Canopy surface technique for parametric blending

A method based on canopy surfaces is presented for blending parametric surfaces. The blend designed using this method gives freedom to the designer in selecting i) the primitive surfaces to be blended, ii) the contact curves lying on them (in which the blend meets the primitives), iii) the endtangent directions along the contact curves (which are used to ensure tangentplane continuity), and iv) the shape of the cross-sectional curve. An important feature of this method, which is not seen in earlier methods, is the use of end tangents to ensure theC ¹ continuity of iso-parametric curves across the junctions between the blend and the primitives in addition to the tangent-plane continuity.     

Designing parametric blends: surface model and geometric correspondence

A model based on the curve-modulation technique is proposed for designing a parametric blend between two parametrically defined surfaces. The modulants are the cross section and the spine curves. The end position and tangency conditions of the cross section are discussed. Its intermediate shape can be left to the designer's choice, and generally depends on the application at hand. Various alternative choices, that fulfill the end conditions are suggested; this provides the flexibility demanded by various applications. The spine is designed as the intersection of two derived surfaces. Two alternative surface derivations are proposed, and both are discussed in some detail with their merits and demerits. The first derivation generates the contact curves automatically and relieves the designer from specifying them; the second one accepts contact curves specified by the designer, and generates the spine accordingly. Both of them are equally important in CAD/CAM and solid modelling applications.     

Parametric circles and spheres

Formulations for parametric circles and spheres in terms of rational Gaussian (RaG) curves and surfaces are introduced. With the proposed formulations, a full circle is generated by interpolating a closed RaG curve to the vertices of an equilateral triangle, and a full sphere is generated by interpolating a closed RaG surface to the vertices of an octahedron with equilateral triangular faces. Generation of circles and spheres in this manner is very intuitive and easy to remember as the weights are all 1 and the nodes are all unique and uniformly spaced.     

Data partitioning for parallel solid modelling

Solid modelling involves processing large amounts of geometric data. Distributed processing is a promising technique for improving the speed of computationally intensive processes. Solid modelling is thus a good candidate for parallel processing. We present a method for distributing entities of solid models in an array of processors for intersection tests in evaluating boolean operations. We employ distributed boundary representation and a recursive spatial subdivision technique for data partitioning. Parallel algorithms distribute entities among the array of processors mapped to a set of 3D rectangular regions in the object space. Entities intersecting or residing in the intersection regions of the objects are distributed. An experimental system was implemented on a DECmpp 12000/Sx/8K distributed memory SIMD computer.     

Adaptive polygonization of parametric surfaces

This paper describes an algorithm for dividing a (u, v)-parametric surface into the least possible number of plane triangular elements that closely follow the surface they approximate. The concept, of closeness defined here is based on a set of three subdivision criteria: unit-normal criterion, chord-distance criterion, and edge-refinement criterion.The triangles generated by the subdivision, together with the unit normals at their vertices, serve as input to the ray-tracing program, which generates a shaded image of the surface with photographic realism.     

Approximate implicitization of parametric surfaces by using compactly supported radial basis functions

In this paper, a new approach is proposed to solve the approximate implicitization of parametric surfaces. It is primarily based on multivariate interpolation of scattered data by using compactly supported radial basis functions. Experimental results are provided to illustrate the proposed method is flexible and effective.     

A comparison of sampling strategies for computing general sweeps

Sweeping moving objects has become one of the basic geometric operations used in engineering design, analysis and physical simulation. Despite its relevance, computing the boundary of the set swept by a non-polyhedral moving object is largely an open problem due to well-known theoretical and computational difficulties of the envelopes.We have recently introduced a generic point membership classification (PMC) test for general solid sweeping. Importantly, this PMC test provides complete geometric information about the set swept by the moving object, including the ability to compute the self-intersections of the sweep itself. In this paper, we compare two recursive strategies for sampling points of the space in which the object moves, and show that the sampling based on a fast marching cubes algorithm possesses the best combination of features in terms of performance and accuracy for the boundary evaluation of general sweeps. Furthermore, we show that the PMC test can be used as the foundation of a generic sweep boundary evaluator in conjunction with efficient space sampling strategies for solids of arbitrary complexity undergoing affine motions.     

A methodology for solid modelling in a virtual reality environment

This paper presents a methodology for solid modelling in a virtual reality (VR) environment. Solid modelling in the VR environment is precisely performed in an intuitive manner through constraint-based manipulations. A hierarchically structured and constraint-based data model is developed to support solid modelling in the VR environment. Constraint-based manipulations are realized by allowable motions for precise 3D interactions in the VR environment. A mathematical matrix is presented for representing allowable motions. A procedure-based degree-of-freedom combination method for 3D constraint solving is presented for deriving the allowable motions from constraints. A rule-based constraint recognition engine is developed for both constraint-based manipulations and implicitly incorporating constraints into the VR environment. A prototype system has been implemented to testify the feasibility of the presented methodology.     

On geometric interpolation of parametric surfaces

By exploiting the freedom in the choice of parametrization of a parametric surface, we show that there exist quadratic parametric surfaces that approximate a given parametric surface with the same approximation order as cubics, namely four, in the neighbourhood of a point where the Gaussian curvature of the surface is nonzero. This provides a first generalization to surfaces of earlier work for curves.     

