Boolean Set Operations Research Articles

Decidability of the Satisfiability Problem for Boolean Set Theory with the Unordered Cartesian Product Operator

We give a positive solution to the decidability problem for the fragment of set theory, dubbed BST ⊗, consisting of quantifier-free formulae involving the Boolean set operators of union, intersection, and set difference, along with the unordered Cartesian product operator ⊗ (where \(s \otimes t := \big \lbrace \lbrace u,v\rbrace \,\texttt {|}\:u \in s \wedge v \in t \big \rbrace\) ), and the equality predicate, but no membership. Specifically, we provide nondeterministic exponential decision procedures for both the ordinary and the finite satisfiability problems for BST ⊗. We expect that these decision procedures can be adapted for the standard Cartesian product and, with added technicalities, to the cases involving membership, providing a solution to a longstanding problem in computable set theory.

BoolSurf

We port Boolean set operations between 2D shapes to surfaces of any genus, with any number of open boundaries. We combine shapes bounded by sets of freely intersecting loops, consisting of geodesic lines and cubic Bézier splines lying on a surface. We compute the arrangement of shapes directly on the surface and assign integer labels to the cells of such arrangement. Differently from the Euclidean case, some arrangements on a manifold may be inconsistent. We detect inconsistent arrangements and help the user to resolve them. Also, we extend to the manifold setting recent work on Boundary-Sampled Halfspaces, thus supporting operations more general than standard Booleans, which are well defined on inconsistent arrangements, too. Our implementation discretizes the input shapes into polylines at an arbitrary resolution, independent of the level of resolution of the underlying mesh. We resolve the arrangement inside each triangle of the mesh independently and combine the results to reconstruct both the boundaries and the interior of each cell in the arrangement. We reconstruct the control points of curves bounding cells, in order to free the result from discretization and provide an output in vector format. We support interactive usage, editing shapes consisting up to 100k line segments on meshes of up to 1M triangles.

A novel algorithm of the digital nervous tissue phantom creation based on 3D Voronoi diagram application

The essential part of mathematical modelling of nutrients convectional reaction-diffusion is creation of a digital phantom of considered biological object. This process becomes an especial problem which needs to be solved before numerical calculations of the concentration gradients will be done. There are two principal ways to get the solution in this case. The first approach is the reconstruction of a digital phantom on the base of the experimental data directly. The second one is the creation of a virtual object according to the experimental evidence and the known principals de novo. The main advantage of the created phantom is a high adaptability to modelling demands and a physical problem formulation. In the present study a new algorithm of a digital phantom creation has been established. The principles of the claimed procedures are demonstrated by the example of a nervous tissue. Initially, one needs to create N 3D objects according to Voronoi diagrams. Each object has 144 edges and 69 boundaries on average. Having chosen M rear objects, a long 3D structure mimicking neurons axons is created according to a loft procedure from the start boundaries to the end ones. Then, the set of Boolean operations has been applied to form continuous smooth objects. The remain (N-(M+s)) objects are combined into several whole bodies using the loft procedures between the closet neighbours. The final structure has a good conformity with a nervous tissue architecture. Furthermore, the obtained phantom is correct to the mesh application and further numerical calculations.

Complexity assessments for decidable fragments of set theory. II: A taxonomy for ‘small’ languages involving membership

We carry on a long-standing investigation aimed at identifying fragments of set theory that are potentially useful in automated verification with proof-checkers, such as ÆtnaNova, based on the set-theoretic formalism. This note provides a complete taxonomy of the polynomial and the NP-complete fragments consisting of all conjunctions that involve, besides variables intended to range over the von Neumann set-universe, a collection of constructs drawn from the Boolean set operators ∪,∩,∖ and the membership relators ∈ and ∉. This is done in sight of combining the aforementioned taxonomy with one recently put together for analogous fragments involving, in place of the relators ∈ and ∉, the Boolean relators ⊆,= and the predicates ‘⋅=∅’ and ‘Disj(⋅,⋅)’ (respectively meaning ‘the argument set is empty’ and ‘the arguments are disjoint sets’), along with their opposites ‘⊈,≠,⋅≠∅’ and ‘¬Disj(⋅,⋅)’.

