Parametric Surface Representation Research Articles

Curvature continuous connections of cones and cylinders

Dupin cyclides are used to obtain connections of cones and cylinders with continuous tangent planes. However, it is impossible to obtain connections with continuous principal curvatures in the same way. In the paper, curvature continuous connections are formed by using so-called normal ringed surfaces. Like cyclides, these surfaces are generated by a circle of variable radius sweeping through space. Further, the curve of the centres is perpendicular to the planes of the circles. A parametric representation of normal ringed surfaces is given which allows a simple algorithm to be developed for the construction of curvature continuous connections of cones and/or cylinders.

Parametrization of Closed Surfaces for 3-D Shape Description

This paper presents procedures for the explicit parametric representation and global description of surfaces of simply connected 3-D objects. The novel techniques overcome severe limitations of earlier methods (restriction to star-shaped objects (D. H. Ballard and Ch. M. Brown, Computer Vision, Prentice-Hall, Englewood Cliffs, NJ, 1981), constraints on positioning and shape of cross-sections (F. Solina and R. Bajcsy, IEEE Trans. Pattern Anal. Much. Intell. 12(2), 1990, 131-147; L. H. Staib and J. S. Duncan, in Visualization in Biomedical Computing 1992 (R. A. Robb, Ed.), Vol. Proc. SPIE 108, pp. 90-104, 1992), and nonhomogeneous distribution of parameter space). We parametrize the surface by defining a continuous, one-to-one mapping from the surface of the original object to the surface of a unit sphere. The parametrization is formulated as a constrained optimization problem. Practicable starting values are obtained by an initial mapping based on a heat conduction model. The parametrization enables us to expand the object surface into a series of spherical harmonic functions, extending to 3-D the concept of elliptical Fourier descriptors for 2-D closed curves (E. Persoon and K. S. Fu, IEEE Trans. Syst. Man Cybernetics 7(3), 1977, 388-397; F. P. Kuhl and Ch. R. Giardina, Comput. Graphics Image Process. 18(3), 1982, 236-258). Invariant, object-centered descriptors are obtained by rotating the parameter net and the object into standard positions. The new methods are illustrated with 3-D test objects. Potential applications are recognition, classification, and comparison of convoluted surfaces or parts of surfaces of 3-D shapes.

Semi-algebraic solids in 3-space: a survey of modelling schemes and implications for view graphs

The formal description of the bounding surfaces of a solid in 3-space represents the input formula for a view (or aspect) graph algorithm (VGA). The design of solids in 3-space and algorithms for computing formal descriptions of the boundaries of such solids are being studied within the field of computer aided (geometric) design (CA(G)D). In this survey, we introduce a specialization hierarchy for the major surface classes in CA(G)D and VGA research, in which the bounding surfaces of semi-algebraic solids represent the most general class. We state upper bounds for the nodes in the view graph for the surface classes in this hierarchy. We also describe possible interactions between symbolic CA(G)D algorithms (like conversion algorithms between parametric and implicit surface representations and boundary-representation algorithms) and symbolic VGAs and the consequences for the efficiency of VGAs.

A data reduction scheme for triangulated surfaces

Given a surface triangulation in three-dimensional space, an algorithm is developed to iteratively remove triangles from the triangulation. An underlying parametric or implicit surface representation is not required. An order is introduced on the set of triangles by considering curvature at their vertices. Triangles in nearly planar surface regions are prime candidates for removal. The degree of reduction can be specified by a percentage or, in the case of bivariate functions, by an error tolerance.

Volumetric error analysis of a multi-axis machine tool machining a sculptured surface workpiece

Abstract The objective of this research is to develop a geometric error model for multi-axis machine tools based on a closed-loop configuration. In this study, the geometric error model of each axis is derived by using 4×4 homogeneous transformation matrix. The ideal cutter location of the sculptured workpiece surface is calculated using the Bezier bicubic parametric surface representation method and machine geometric errors. The actual cutter locations is calculated by considering runout error of the cutting tool and machine geometric errors. Then, the step-by-step volumetric error analysis method is suggested on the basis of the closed-loop configuration of the multi-axis machine tools. The simulation study shows the simplicity and effectiveness of the proposed strategy.

