R. B. Tilove and A. A. G. Requicha. Closure of Boolean operations on geometric entities. Computer-Aided Design, September 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....points that have some interior points nearby; all other points with eroded lower dimensional neighborhoods indicate the lack of solidity. See example in Figure 2. Formally, set X is called closed regular if X = closure(interior(X) 1) Based on this definition, introduced and studied in [80, 84, 120], we can now check the neighborhoods of individual points in the set X to see if they pass the neighborhood test. If all points pass, X is indeed solid; otherwise the set is not solid, but we can regularize (and therefore solidify) any set X by taking the closure of its interior, as shown in ....

.... constructions require substantially more work than is implied by Equation (5) Specifically, combining results of two classifications with respect to sets A and B requires not only the logical information, but also representing and combining the neighborhoods of p with respect to A, B, and Aop # B [118, 120] (see the example in Figure 5. The constructive representation scheme relying on closed regular primitives, rigid body motions, and the regularized set operations is called Constructive Solid Geometry or CSG [85] By design, the use of CSG is limited by availability of solid primitives and by ....

R. B. Tilove and A. A. G. Requicha. Closure of Boolean operations on geometric entities. Computer-Aided Design, September 1980.

....more complex primitives can be developed on a case by case basis. The second problem is more difficult because many useful modeling constructions and operations do not guarantee or preserve solidity. In particular, it is well known that intersection of two solids (regular sets) need not be solid[47] and must be regularized. In practical terms, this means that given a classification of a point p with respect to solids A and B, classification of p with respect to k(i(A N B) may still require computing neighborhoods of p with respect to A, B and A N B. If the neighborhood of p does not contain ....

....treatment of this subject, as well as analysis of neighborhood computations, was first carried out by members of the Production Automation Project at the University of Rochester. They studied topological properties of solids[30] pointed out that solids are not closed under standard set operations[47], proposed regularized set operations and Constructive Solid Geometry (CSG) 31] unified the concepts of the set membership classification[45] and developed sound algorithms for boundary evaluation and merging[32] More recently, there appears to be a renewed interest in the standard ....

R. B. Tilove and A. A. G. Requicha. Closure of boolean operations on geometric entities. ComputerAided Design, 12(5):219 220, September 1980.

....conditions for any component based software architecture to support the engineering design and simulation process. 2 Regular Closed Sets and Intersection Gaps The design of algorithms for engineering simulations and analyses usually assumes that geometric algorithms deliver regular closed sets [32]. We begin by describing how the algebra of regular closed sets is never realized, even ideally, with today s geometric modelers. We discuss the resulting consequences for software architecture. Surface intersection procedures are invoked for all Boolean operations on solid models. But surface ....

Tilove, R. B., and Requicha, A. A. A. G., Closure of Boolean operations on geometric entities, CAD 12 (5), 1980.

....thus, could be considered as an operand for another binary operation. The set intersection operation formed the basic kernel of algorithms for all three operations and it was readily apparent that the set theoretic intersection operator could easily yield resultants which were not valid solids [26] (e.g. consider two closed unit cubes aligned so that their intersection is merely a unit square) The Boolean algebra of regular closed sets was seen as a formal expression of the necessary binary operators [26] The intent of intersection was replaced by the Boolean meet. The intent of ....

.... intersection operator could easily yield resultants which were not valid solids [26] e.g. consider two closed unit cubes aligned so that their intersection is merely a unit square) The Boolean algebra of regular closed sets was seen as a formal expression of the necessary binary operators [26]. The intent of intersection was replaced by the Boolean meet. The intent of union was replaced by the Boolean join. If A and B are solids, the intent of the subtraction of B from A, can be specified notationally as A Gamma B = cl X int X (A B 0 ) 1) where B 0 indicates the set ....

Tilove, R. B., and Requicha, A. A. A. G., Closure of Boolean Operations on Geometric Entities, CAD 12 (5), 1980.

....are used; these are equivalent to performing a standard set operation, followed by taking the closure of the interior of the resultant set. Informally: we perform the set operation, and stretch a tight skin over the resultant set. These operations can be shown to form a Boolean algebra [TR80], and the main problem in using them is that we have to be a careful when considering the boundaries of sets formed. However, the following result can be shown [Cam84] Pi Distribution Theorem (for regularised sets) The extrusion operator Ex distributes over the (closed) regularised set ....

R. B. Tilove and A. A. G. Requicha. Closure of Boolean operations on geometric entities. CAD J., 12(5):219--220, September 1980.

....either inside, or outside, or on the solid s boundary. This problems is similar to the point location problem in computational geometry [14, 18] and to the clipping problem in computer graphics [23] The line polygon problem has been investigated for the case of constructive solid geometry; e.g. [25, 27]. A solution for octrees appears in Samet s books [21, 22] Naylor solves the point solid classification using binary space partition trees in [24] The point solid and the line solid classification methods have uses in solving a wide range of diverse problems. One such problem is the construction ....

R. B. Tilove and A. A. G. Requicha. Closure of boolean operations on geometric entities. Computer Aided Design, pages 219--220, September 1980.

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R. B. Tilove and A. A. G. Requicha. Closure of boolean operations on geometric entities. Computer-Aided Design, 12(5):219--220, September 1980.

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