Rapid Identification of Post-Earthquake Collapsed Buildings via Multi-Scale Morphological Profiles With Multi-Structuring Elements

Rapid identification of post-earthquake collapsed buildings can be conducive to capturing an immediate disaster assessment, which can help design effective emergency response strategies. At present, earthquake disaster assessment in practice mainly relies on time-consuming artificial field investigations, which cannot adapt to the demands of a timely rescue. However, morphological methods have proved to be of great potential in describing the disaster characteristics of building as for size, shape, directionality and contrast. Conventionally, boundaries of intact building appear flat and smooth, thus it can be reasonably depicted by linear structural element, while collapsed buildings generally emerge dispersedly distributed without regular geometry which can be suitably described by disk structural element. Based on these intuitive findings, we propose an approach for extracting intact and collapsed buildings via multi-scale morphological profiles with multi-structuring elements from post-earthquake satellite imagery. This approach consists of three core components: 1) Linear and disk structuring elements are established through top-hat reconstruction for extracting the initial intact and collapsed building, respectively. 2) Purified intact and collapsed buildings can be obtained by a straightforward threshold segmentation and further post processing, such as area, normalized green plant index (NGPI) and the length-width ratio for image noise suppression. 3) Boolean set operations are adopted to distinguish the intact and collapsed buildings. Experiments demonstrate that the proposed method has achieved satisfactory results and exerted great superiority in computational efficiency.

Higher-Order Accurate Meshing of Nonsmooth Implicitly Defined Surfaces and Intersection Curves

A higher-order accurate meshing algorithm for nonsmooth surfaces defined via Boolean set operations from smooth surfaces is presented. Input data are a set of level-set functions and a bounding box containing the domain of interest. This geometry definition allows the treatment of edges as intersection curves. Initially, the given bounding box is partitioned with an octree that is used to locate corners and points on the intersection curves. Once a point on an intersection curve is found, the edge is traced. Smooth surfaces are discretized using marching cubes and then merged together with the advancing-front method. The piecewise linear geometry is lifted by projecting the inner nodes of the Lagrangian elements onto the surface or intersection curve. To maintain an accurate mesh, special attention is paid to the accurate meshing of tangential intersection curves. Optimal convergence properties for approximation problems are confirmed in numerical studies.

Cardinality of generating sets for operations from the Post lattice classes

Abstract We obtain precise values of cardinality of minimal generating sets for all Cartesian products of two-element set with respect to an arbitrary set of Boolean operations from the central part of the Post lattice. For the case of sets containing operations from the remaining classes of the Post lattice we obtain cardinality estimations that are accurate up to one.

On an algorithmic definition for the components of the minimal cell.

Living cells are highly complex systems comprising a multitude of elements that are engaged in the many convoluted processes observed during the cell cycle. However, not all elements and processes are essential for cell survival and reproduction under steady-state environmental conditions. To distinguish between essential from expendable cell components and thus define the ‘minimal cell’ and the corresponding ‘minimal genome’, we postulate that the synthesis of all cell elements can be represented as a finite set of binary operators, and within this framework we show that cell elements that depend on their previous existence to be synthesized are those that are essential for cell survival. An algorithm to distinguish essential cell elements is presented and demonstrated within an interactome. Data and functions implementing the algorithm are given as supporting information. We expect that this algorithmic approach will lead to the determination of the complete interactome of the minimal cell, which could then be experimentally validated. The assumptions behind this hypothesis as well as its consequences for experimental and theoretical biology are discussed.

The Satisfiability Problem for Boolean Set Theory with a Choice Correspondence

Given a set U of alternatives, a choice (correspondence) on U is a contractive map c defined on a family Omega of nonempty subsets of U. Semantically, a choice c associates to each menu A in Omega a nonempty subset c(A) of A comprising all elements of A that are deemed selectable by an agent. A choice on U is total if its domain is the powerset of U minus the empty set, and partial otherwise. According to the theory of revealed preferences, a choice is rationalizable if it can be retrieved from a binary relation on U by taking all maximal elements of each menu. It is well-known that rationalizable choices are characterized by the satisfaction of suitable axioms of consistency, which codify logical rules of selection within menus. For instance, WARP (Weak Axiom of Revealed Preference) characterizes choices rationalizable by a transitive relation. Here we study the satisfiability problem for unquantified formulae of an elementary fragment of set theory involving a choice function symbol c, the Boolean set operators and the singleton, the equality and inclusion predicates, and the propositional connectives. In particular, we consider the cases in which the interpretation of c satisfies any combination of two specific axioms of consistency, whose conjunction is equivalent to WARP. In two cases we prove that the related satisfiability problem is NP-complete, whereas in the remaining cases we obtain NP-completeness under the additional assumption that the number of choice terms is constant.

