CPSC 645/VIZA 675 Geometric Modeling

Homework 5 due 12/10/02

1. [32 points] Assume you want to model the following solid:

a) [8 points] Describe the solid in two ways by different CSG trees.

b) [8 points] Describe the solid as a B-rep.

c) [8 points] Describe the solid using a volumetric representation.

d) [8 points] Describe the solid using a medial representation.

You do not need to give precise equations or numbers for these cases, but should indicate all of the information that should be specified for each (using pictres if necessary).

2. [13 points] Assume you have two cubes for which you want to find their union. List with brief descriptions or drawings the types of degenerate configurations that lead to non-manifold results.

3. [15 points] Assume you are given two lines that intersect in an acute angle, a. Call the intersection point of the two lines x. Now, assume that there is numerical error in one of the two lines, causing it to be translated a distance y perpendicular to the original line (i.e. the new line is parallel to the old line, a distance y apart). The new intersection point between the lines is called x’.

a) Express (in terms of a and y) the distance x’ is from x.

b) Is this distance more than, less than, or the same as the distance of the line’s

translation, y, or can you tell?

c) What happens as a becomes smaller?

d) What conclusion can you draw about the types of geometric modeling situations

that are prone to numerical error?