Approximation and characterization of molecular surfaces

The representation and characterization of molecular surfaces are important in many areas of molecular modeling. Parametric representations of protein molecular surfaces are a compact way to describe a surface, and are useful for the evaluation of surface properties such as the normal vector, principal curvatures, and principal curvature directions. Simplified representations of molecular surfaces are useful for efficient rendering and for the display of large-scale surface features. Several techniques for representing surfaces by expansions of spherical harmonic functions have been reported, but these techniques require that the radius function is single valued, that is, each ray from an origin inside the surface intersects the surface at one and only one point. A new technique is described that removes this limitation and can be used to compute surface shape properties.

Bézier representation for cubic surface patches

The paper describes a new method for creating rectangular Bézier surface patches on an implicit cubic surface. Traditional techniques for representing surfaces have relied on parametric representations of surfaces, which, in general, generate surfaces of implicit degree 8 in the case of rectangular Bézier surfaces with rational biquadratic parameterization. The method constructs low-degree algebraic surface patches by reducing the implicit degree from 8 to 3. The construction uses a rectangular biquadratic Bézier control polyhedron that is embedded within a tetrahedron and satisfies a projective constraint. The control polyhedron and the resulting cubic surface patch satisfy all of the standard properties of parametric Bézier surfaces, including interpolation of the corners of the control polyhedron and the convex-hull property.

Implicit representation of rational parametric surfaces

In many applications we need to compute the implicit representation of rational parametric surfaces. Previously, resultants and Gröbner bases have been applied to this problem. However, these techniques at times result in an extraneous factors along with the implicit equation and fail altogether when a parametrization has base points. In this paper we present algorithms to implicitize rational parametric surfaces with and without base points. One of the strength of the algorithms lies in the fact that we do not use multivariate factorization. The base points blow up to rational curves on the surface and we present techniques to compute the rational parametrization of the blow up curves.

Parametric representation of quadric surfaces

Parametric representation of quadric surfaces

Feature-based design and finite element mesh generation for functional surfaces

This paper describes an approach to feature-based design and feature-based mesh generation for multi-featured functional surfaces. Unlike standard free-form parametric surface representation where the parametric surface patch plays the key role in both surface design and finite element mesh generation, we propose an approach to these two tasks which proceeds from the level of a complete feature (for example, a pocket or channel). The result is a more direct method for modeling functional surface characteristics and a more efficient feature-based implementation of Delaunay surface triangulation.

Geometry-adaptive surface grid generation using a parametric projection

A general surface grid generation technique is presented that produces grids on arbitrary three-dimensional surface patches, with geometry-adaptive grid control. The method uses a two-dimensional parametric representation of the surface patch obtained by projecting the surface onto a locally defined screen. Arbitrary surfaces, including those with geometric singularities, are transformed into simple domains in the parametric space. A grid is generated on the parametric domain using a two-dimensional grid generation technique and the grid points are mapped back to the original surface. The method is applied to a variety of configuration s with different geometrical difficulties. ECENT advances in computational technology are mak- ing it possible to obtain numerical solutions to problems with increasing degrees of physical and geometrical complex- ity. As the geometry becomes more complicated and the desired level of confidence in numerical results increases, the demand for suitable grids also increases. The accuracy and efficiency of numerical solutions depend on certain grid prop- erties, such as grid topology, grid density, grid skewness, grid smoothness, cell aspect ratio, and variations of these proper- ties throughout the grid. Also, the total number of grid points should be kept to a minimum for computational economy. The optimum grid qualities suggested by these factors are often conflicting and compromises must be made. Improved grid generation techniques are needed as physical and geomet- rical complexities increase. After choosing a grid topology, surface grid generation is the next step in providing field grids for three-dimensional flow calculations. However, few techniques have been devel- oped specifically for surface grid generation. Algebraic meth- ods are often used, but most of these methods are based on assumed configurations or require excessive amounts of input. Other approaches include a method discussed by Thompson et al.1 using a three-dimensional curvilinear coordinate system with one coordinate constant on the surface. Also, Thomas2 proposed a surface grid generation scheme that uses a quasi- two-dimensional elliptic system for analytically defined, smooth surfaces. The method was later extended by Takagi and Miki3 for arbitrary curved surfaces using a parametric surface representation. Eiseman4 has demonstrated an adap- tive surface grid generation scheme that distributes grid points to optimal positions based on flow solutions. The purpose of this paper is to describe and demonstrate a surface grid generation technique that can be applied to arbitrary surface patches. The method uses a projection con- cept to define a parametric representation of the surface patch and a two-dimensional grid generation technique on the para- metric domain. Geometry-adaptive grid control is achieved through the definition of the parametric coordinates. First, an overall description of the method is presented, followed by discussions of specific items used in the implementation of the method. Then, several example applications are presented to demonstrate the capabilities of the method.

