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The Detailed presentation on Applying Regularized Boolean set operations on to the solid objects in Computer Graphics.
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Solid Modeling
Regularized Boolean Operations
Prepared by :- Hitesh H. Parmar [ MEFGI 1st PG-CE ]Contact :- [email protected]
Prepared by Hitesh H. Parmar ([email protected]) 2
Topics that we will cover today.
▪ Representation of Solid Model.
▪ Definition of Solid Model
▪ Boolean set Operation– Ordinary Boolean Operation on Solids– Regularized Boolean Operation on Solids
▪ Examples
Prepared by Hitesh H. Parmar ([email protected]) 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
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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.
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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).
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Regularized Boolean Set Operations
Using regularized booleanoperators:All 3 intersections = NULL
Effectively, we throw away anyresults from an operation that is oflower dimensionality than theoriginal solids.
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▪ 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
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▪ 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
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▪ 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
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Ordinary & Regularized Boolean Set Operations
[ Example 1 ]
[ O1][ O2]
[ O1][ O2]
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Regularized Boolean Set Operations
[ Example 3 ]
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• Regularized Boolean Operations
Source : University of Manitoba
