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# regularized boolean set operations

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

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### regularized boolean set operations

1. 1. Solid Modeling Regularized Boolean Operations Prepared by :- Hitesh H. Parmar [ MEFGI 1st PG-CE ] Contact :- Hitesh.parmar915@gmail.com
2. 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. 3. Solid Representation
4. 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. 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. 6. Ordinary Boolean Set Operations on Solid Objects
7. 7. Ordinary Boolean Set Operations on Solid Objects
8. 8. Ordinary Boolean Set Operations on Solid Objects
9. 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. 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. 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. 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. 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. 14. Ordinary & Regularized Boolean Set Operations [ Example 1 ] [ O1] [ O2] [ O1] [ O2]
15. 15. Ordinary Boolean Set Operations [ Example 2 ]
16. 16. Regularized Boolean Set Operations [ Example 3 ]
17. 17. • Regularized Boolean Operations Source : University of Manitoba
18. 18. Thanks