Coefficient formula and matrix of nonuniform B-spline functions

The paper derives a coefficient formula of nonuniform B-spline functions of arbitrary degree from the Coxde Boor recursive algorithm. An efficient numerical algorithm for the coefficient matrix of nonuniform B-spline functions is also presented that is based on this formula. The results in the paper are useful for the evaluation and conversion of NURBS curves and surfaces in CAD/CAM systems. They will promote the application of product data-exchange standards in industry.     

Angular interpolation of bi-parameter curves

This paper presents an approach to the interpolation of angular feedrate for bi-parameter curve paths in multi-axis machining. A bi-parameter curve is the intersection of a parametric surface and an implicit surface. A tool path is identified by a position curve and an orientation curve, both of which are generated based on the bi-parameter curve. The angular feedrate interpolator calculates the tool position and orientation at each sampling cycle according to the specified angular feedrates and the given tool path. The paper analytically relates the angular arc-length derivatives to the time derivatives of the parameters along the path making use of both angular feedrates and angular feed acceleration. The results are then used to interpolate the parameters of the bi-parameter curve leading to an accurate calculation of the position and orientation of the cutting tool. A general parametric surface has been used to verify the effectiveness of the algorithm. The bi-parameter curves of the surface have been computed for arbitrarily selected intersecting cylinders.     

Hierarchical structure to winged-edge structure: a conversion algorithm

The winged-edge data structure is advantageous for traversing the topological graph of the boundary representation of a solid object. This paper presents an algorithm for converting hierarchical boundary representations into representations in the winged-edge data structure. As a result of the conversion, the adjacency relationships of geometric entities embedded in hierarchical boundary representations,-which may be evaluated through boundary evaluation on solid objects defined via Boolean set-operations, can be easily and efficiently accessed.     

Recovering a Smooth Boundary Representation from an Edge Quadtree and from a Face Octree

The Face Octrees are a representation scheme which is very suitable to represent both smooth free form surfaces and solids with smooth boundary, as well as, to perform operations among them. However, the Face Octree representation is non continuous, thus an algorithm to convert this model to a boundary representation can be necessary. The conversion algorithm makes the advantages of this scheme really significant. The current paper is involved in the conversion from the Face Octrees model to a smooth boundary representation. However, first the planar case is treated.     

Integration of parametric modelling in web-based knowledge-based engineering applications

Knowledge-based Engineering (KBE) is a methodology to efficiently handle complex and iterative processes as well as simple recurring tasks in CAD-based product design. Even though Knowledge-based Engineering’s roots go back as far as the late 20th century, potential of attractive research is still given. As geometry handling in native full Knowledge-based Engineering approaches can still be a challenging and time-consuming task, approaches for utilizing the parametric modeling capabilities of high-level CAD packages were developed. With the rise of web-based technologies, bringing these utilizing approaches to the Web, seems to be the next step in the technological development of Knowledge-based Engineering applications. Even though KBE has found its way to the Web already in many different sectors, planning of large systems based on existing machinery and subsystems, is still at least one open field left. This article connects to a newly developed KBE methodology, tailored for the use within intralogistic system planning, but widely usable in other systems and proposes a software architecture to utilize CAD’s capabilities regarding geometry creation and manipulation by the parametric modelling method in a web-based environment. The architecture is especially designed for implementations of the Knowledge-based Engineering method described within the research context in the field of intralogistics system planning.     

Bridging qualitative spatial constraints and feature-based parametric modelling: Expressing visibility and movement constraints

We present a concept for integrating state-of-the-art methods in geometric and qualitative spatial representation and reasoning with feature-based parametric modelling systems. Using a case-study involving a combination of topological, visibility, and movement constraints, we demonstrate the manner in which a parametric model may be constrained by the spatial aspects of conceptual design specifications and higher-level semantic design requirements. We demonstrate the proposed methodology by applying it to architectural floor plan layout design, where a number of spaces with well defined functionalities have to be arranged such that particular functional design constraints are maintained. The case-study is developed by an integration of the declarative spatial reasoning system CLP(QS) (CLP(QS) – a declarative spatial reasoning system. www.spatial-reasoning.com.) with the parametric CAD system FreeCAD.     

Fairing of Parametric Cubic B-spline Curves and Bicubic B-spline Surfaces

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Trimming of free-form objects for a B-Rep solid modeller

This paper describes a technique for trimming solid objects in a B-Rep solid modelling system. A B-Rep scheme based on the split-edge structure, which is suitable for the representation of trimmed surfaces, is first discussed. Algorithms for graph traversal, graph merging and graph splitting are then given in detail. Based on this B-Rep scheme, a technique for trimming objects is developed. The trimming process involves the evaluation of surface intersections, graph cutting along intersections and merging of appropriate graphs to obtain the required trimmed objects.     

Analysis and verification of the boundary surfaces of solid objects

Triangulation of complex three-dimensional objects is a time-consuming process, and the boundary surfaces of the object have to be checked very carefully to ensure no crror is made. With the increased use of computer graphics in the generation of complicated object shapes, such as aircraft, space vessels, machine parts, physical models, etc., visual inspection alone is not good enough to affirm the validity of the object, and a thorough verification of the output from a CAD system is necessary to ensure that the object is well defined and complies with the input requirements of the mesh generator.This paper describes such a data verification procedure for general curved surfaces and objects defined by the boundary surface modelling technique. The quality of individual elements, the overall topological structures, and geometrical correctness in terms of intersections, close touches and sharp angles will all be studied and verified. Several engineering objects are analysed to illustrate the practical applications of the procedure.