Dualities in the class of extended Boolean functions

We introduce and discuss duality operators on the set of binary extended Boolean functions, i.e., on the set of binary operations on the real interval [0,1] whose restrictions to Boolean inputs yield Boolean functions. These dualities have been divided into seven classes, and the majority of their properties depend on the class they belong to. We study composition of dualities, properties of dual classes, duality of properties of extended Boolean functions and invariantness of extended Boolean functions with respect to particular dualities. Our approach allows to transfer the results known for some studied class of extended Boolean functions into the results for the corresponding dual classes. As typical examples, one can recall the standard duality of the classes of all t-norms and t-conorms and the duality of implication functions and conjunctive (disjunctive) aggregation functions.

The logical primitives of thought: Empirical foundations for compositional cognitive models.

The notion of a compositional language of thought (LOT) has been central in computational accounts of cognition from earliest attempts (Boole, 1854; Fodor, 1975) to the present day (Feldman, 2000; Penn, Holyoak, & Povinelli, 2008; Fodor, 2008; Kemp, 2012; Goodman, Tenenbaum, & Gerstenberg, 2015). Recent modeling work shows how statistical inferences over compositionally structured hypothesis spaces might explain learning and development across a variety of domains. However, the primitive components of such representations are typically assumed a priori by modelers and theoreticians rather than determined empirically. We show how different sets of LOT primitives, embedded in a psychologically realistic approximate Bayesian inference framework, systematically predict distinct learning curves in rule-based concept learning experiments. We use this feature of LOT models to design a set of large-scale concept learning experiments that can determine the most likely primitives for psychological concepts involving Boolean connectives and quantification. Subjects' inferences are most consistent with a rich (nonminimal) set of Boolean operations, including first-order, but not second-order, quantification. Our results more generally show how specific LOT theories can be distinguished empirically. (PsycINFO Database Record

On language equations with concatenation and various sets of Boolean operations

Systems of equations of the form X i = ϕ i (X 1 , ... , X n ), for 1 ⩽ i ⩽ n , in which the unknowns X i are formal languages, and the right-hand sides ϕ i may contain concatenation and union, are known for representing context-free grammars. If, instead of union only, another set of Boolean operations is used, the expressive power of such equations may change: for example, using both union and intersection leads to conjunctive grammars [A. Okhotin, J. Automata, Languages and Combinatorics 6 (2001) 519–535], whereas using all Boolean operations allows all recursive sets to be expressed by unique solutions [A. Okhotin, Decision problems for language equations with Boolean operations, Automata, Languages and Programming, ICALP 2003, Eindhoven, The Netherlands, 239–251]. This paper investigates the expressive power of such equations with any possible set of Boolean operations. It is determined that different sets of Boolean operations give rise to exactly seven families of formal languages: the recursive languages, the conjunctive languages, the context-free languages, a certain family incomparable with the context-free languages, as well as three subregular families.

Automating Boolean Set Operations in Mizar Proof Checking with the Aid of an External SAT Solver

In this paper we present the results of an experiment with employing an external SAT solver to strengthen the notion of obviousness of the Mizar proof checker. The presented extension of the Mizar system is based on a version of MiniSAT, called Logic2CNF. The SAT-enhanced Mizar checker is programmed to automatically spawn a new Logic2CNF process whenever it needs to justify any goal that can be solved by reducing it into a corresponding propositional satisfiability problem (equalities based on Boolean operations or set inclusion). The external tool is interfaced within the implementation of Mizar's requirements directives.

Sequence binary decision diagram: Minimization, relationship to acyclic automata, and complexities of Boolean set operations

The manipulation of large sequence data is one of the most important problems in string processing. In this paper, we discuss a new data structure for storing and manipulating sets of strings, called Sequence Binary Decision Diagrams (sequence BDDs), which has recently been introduced by Loekito et al. (2010) as a descendant of both acyclic DFAs (ADFAs) and binary decision diagrams (BDDs). Sequence BDDs can compactly represent sets of sequences similarly to minimal ADFAs, and allow efficient set operations inherited from BDDs. We study fundamental properties of sequence BDDs, such as the characterization of minimal sequence BDDs by reduced sequence BDDs, non-trivial relationships between sizes of minimal sequence BDDs and minimal ADFAs, the complexities of minimization, Boolean set operations, and sequence BDD construction. We also show experimental results for real and artificial data sets.

