Final-year Geometric Modelling Course

1997 Exam

1

Answers to each part of this question should be brief: a sentence or two, or a quick sketch. (1 mark for each part)

1. Show how a wire-frame `model' can be ambiguous.

2. Why is a simple list of polygonal faces not very good as a geometric model of an object with flat faces?

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

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

5. What is the difference between a boundary-representation model and a set-theoretic (that is a CSG) model?

6. What is an Euler operator in a B-rep geometric modelling program?

7. Why are some curved shapes slightly problematical for B-rep modellers?

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

9. What are the differences between a Bézier curve and a NURBS?

10. What are the twist vectors in the definition of a bicubic parametric patch?

11. Why is it hard to compute the volume of a boundary model, but easy to compute its surface area?

12. 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?

13. What is a membership test?

14. Why is it important that the Bernstein basis functions should add up to 1?

15. How can the difference operator be removed from set-theoretic models?

16. How does the Monte Carlo method of finding the volume of a geometric model work?

17. In feature recognition, what are the two distinct meanings of destructive solid geometry?

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

19. A slot-drill (a type of cylindrical cutter) moves in a horizontal straight line in a vertical-axis milling machine. What shape would it sweep out?

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

2

The formula for a Bézier patch is:

where

A Bézier patch has 9 control points, , arranged in a grid in space. The control points are conventionally numbered up to . Sketch the weighting functions for the point (10 marks).

A ray-trace rendering program is depicting such a surface. A procedure exists for finding the intersection of a ray with the surface; it returns the s and t values at the place where the ray strikes. Describe how, given a parametric ray where u is the ray parameter, the s and t values of the point hit, the control points, and the Bézier formula, a reflected ray could be generated so that the surface could be made to appear shiny. Do not do the algebra, but describe exactly what steps need to be taken (10 marks).

3

A set theoretic geometric model of an object has the following implicit inequalities as primitives:

and the set-theoretic expression describing the object is:

Draw a sketch of the object (6 marks). What is the length, width, and height of the object in the x, y, and z directions (4 marks)? Is the point (6, 1, 0.5) inside the object or outside it (prove this - don't just rely on plotting on your sketches) (4 marks)? What is the volume of the object (work this out analytically; don't try to use a Monte Carlo method) (6 marks)?

4

A series of differing components are being moved along a conveyor. They are to be picked up by a robot as the first stage in assembling them into a finished product. The robot has a gripper consisting of two fingers that move together whilst staying parallel. This gripper needs to grasp two parallel flat faces on each object in order to pick the object up. Geometric models of all the components are held in the robot's controlling computer. Describe how a feature recognition program might interrogate the models looking for suitable faces on them for the gripper to grasp (10 marks).

In order for the components not to slip once gripped the torque and shear on the gripper's surfaces must be kept below certain values. How could the torque and shear for each component be predicted from the geometric models (10 marks)?

5

It is proposed to erect a modest 20m-high statue in the centre of Manchester to celebrate an obscure rock band. The statue will be of two naked human figures frozen in the act of trying to strike each other with sacks filled with money. A geometric model of the statue is to be created in order to judge its visual impact before it is erected. What type of geometric model would you use to represent a human figure and what primitives would you employ (give your reasons---5 marks)? Musical instruments will be scattered at the figures' feet; how might they be represented (contrast this with the requirements for the human form---3 marks)? Which components of the statue would be the most difficult to represent and why (2 marks)?

Given a geometric model of the buildings that will surround the statue, how might you compute all the areas of the surroundings on which the statue will cast shadows during the course of a complete day's sunlight (10 marks---no smart answers about the amount of sun in Manchester)?

© Adrian Bowyer 1996