight line in a vertical-axis milling machine. What shape would it sweep out?

20. Are most commercial geometric modellers B-rep or set-theoretic? Give a reason for your answer.

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Figure Q2

Figure Q2 shows a traffic surveillance camera, the road that it is surveying, and a bank that does not yet exist but which it is proposed to build. The bank's architects are investigating potential obstructions to the camera's lines of sight. The bank is held as a set-theoretic geometric model in a CAD system. The outside walls and roof are extracted from this; they are described by the inequalities:

All units are in metres. The set-theoretic expression for the bank building (ignoring the floor) is . The camera is at the point . Calculate if it can see the point on the road. (11 marks)

How would things become more complicated if some walls of the bank were curved? (Give a brief numerical example in your answer.) (4 marks)

The bank is built. A gang of bank robbers hack into the architects' CAD system and copy the complete detailed final geometric model, which includes data on the positions of the bank's internal security cameras. Describe how the gang could make pictures of the room containing the bank vault that would show them what parts of the room the security cameras could and couldn't see, and what parts of the room the traffic camera could and couldn't see through the windows. (5 marks)

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Figure Q3

Figure Q3 shows a block with numbered faces (faces 4, 5, and 6 being invisible from the viewpoint) that is held in a B-rep modeller. Construct a face-adjacency graph for the block (4 marks). Construct a template graph for the indentation and mark where it appears on the larger graph (2 marks). Construct a template graph for a rectangular hole that might go right through a component (6 marks). Explain how these sorts of graphs and their closely-related face-adjacency matrices (give examples of these) help with the problems of feature recognition (8 marks).

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Two nearly-parallel straight lines are represented by the equations Ax + By + C = 0 and . The epsilons are small, and may be positive or negative. What are the coordinates of the lines' intersection point (5 marks)? What effect will small changes in the values of the epsilons have on them? (Draw pictures to illustrate your answer.) (7 marks) Comment on the difficulties that problems of this sort pose in geometric modellers and how they may be alleviated by both careful programming and by changes in the way that the computer is instructed to do arithmetic (8 marks).

5

Explain how the volume and the centroid of both a set-theoretic and a B-rep geometric model might be found (6 marks).

Components which are being represented using a geometric modeller will be brought to a robot on a conveyor belt. Before they are picked up by the robot, a TV camera sends an image of them taken directly from above to the robot's controlling computer.

Given a procedure that will compute the 3D convex hull of the models, how could the stable configurations that each object that they represent might rest in on the flat conveyor be found? (6 marks) Give two ways for generating synthetic images of the models that the controlling computer could compare with the TV images to find out which way up the real components were resting. (4 marks)

What problems would there be with models with curved surfaces? (4 marks)

© Adrian Bowyer 1996