Abstract 5322: Collections of simultaneously altered genes as highly predictive markers of cancer cell drug response

Abstract Introduction: Many genetic markers of drug response have been reported in cancer pharmacogenomics studies. However, most of these markers fail to achieve clinical utility because they capture response in only a small fraction of the patient population; this shortcoming might result from a reliance on single-gene markers in contemporary personalized medicine. While there is growing evidence that therapeutic response can be modulated by the concerted impact of multiple genetic alterations, the combinatorics associated with many-body interactions has mostly prohibited the discovery of multi-gene biomarkers, even using computational methods. We developed a computational approach, MOCA (Multivariate Organization of Combinatorial Alterations), to partially address these challenges. Extending our method that accounts for pairwise interactions, MOCA combines many genomic alterations into biomarkers of drug response, using Boolean set operations coupled with optimization; in this framework the union, intersection, and difference Boolean set operations are proxies of molecular redundancy, synergy, and interference. The algorithm is fast, broadly applicable to genomics data, of immediate utility for prioritizing cancer pharmacogenomics experiments, and recovers known clinical findings without bias. Furthermore, results from this work connect many important, previously isolated observations. Major Findings: When applied to 416 pharmacogenomically characterized cancer cell lines from the Cancer Cell Line Encyclopedia, MOCA identified many known and potential markers of drug response. For example, correlation of ERBB inhibitor response drastically increased when considering EGFR (ERBB1), ERBB2, ERBB3, ERBB4, and KRAS alterations in a single marker. Similarly, a marker combining IGF1, IGF1R, and RAD51 alterations drastically increased correlation with IGF1R inhibitor response, relative to any of these three genetic markers considered in isolation. This approach is also powerful for determining subsets of gene-specific mutations that increase correlation with drug response. For instance, MOCA accurately captures the differential EGFR inhibitor response conferred by known activating EGFR mutations. Similarly, we find specific HDAC1 mutations cooperate with HDAC5 overexpression to potentiate cells to the HDAC inhibitor panobinostat. Additionally, considering all pairwise gene-drug interactions, MOCA recovers known and compelling correlations, including: RTK inhibitor resistance via c-MET, EGFR, ERBB2, and PDGFRB kinase switching; mutual exclusivity of TP53 mutation and response to the MDM2 inhibitor nutlin-3; greater nutlin-3 potentiation via MDM4, rather than MDM2, overexpression; MEK and RAF inhibitor response in BRAF mutated cell lines; and MEK inhibitor potentiating NRAS mutations. Masica, David L., & Rachel Karchin. Cancer Research 73 (2013): 1699-1708. Citation Format: David L. Masica, Rachel Karchin. Collections of simultaneously altered genes as highly predictive markers of cancer cell drug response. [abstract]. In: Proceedings of the 105th Annual Meeting of the American Association for Cancer Research; 2014 Apr 5-9; San Diego, CA. Philadelphia (PA): AACR; Cancer Res 2014;74(19 Suppl):Abstract nr 5322. doi:10.1158/1538-7445.AM2014-5322

