V. Shapiro. Real functions for representation of rigid solids. Computer-Aided Geometric Design, 11(2):153--175, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....points x in space satisfying f x z 0 for a continuous function f . Such a representation is usually called a function representation. Set theoretic solids have been successfully included in this type of representation with the application of R functions and their modifications (see [1], 2] A vast volume of literature is devoted to the subject of scattered data reconstruction and interpolation. In most applications, Delaunay triangulation is used for 3D reconstruction. The main idea of Delaunay decomposition is to reconstruct a surface from non uniform samples by connecting ....

V. Shapiro, Real Functions for Representation of Rigid solids, Computer Aided Geometric Design, 11(2): 153-175, 1994.

.... the mesh evolution process (right) Sharp features, edges and corners, appear naturally if the implicit surface is constructed using set operations (union, intersection, difference) The defining function for such surface can be obtained by applying min max functions or more general functions [25, 26, 29, 23] to the defining functions of the arguments of set operations. If the binary tree of operations is available, then the sharp features can be found by a numerical method that analyzes the functions of both arguments for each operation [36] However, if the resulting function is evaluated by a ....

V. Shapiro. Real functions for representation of rigid solids. Computer Aided Geometric Design, 11(2):152--175, 1994.

....solids can be collected into a hierarchical organization with the help of Boolean operations. More complicated operations through the use of functional composition are also possible to generate more interesting shapes. The common feature essential to all implicit solid modeling methods [18][19] is the creation of an oriented three dimensional boundary surface which partitions the entire 3 space into two distinct regions, namely the one occupied by the solid interior and the one outside of the defined solid. 2.3 Haptic Rendering Haptic rendering is the process of applying forces ....

V. Shapiro. Real functions for representation of rigid solids. Computer Science Tech. Report TR91-1245, Cornell Univ., Ithaca, NY, 1991.

.... sets can also be defined, so for example if A X and B Y , the fuzzy subset A B of X Y is their direct product where AB (x, y) min(A (x) B (y) Note that the connection between union and intersection operators and max and min was observed by Ricci[20] see below) and also by Shapiro[26] in his description of the use of R functions for defining solids. One particular type of fuzzy set is the fuzzy number,sofor example the membership function of the set about 4.2 would be some convex function peaking at the value 4.2. If we choose to consider a particular grade of membership # ....

V. Shapiro, Real functions for representation of rigid solids, to appear, Computer Aided Geometric Design.

....tools that provide the user a means to: Specify a functionally based model in the HyperFun [1] or C languages. HyperFun is a specialized highlevel language for FRep object specification with a simplified C type syntax and several additional settheoretic operators implemented using R functions [7,8,6]. One can use the library of FRep specific geometric objects and transformations as well as the user s own library written in HyperFun or C. Define mappings of object space to multimedia space by assigning multimedia types to object coordinates. At least one object space coordinate has to ....

V. Shapiro, "Real functions for representation of rigid solids," Computer-Aided Geometric Design, vol. 11, pp. 153-175, 1994.

....with set theoretic operations by CSG like scheme. The exact analytical definitions of set theoretic operations have been proposed by Rvachev in the theory of R functions and applied for solving problems of mathematical physics. Applications of R functions in solid modeling are described in [9, 10]. In the following considerations, f f 1 2 stands for R intersection, f f 1 2 means R union, f f 1 2 stands for R subtraction and f for R complement. For example, if geometric object G 1 is defined as f 1 (x,y,z) 0 and geometric object G 2 is defined as f 2 (x,y,z) 0 then the ....

Shapiro V., "Real functions for representation of rigid solids", Computer Aided Geometric Design, vol.11, No.2, 1994, pp.153-175.

....The exact analytical definitions of set theoretic operations have been proposed by Rvachev in the theory of R functions (Rvachev[10] and applied for solving problems of mathematical physics. Applications of R functions in solid modelling are described in Shapiro[11] Pasko[12] Pasko[13] Shapiro[14], Pasko[15] In following considerations, f f 1 2 stands for R intersection, f f 1 2 means R union, f f 1 2 stands for R subtraction and f for R complement. For example, if geometric object G 1 is defined as f 1 (x,y,z) 0 and geometric object G 2 is defined as f 2 (x,y,z) 0 then the ....

V.Shapiro, Real functions for representation of rigid solids, Computer Aided Geometric Design, 11:2, 153-175 (1994). 13

....(from 8 to 35 hours on an HP 700 series workstation) 1.3 Set theoretic representations A convex polygon can be represented as the intersection of half planes. Then, to convert this set theoretic representation to a single real function one can use min max [12] or more complex R functions [13, 14]. A concave polygon needs to be represented by set theoretic operations on convex polygons or halfplanes. There are three basic approaches to do this: cell partition, convex decomposition, and monotone boolean formula. The cell partition [15, 6] results in a concave polygon represented by the ....

V. Shapiro. Real functions for representation of rigid solids, Computer Aided Geometric Design, Vol. 11, No. 2, 1994, pp. 151-160.

.... implicits , skeleton based implicits , set theoretic solids, sweeps, volumetric objects, parametric and procedural models [6] 9] Set theoretic operations are closed on this representation with the use of R functions C k continuous definitions introduced by Rvachev [8] see a survey in [10]) Many geometric operations are also closed on F rep. They are blending, offsetting, Cartesian product, bijective mapping, metamorphosis and others (see [6] for details) These operations generate new real continuous defining functions and provide the closure property of the representation. One ....

Shapiro V. Real functions for representation of rigid solids, Computer Aided Geometric Design, vol. 11, No.2, 1994, pp.153-175.