Rectangular v-Splines

This article describes and presents examples of some techniques for the representation and interactive design of surfaces based on a parametric surface representation that user v-spline curves. These v-spline curves, similar in mathematical structure to v-splines, were developed as a more computationally efficient alternative to splines in tension. Although splines in tension can be modified to allow tension to be applied at each control point, the procedure is computationally expensive. The v-spline curve, however, uses more computationally tractable piecewise cubic curves segments, resulting in curves that are just as smoothly joined as those of a standard cubic spline. After presenting a review of v-splines and some new properties, this article extends their application to a rectangular grid of control points. Three techniques and some application examples are presented.

Recursive algorithms for the representation of parametric curves and surfaces

Recursive algorithms for the representation of parametric curves and surfaces are presented which are based upon a geometric property of the de Casteljau algorithm. The algorithms work with triangular and pyramidal arrays that provide an easy handling of the curve and the surface ‘in a large’ design.

Implicit representation of parametric curves and surfaces

The following two problems are shown to have closed-form solutions requiring only the arithmetic operations of addition, subtraction, multiplication and division: (1) Given a curve or surface defined parametrically in terms of rational polynomials, find an implicit polynomial equation which defines the same curve or surface. (2) Given the Cartesian coordinates of a point on such a curve or surface, find the parameter(s) corresponding to that point. It is shown that a two-dimensional curve defined parametrically in terms of rational degree n polynomials in t can be expressed implicitly as a degree n polynomial in z and y. It is also demonstrated that a “bi- m-ic” parametric surface (where e.g., m = 3 for bicubic) can be expressed implicitly as a polynomial in x, y, z of degree 2 m 2. The degree of a rational bi- m-ic surface is also shown to be 2 m 2. The application of these results to finding curve and surface intersections is discussed.

Implicit representation of parametric curves and surfaces

Implicit representation of parametric curves and surfaces

Accurate field computation with the boundary element method

Interface problems in magnetostatics are formulated as boundary integral equations of the second kind involving the appropriate scalar (no current sources) or vector potentials. The boundary element method (BEM), which employs parametric representation of surfaces and sources, is used to solve some two-dimensional examples by way of illustration. A novel approach, automated to address Green's functions singularities over arbitrarily-shaped geometries, is introduced.

On bounded variation and absolute continuity for parametric representations of continuous surfaces

where each of the functions x(u), y(u), z(u) is continuous on the closed interval [a, b]. The following facts are known(') (see Rado [3, chap. I]; Saks [1, chap. IV]). (1) A necessary and sufficient condition that the length L(C) of C be finite is that each of the functions x(u), y(u), z(u) be of bounded variation on [a, b]. (2) If the length L(C) is finite, then each of the derivatiVes x'(u), y'(u), z'(u) exists almost everywhere in [a, b], is summable on [a, b], and