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Efficient Computation of Regularized Boolean Operations on n-Dimensional Orthogonal Polytopes Ricardo Pérez-Aguila Universidad Tecnológica de la Mixteca (UTM) Carretera Huajuapan-Acatlima, km. 2.5 Huajuapan de León, Oaxaca 69000, México [email protected] Abstract The first part of this work is devoted to describe the Extreme Vertices Model in the n-Dimensional Space (nD-EVM) and the way it represents nD Orthogonal Pseudo-Polytopes (nD-OPP’s) by considering only a subset of their vertices: the Extreme Vertices. This model has enabled the development of simple and robust algorithms for performing the most usual and demanding tasks on Polytopes Modeling. Although the EVM of an nD-OPP has been defined as a subset of its vertices, there is much more information about the polytope hidden within this subset of vertices. In the second part it is presented the way the nD-EVM performs some of the key operations between polytopes: Regularized Boolean Operations. These operations provide a way for combining polytopes in order to compose new ones. There are described the Regularized Boolean Operators and the way they assure the dimensional homogeneity. Algorithms for performing Boolean Operations under the nD-EVM will be described, and finally, details about an experimental time complexity analysis will be given in order to sustain the efficiency shared by the nD-EVM when these operations take place through the described procedures. Some practical applications of the nD-EVM are also commented. Keywords. Polytopes Representation Schemes, Computational Geometry, Geometrical and Topological Modeling, Regularized Boolean Operations. Resumen La primera parte de este trabajo está dirigida a describir al Modelo de Vértices Extremos en el Espacio n-Dimensional (nD-EVM) y la manera en que representa Pseudo-Politopos Ortogonales nD (nD-OPP’s) al considerar únicamente un subconjunto de sus vértices: los Vértices Extremos. Este modelo ha permitido el desarrollo de algoritmos simples y robustos a fin de efectuar las tareas más comunes y demandantes en el Modelado de Politopos. Aunque el EVM de un nD-OPP ha sido definido como un subconjunto de sus vértices, existe información importante acerca del politopo que se encuentra oculta dentro de este subconjunto de vértices. En la segunda parte es presentada la manera en la que el nD-EVM lleva a efecto algunas de las operaciones clave entre politopos: las Operaciones Booleanas Regularizadas. Estas operaciones proveen de una metodología para combinar politopos a fin de obtener nuevos objetos. Se describirán a los Operadores Booleanos Regularizados y la manera en la que éstos aseguran la homogeneidad dimensional. Se proporcionarán detalles acerca de dos algoritmos para efectuar Operaciones Booleanas bajo el nD-EVM. Finalmente, se describirá un análisis experimental de complejidad temporal a fin de sustentar la eficiencia proporcionada por el nD-EVM cuando estas operaciones se efectúan a través de los procedimientos presentados. También se comentarán algunas aplicaciones prácticas del nD-EVM. Palabras Clave. Esquemas para la Representación de Politopos, Geometría Computacional, Modelado Geométrico y Topológico, Operaciones Booleanas Regularizadas.

Introduction Polytopes Modeling is a recent area of wide development: If a polytope can be modeled in a way that its geometry is appropriately captured, then it will be possible to apply, on such polytope, a range of useful operations. Due to the need of modeling objects or phenomena as polytopes (for example in Computer Aided Design and Manufacturing, electronic prototypes, medical datasets, animation planning, collision detection, etc.) the development of a variety of specialized mechanisms to represent them has arisen. The representation schemes for polytopes are frequently divided in two large categories (although not all the representations are completely inside in one of them): n-Dimensional Boundary Representations and Hyperspatial Partitioning Representations. The first part of this work is devoted to describe the Extreme Vertices Model in the n-Dimensional Space (nD-EVM) and the way it represents nD Orthogonal Pseudo-Polytopes by considering only a subset of their vertices: the Extreme Vertices. The Extreme Vertices Model (3D-EVM) was originally presented, and widely described, by Aguilera & Ayala for representing 2-manifold Orthogonal Polyhedra (Aguilera,1997) and later considering both Orthogonal Polyhedra (3D-OP’s) and Pseudo-Polyhedra (3D-OPP’s) (Aguilera,1998). This model has enabled the development of simple and robust algorithms for performing the most usual and demanding tasks on solid modeling, such as solid splitting, set membership classification operations, and measure operations on 3D-OPP’s. It is natural to ask if the EVM can be extended for modeling n-Dimensional Orthogonal Pseudo-Polytopes (nD-OPPs). In this sense, some experiments were made, by Pérez-Aguila & Aguilera (2003), where the validity of the model was assumed true in order to represent 4D and 5D-OPPs. Finally, in (Pérez-Aguila,2006) was formally proved that the nD-EVM is a complete scheme for the representation of nD-OPPs. The meaning of complete scheme was based in Requicha's set of formal criteria that every scheme must have rigorously defined: Domain, Completeness, Uniqueness and Validity. The second part of this work is oriented to present the way the nD-EVM performs some of the key operations between polytopes: Closed and Regularized Boolean Operations. These operations provide a way for combining polytopes in order to compose new ones. There are described the Regularized Boolean Operators and the way they assure the dimensional homogeneity, that is, it is expected the regularized operations between nD polytopes always will generate nD polytopes, or the null object (the empty set). Algorithms for performing Boolean Operations under the nD-EVM will be described, and finally, details about an experimental time complexity analysis will be given in order to sustain the efficiency shared by the nD-EVM when these operations take place through the described procedures.

