Download Presentation
## Solutions to homework #4 and #5

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**Solutions to homework #4 and #5**Wu Xiaomao May 5, 2004**12.1. Define the results of performing and**for two polyhedral objects in the same way as the result of performing was defined in section 12.2. Explain how the resulting obje-cts is constrained to be a regular set, and sp-ecify how the normal is determined for each of the object’s faces.**Solution:**• The union operation is defined as: A B=closure(interior(A B)) Nb A B B A Na Nb B A B A Na Nb A B B A Na**The union operation is defined as:**A B=closure(interior(A B)) Nb A B N A B Na Nb A A B B Na Nb A B A B Na**9.13 Explain why an implementation of Boo-lean set**operations on quadtrees or on octr-ees does not need to address the distinction between the ordinary and regularized opera-tions described in Section 12.2**Solution:**The reason why we need to distinguish between ordinary and regularized Boolen set operations is that we want to resulting objects to be a regularized objects( no dangling edges or dangling surfaces). In quadtrees or octrees representation, objects are expressed in a discrete way. Here no boundary or surface concept is concerned. The Boolean set operation is done by compare the nodes one by one.**12.8 Describe how to perform point classifi-cation in**primitive instancing, b-rep, spatial occupancy enumeration, and CSG**Solution:**• Primitive instancing: • As primitive instance is created by codes, the programmer should design a subroutine for point classification and embed it in the primitive instance. The user can load these subroutines when he/she what to do a point classification.**2. b-rep**A B 1 Odd intersection points: In Even intersection points: Out 2 3 4 E C F D**3. Spatial occupancy enumeration**Search the quadtree/octree, if point A belongs to one of the empty node, then point A is outside. If point A belongs to one of the full node, continue searching the neighbors of point A (minimum unit). A is on the boundary if one of A’s neighbor is a empty node. A is inside if all its neighbors are full nodes or within a full node.**4. CSG**Determine whether the point is inside/outside/ on the boundary for each primitive. Let that point join the Boolean set operation to determine its relation to the object.**Supplemental problem:**V=16 E=24 F=11 V-E+F=3 V=16 E=28 F=14 V-E+F=2**Supplemental problem:**V=16 E=24 F=11 H=1 C=1 G=0 V-E+F-H=2 2(C-G)=2 V-E+F-H=2(C-G)