....consuming (from 8 to 35 hours on an HP700 series workstation) 1.4 Set theoretic representations A convex polygon can be represented as the intersection of half planes. Then, to convert this settheoretic representation to a single real function one can use min max [13] or more complex Rfunctions [14, 17]. A concave polygon needs to be represented by set theoretic operations on convex polygons or half planes. There are three basic approaches to do this: cell partition, convex decomposition, and monotone boolean formula. The cell partition [18, 6] results in a concave polygon represented by the ....

V. Shapiro. Real functions for representation of rigid solids, Computer Aided Geometric Design, Vol. 11, No. 2, 1994, pp. 151-160.

....one has to apply inverse mapping of a given point to evaluate the defining function. The B ezier clipping method [3] does not work for points outside the domain and is too slow for inside points. We propose here extended B ezier clipping to deal with these problems. The application of R functions [7, 10] makes set theoretic operations closed on the function representation. It means that a complex constructive object can be described by a single real function. Although to evaluate such a function is time consuming, this approach gives a basis for quite general solutions to blending, offsetting, ....

....we mentioned above, after functional clipping is applied a free form primitive has a defining function suitable to set theoretic operations using R functions. Figure 4(b) shows the result of settheoretic operations on the free form primitive. Details on these and other operations can be found in [5, 10]. Note that objects resulting from set theoretic operations with R functions are not guaranteed to be regular solids. They can have dangling low dimensional portions and their defining functions can have internal zeroes . In some practical applications non regular objects are required. If ....

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V. Shapiro, "Real functions for representation of rigid solids", Computer Aided Geometric Design 11, 2(152-175), 1994.

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V. Shapiro. Real functions for representation of rigid solids. Computer-Aided Geometric Design, 11(2):153--175, 1994.

....exist for a given solid and a fired set of primitives, and develops a test to detect such conditions. Finally, we explain in section 5 that well formed set theoretic solid representations can be trivially (syntactically translated into representations of solids by real valued implicit functions[39]; such functions facilitate novel methods of solving boundary value problems without meshing and have many other engineering applications. 1.2 Fundamental Difficulties There are several mathematically equivalent ways to define solidity[29] a common set theoretic model re quires that any solid ....

.... way to specify many other useful representations of point sets; for example, the recent surge in popularity of implicit real functions to represent solids[27, 21, 44] can be attributed to the rediscovery of R functions, which are directly constructed from the logic and set theoretic expressions[34, 39]. An attempt to unify standard, regularized and other set theoretic operations based on a hierarchy of algebras and space decompositions is described in[37, 38] These methods are also used in sections 4 and 5 of this paper. 2 Well formed expressions 2.1 Expressions and sets Representations of ....

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V. Shapiro. Real functions for representation of rigid solids. Computer-Aided Geometric Design, 11(2):153 175, 1994.

....but the theory of R functions[11, 12, 15] gives an algorithmic method for constructing functions that exactly represent virtually any geometric shape of interest in engineering. Specific constructions for solid modeling and computer graphics are described in numerous references, for example see [16, 6, 4]. In its simplest and most commonly used form, a function f serves as a characteristic function for the point set in the sense that its sign can be used to distinguish points belonging to the set from those points that are not in the set. For instance, one can assume that f 0 is true for ....

....and exterior of the solid, but this information is available from the boundary representation. In other words, if a PMC function ##p# returns 1, 0,or,1 depending on whether point p is in, on, or out of the solid respectively, then the function f = # is the usual solid defining implicit function [16]. Figures 17(b) and (c) show two functions for the same polygon constructed directly from the boundary representation of Washington Island: using the simple trimming in Figure 17(b) and using the normalized trimming in Figure 17(c) By comparison, the functions constructed from the boundary ....

V. Shapiro. Real functions for representation of rigid solids. Computer-Aided Geometric Design, 11(2):153--175, 1994.

....value problems stems from the recognition that the solution u can be written in the form of equation (5) and therefore depends on the ability to construct such a function for the specified geometric domain. Fortunately, this construction problem has been solved using the theory of R functions [22, 28, 25, 29]. Briefly, an R function is a real valued function whose sign is completely determined by the signs of its arguments. For example, the function xyz can be negative only when the number of its negative arguments is odd. Such functions encode Boolean logic functions and are called R functions. ....

....related to these tasks. 4.1 Constructing implicit functions The scope and applicability of RFM is largely determined by the ability to construct implicit functions for geometric domains. Fortunately, the required functions exist for virtually all geometric objects of interest in engineering [29], 14 Figure 10: Convergence test temperature distributions computed at the same time point using different time step and degrees of freedom and their construction can be completely automated using the theory of R functions [22, 28, 31] Any geometric object that can be represented by a ....

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V. Shapiro. Real functions for representation of rigid solids. Computer-Aided Geometric Design, 11(2):153--175, 1994.

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V. Shapiro. Real functions for representation of rigid solids. Computer-Aided Geometric Design, 11(2):153--175, 1994.

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V. Shapiro. Real functions for representation of rigid solids. Computer-Aided Geometric Design, 11(2):153--175, 1994.

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V. Shapiro, Real Functions For Representation Of Rigid Solids, Computer Aided Geometric Design, 11(2): 153-175, 1994.

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V. Shapiro. Real functions for representation of rigid solids. Computer-Aided Geometric Design, 11(2):153--175, 1994.

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V. Shapiro. Real functions for representation of rigid solids. Computer Aided Geometric Design, 11(2):152--175, 1994.

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Shapiro, V., "Real Functions for representation of rigid solids" Computer Aided Geometric Design, Vol. 11, 1994, PP. 153-175.

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Shapiro V (1994a) Real functions for representation of rigid solids, Computer Aided Geometric Design, 11(2):153-175.

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