regularized boolean set operations in Computer Graphics to perform the boolean operations on the 3D object which will yield the 3D object only.

1. Solid Modeling
Regularized Boolean Operations
Prepared by :- Hitesh H. Parmar [ MEFGI 1st PG-CE ]
Contact :- Hitesh.parmar915@gmail.com

2. Topics that we will cover today.
▪ What do you understand by the solid object?
▪ Method to represent solid object
▪ Boolean set Operation
– Ordinary Boolean Operation on Solids
– Regularized Boolean Operation on Solids
▪ Examples

3. Solid Representation

4. Definition of a Solid Model
▪ A solid model of an object is a more complete representation than its surface
(wireframe) model
▪ Solid is bound by surfaces. So need to also define the polygons of vertices, which
form the solid. It must also be a valid representation.
Wireframe Model Solid Model

5. Ordinary Boolean Operation on Solids
▪ One of the most popular methods for combining objects is by Boolean set
operations, such as union, difference, and intersection
▪ Applying an ordinary Boolean set operation to two solid
objects, however, does not necessarily yield a solid object. For
example, the ordinary Boolean intersections of the cubes in Fig. 12.3(a)
through (e) are a solid, a plane, a line, a point, and the null
object, respectively.

6. Ordinary Boolean Set Operations
on Solid Objects

7. Ordinary Boolean Set Operations
on Solid Objects

8. Ordinary Boolean Set Operations
on Solid Objects

9. Regularized Boolean Operation on Solids
▪ Rather than using the ordinary Boolean set operators, we will instead use
the regularized Boolean set operators, denoted ∪*, ∩*, and −*, and defined
such that operations on solids always yield solids.
▪ For example, the regularized Boolean intersection of the objects shown in
Fig. 12.3 is the same as their ordinary Boolean intersection in cases (a) and
(e), but is empty in (b) through (d).

10. Regularized Boolean Set Operations
Using regularized boolean
operators:
All 3 intersections = NULL
Effectively, we throw away any
results from an operation that is of
lower dimensionality than the
original solids.

11. ▪ boundary / interior points :
– points whose distance from the object and the object’s complement is zero / other
points
▪ closed set
– a set contains all its boundary points
▪ open set
– a set contains none of its boundary points
Regularized Boolean Set Operations

12. ▪ Closure :
– the union of a set with the set of its boundary points
– is a closed set
▪ Boundary :
– the set of closed set’s boundary points
▪ Interior :
– the complement of the boundary with respect to the object
Regularized Boolean Set Operations

13. ▪ regularization :
– the closure of a set’s interior points
▪ regularized Boolean set operator :
– A op* B= closure (interior (A op B))
– only produce the regular set when applied to regular sets
Regularized Boolean Set Operations
Object Closure Interior Regularized Object

14. Ordinary & Regularized Boolean Set
Operations
[ Example 1 ]
[ O1]
[ O2]
[ O1]
[ O2]

15. Ordinary Boolean Set Operations
[ Example 2 ]

16. Regularized Boolean Set Operations
[ Example 3 ]

17. • Regularized Boolean Operations
Source : University of Manitoba

18. Thanks