Computer graphics: principles and practice

1. Introduction. Image Processing as Picture Analysis. The Advantages of Interactive Graphics. Representative Uses of Computer Graphics. Classification of Applications. Development of Hardware and Software for Computer Graphics. Conceptual Framework for Interactive Graphics. 2. Programming in the Simple Raster Graphics Package (SRGP)/. Drawing with SRGP/. Basic Interaction Handling/. Raster Graphics Features/. Limitations of SRGP/. 3. Basic Raster Graphics Algorithms for Drawing 2d Primitives. Overview. Scan Converting Lines. Scan Converting Circles. Scan Convertiing Ellipses. Filling Rectangles. Fillign Polygons. Filling Ellipse Arcs. Pattern Filling. Thick Primiives. Line Style and Pen Style. Clipping in a Raster World. Clipping Lines. Clipping Circles and Ellipses. Clipping Polygons. Generating Characters. SRGP_copyPixel. Antialiasing. 4. Graphics Hardware. Hardcopy Technologies. Display Technologies. Raster-Scan Display Systems. The Video Controller. Random-Scan Display Processor. Input Devices for Operator Interaction. Image Scanners. 5. Geometrical Transformations. 2D Transformations. Homogeneous Coordinates and Matrix Representation of 2D Transformations. Composition of 2D Transformations. The Window-to-Viewport Transformation. Efficiency. Matrix Representation of 3D Transformations. Composition of 3D Transformations. Transformations as a Change in Coordinate System. 6. Viewing in 3D. Projections. Specifying an Arbitrary 3D View. Examples of 3D Viewing. The Mathematics of Planar Geometric Projections. Implementing Planar Geometric Projections. Coordinate Systems. 7. Object Hierarchy and Simple PHIGS (SPHIGS). Geometric Modeling. Characteristics of Retained-Mode Graphics Packages. Defining and Displaying Structures. Modeling Transformations. Hierarchical Structure Networks. Matrix Composition in Display Traversal. Appearance-Attribute Handling in Hierarchy. Screen Updating and Rendering Modes. Structure Network Editing for Dynamic Effects. Interaction. Additional Output Features. Implementation Issues. Optimizing Display of Hierarchical Models. Limitations of Hierarchical Modeling in PHIGS. Alternative Forms of Hierarchical Modeling. 8. Input Devices, Interaction Techniques, and Interaction Tasks. Interaction Hardware. Basic Interaction Tasks. Composite Interaction Tasks. 9. Dialogue Design. The Form and Content of User-Computer Dialogues. User-Interfaces Styles. Important Design Considerations. Modes and Syntax. Visual Design. The Design Methodology. 10. User Interface Software. Basic Interaction-Handling Models. Windows-Management Systems. Output Handling in Window Systems. Input Handling in Window Systems. Interaction-Technique Toolkits. User-Interface Management Systems. 11. Representing Curves and Surfaces. Polygon Meshes. Parametric Cubic Curves. Parametric Bicubic Surfaces. Quadric Surfaces. 12. Solid Modeling. Representing Solids. Regularized Boolean Set Operations. Primitive Instancing. Sweep Representations. Boundary Representations. Spatial-Partitioning Representations. Constructive Solid Geometry. Comparison of Representations. User Interfaces for Solid Modeling. 13. Achromatic and Colored Light. Achromatic Light. Chromatic Color. Color Models for Raster Graphics. Reproducing Color. Using Color in Computer Graphics. 14. The Quest for Visual Realism. Why Realism? Fundamental Difficulties. Rendering Techniques for Line Drawings. Rendering Techniques for Shaded Images. Improved Object Models. Dynamics. Stereopsis. Improved Displays. Interacting with Our Other Senses. Aliasing and Antialiasing. 15. Visible-Surface Determination. Functions of Two Variables. Techniques for Efficient Visible-Surface Determination. Algorithms for Visible-Line Determination. The z-Buffer Algorithm. List-Priority Algorithms. Scan-Line Algorithms. Area-Subdivision Algorithms. Algorithms for Octrees. Algorithms for Curved Surfaces. Visible-Surface Ray Tracing. 16. Illumination And Shading. Illumination Modeling. Shading Models for Polygons. Surface Detail. Shadows. Transparency. Interobject Reflections. Physically Based Illumination Models. Extended Light Sources. Spectral Sampling. Improving the Camera Model. Global Illumination Algorithms. Recursive Ray Tracing. Radiosity Methods. The Rendering Pipeline. 17. Image Manipulation and Storage. What Is an Image? Filtering. Image Processing. Geometric Transformations of Images. Multipass Transformations. Image Compositing. Mechanisms for Image Storage. Special Effects with Images. Summary. 18. Advanced Raster Graphic Architecture. Simple Raster-Display System. Display-Processor Systems. Standard Graphics Pipeline. Introduction to Multiprocessing. Pipeline Front-End Architecture. Parallel Front-End Architectures. Multiprocessor Rasterization Architectures. Image-Parallel Rasterization. Object-Parallel Rasterization. Hybrid-Parallel Rasterization. Enhanced Display Capabilities. 19. Advanced Geometric and Raster Algorithms. Clipping. Scan-Converting Primitives. Antialiasing. The Special Problems of Text. Filling Algorithms. Making copyPixel Fast. The Shape Data Structure and Shape Algebra. Managing Windows with bitBlt. Page Description Languages. 20. Advanced Modeling Techniques. Extensions of Previous Techniques. Procedural Models. Fractal Models. Grammar-Based Models. Particle Systems. Volume Rendering. Physically Based Modeling. Special Models for Natural and Synthetic Objects. Automating Object Placement. 21. Animation. Conventional and Computer-Assisted Animation. Animation Languages. Methods of Controlling Animation. Basic Rules of Animation. Problems Peculiar to Animation. Appendix: Mathematics for Computer Graphics. Vector Spaces and Affine Spaces. Some Standard Constructions in Vector Spaces. Dot Products and Distances. Matrices. Linear and Affine Transformations. Eigenvalues and Eigenvectors. Newton-Raphson Iteration for Root Finding. Bibliography. Index. 0201848406T04062001