The n-Dimensional Orthogonal Pseudo-Polytopes Definition 1.1: A Singular n-Dimensional Hyper-Box in
n is the continuous function I n : [0,1]n x

→

[0,1]n

∼

I n ( x) = x

Definition 1.2: For all i, 1 ≤ i ≤ n, the two singular (n-1)D hyper-boxes I (ni ,0) and I (ni,1) are defined as

follows:

If

then

x ∈[0,1]n−1

I (ni ,0) ( x ) = I n ( x1 ,..., xi −1 , 0, xi ,..., xn −1 ) = ( x1 ,..., xi −1 , 0, xi ,..., xn −1 )

and

I (ni ,1) ( x ) = I n ( x1 ,..., xi −1 ,1, xi ,..., xn −1 ) = ( x1 ,..., xi −1 ,1, xi ,..., xn −1 )

Definition 1.3: In a general singular nD hyper-box c we define the (i,α)-cell as c( i ,α ) = c  I (ni ,α ) Definition 1.4: The orientation of a cell c  I (ni ,α ) is given by (−1)α +i . Definition 1.5: An (n-1)D oriented cell is given by the scalar-function product (−1)i +α ⋅ c  I (ni ,α ) Definition 1.6: A formal linear combination of singular general kD hyper-boxes, 1 ≤ k ≤ n, for a closed set A is called a k-chain. Definition 1.7 (Spivak,1965): Given a singular nD hyper-box In we define the (n-1)-chain, called the boundary of In, by ∂ ( I n ) = ∑  ∑ ( −1)i +α ⋅ I (ni ,α )  n

i =1

 α =0,1



Definition 1.8 (Spivak,1965): Given a singular general nD hyper-box c we define the (n-1)-chain, called the boundary of c, by ∂ (c ) = ∑  ∑ (−1)i +α ⋅ c  I n  ( i ,α ) n

i =1

 α = 0,1



Definition 1.9 (Spivak,1965): The boundary of an n-chain

∑c , i

where each ci is a singular

general nD hyper-box, is given by ∂ ( ∑ ci ) = ∑ ∂ (ci ) Definition 1.10: A collection c1, c2, …, ck, 1 ≤ k ≤ 2n, of general singular nD hyper-boxes is a combination of nD hyper-boxes if and only if k  n n n   ∩ cα ([0,1] ) = (0,...,0)   ∧ ( ∀i, j, i ≠ j , 1 ≤ i, j ≤ k ) ci ([0,1] ) ≠ c j ([0,1] )  = 1 α n  

(

)

The first part of the conjunction establishes that the intersection between all the hyper-boxes is the origin, while the second part establishes that there are not overlapping nD hyper-boxes. Definition 1.11: We say that an n-Dimensional Orthogonal Pseudo-Polytope p, or just an nD-OPP p, will be an n-chain composed by nD hyper-boxes arranged in such way that by selecting a vertex, in any of these hyper-boxes, we have that such vertex describes a combination of nD hyper-boxes (Definition 1.10) composed up to 2n hyper-boxes.

The Extreme Vertices Model in the n-Dimensional Space (nD-EVM) Definition 2.1: Let c be a combination of hyper-boxes in the nD space. An Odd Edge will be an edge with an odd number of incident hyper-boxes of c. Definition 2.2: A brink or extended edge is the maximal uninterrupted segment, built out of a sequence of collinear and contiguous odd edges of an nD-OPP. Definition 2.3: We will call Extreme Vertices of an nD-OPP p to the ending vertices of all the brinks in p. EV(p) will denote to the set of Extreme Vertices of p. Definition 2.4: Let p be an nD-OPP. EVi(p) will denote to the set of ending or extreme vertices of the brinks of p which are parallel to Xi-axis, 1 ≤ i ≤ n. Theorem 2.1 (Pérez-Aguila,2006): A vertex of an nD-OPP p, n ≥ 1, when is locally described by a set of surrounding nD hyper-boxes, is an extreme vertex if and only if it is surrounded by an odd number of such nD hyper-boxes. Theorem 2.2 (Pérez-Aguila,2006): Any extreme vertex of an nD-OPP, n≥1, when is locally described by a set of surrounding nD hyper-boxes, has exactly n incident linearly independent odd edges. Definition 2.5: Let p be an nD-OPP. A kD couplet of p, 1