Lecture 3

scheduled: Saturday, Jan 15, 2005

We discussed:

1. The CGS (Constructive Solid geometry): We follow a binary-tree approach towards the construction of a complex 3D object.

• CGS-tree (the binary tree)
  
• Operations: union, intersect and difference and their regularized versions.
• The essence of world co-ordinate system.
  

2. Why and how of Regularization. How artifacts get created in CG. Finding the interior points and putting the shell/boundary points.

3. The practical approach -- Classification! The aim is to determine whether a point (or a curve or a surface) of interest lies IN, ON or OUT of the given solid (whose CSG-Tree we already know). The signature of the method CLASSIFY:

CLASSIFY CSG-tree * {point, curve, surface} --> {IN, ON, OUT}

4. The recursive implementation of the CLASSIFY(...) method. Note that the leaves of the CSG-Tree are the primitives (and the terminating condition for the recurtion). When we reach a primitive, the native implementation of the primitive's CLASSIFY is summoned.

5. The root-operation over the two returned results (corrresponding to its two child-nodes), depends on the operation.

if X is one of the {IN, ON, OUT} then for the Regularized-Union:

(IN, X) --> IN
(OUT, OUT) --> OUT
(OUT, ON) --> ON
(ON, ON) --> {IN | ON}

Now for the last case, we use the Neighborhood-method to determine whether it's IN or ON.

The aim is to see if all the points in the epsilon-neighborhood (an open-ball of radius equalling epsilon; for epsilon being as small you can think of) of the point (under consideration, where the center of the open-ball lies) are in the interior of the resulting structure. If so, we declare that the point (under consideration) is IN the resulting structure. This method helps in Regularization.

6. The other (regularized) operations are left as an exercise.