Isolating Influential Regions of Electrostatic Focusing in Protein and DNA Structure

Electrostatic focusing is a general phenomenon that occurs in cavities and grooves on the molecular surface of biomolecules. Narrow surface features can partially shield charged atoms from the high-dielectric solvent, enhancing electrostatic potentials inside the cavity and projecting electric field lines outward into the solvent. This effect has been observed in many instances and is widely considered in the human examination of molecular structure, but it is rarely integrated into the digital representations used in protein structure comparison software. To create a computational representation of electrostatic focusing, that is compatible with structure comparison algorithms, this paper presents an approach that generates three-dimensional solids that approximate regions where focusing occurs. We verify the accuracy of this representation against instances of focusing in proteins and DNA. Noting that this representation also identifies thin focusing regions on the molecular surface that are unlikely to affect binding, we describe a second algorithm that conservatively isolates larger focusing regions. The resulting 3D solids can be compared with Boolean set operations, permitting a new range of analyses on the regions where electrostatic focusing occurs. They also represent a novel integration of molecular shape and electrostatic focusing into the same structure comparison framework.

Collections of Simultaneously Altered Genes as Biomarkers of Cancer Cell Drug Response

Computational analysis of cancer pharmacogenomics data has resulted in biomarkers predictive of drug response, but the majority of response is not captured by current methods. Methods typically select single biomarkers or groups of related biomarkers but do not account for response that is strictly dependent on many simultaneous genetic alterations. This shortcoming reflects the combinatorics and multiple-testing problem associated with many-body biologic interactions. We developed a novel approach, Multivariate Organization of Combinatorial Alterations (MOCA), to partially address these challenges. Extending on previous work that accounts for pairwise interactions, the approach rapidly combines many genomic alterations into biomarkers of drug response, using Boolean set operations coupled with optimization; in this framework, the union, intersection, and difference Boolean set operations are proxies of molecular redundancy, synergy, and resistance, respectively. The algorithm is fast, broadly applicable to cancer genomics data, is of immediate use for prioritizing cancer pharmacogenomics experiments, and recovers known clinical findings without bias. Furthermore, the results presented here connect many important, previously isolated observations.

A Framework for Developing Boolean Set Implicit Blends from an Existing Union or Intersection Blend in Soft Object Modeling

In implicit surface, soft object modeling is getting popular because a complex soft object is able to be constructed easily by performing a soft (union) blend on primitive soft objects. Especially soft blend has lower computing complexity than other existing implicit blends because it dose addition operation only. However, in soft object modeling existing implicit blends still do not provide intersection blend containing blending range parameters for generating sequential blends, and precisely only intersection and union blends are developed. In order to solve this problem, this paper proposes two frameworks that can transform an existing union or intersection blend into a new family of Boolean set blending operations, including union, intersection, and difference blends. Based on the proposed frameworks, this paper has transformed scale function, developed from the scale method for an intersection blend, and soft blend, respectively, into two new groups of Boolean set blending operations with blending range parameters. The newly proposed Boolean set operations not only offer blending range parameters for blending range control but also have lower computing complexity and they especially can do bulge elimination in a union blend. 

Tool path generation by offsetting curves on polyhedral surfaces based on mesh flattening

Polyhedral surfaces are used as representation model for CAM and process planning purposes because of its simplicity for data exchange and geometric computation. However, there are few tool path planning strategies for such surfaces but isoplanar method. This paper presents a contour offset approach to tool path generation for three-axis ball-end milling of polyhedral surfaces, based on a novel method for offsetting curves on polyhedral surfaces. One of its salient features is to reduce the task of removing complex interfering of offsets from 3D physical surfaces to 2D plane by flattening mesh surfaces and avoid costly 3D Boolean set operations and relatively expensive distance calculation. Moreover, in practical implement, the procedures of calculating offset points and removing interfering loops are merged and carried out simultaneously results in an efficient tool path generation method. Empirical examples illustrate the feasibility of the proposed method.