Rational Polynomial Surface Intersections

Many modern geometric modelers use nonuniform rational B-spline curves and surfaces as their canonical representations. Rational B-splines are a versatile representation, encompassing integral B-splines and the basic classical primitives such as conics, quadrics, and torii. However, rational B-splines representations other than these classical primitives have found little application in surface modeling. In this paper we develop approximation algorithms based on the general rational B-spline formulation. Numerical experiments indicate that rational B-splines allow a significantly more compact approximation of two classes of parametric surfaces in comparison to integral B-splines. The two classes of surfaces studied are generalized cylinders and offsets of a rational B-spline surface patch progenitor.

An accurate creeping ray-tracing algorithm is presented in this paper to determine the tracks of creeping waves (or creeping rays) on arbitrarily shaped free-form parametric surfaces [nonuniform rational B-splines (NURBS) surfaces]. The main challenge in calculating the surface diffracted fields on NURBS surfaces is due to the difficulty in determining the geodesic paths along which the creeping rays propagate. On one single parametric surface patch, the geodesic paths need to be computed by solving the geodesic equations numerically. Furthermore, realistic objects are generally modeled as the union of several connected NURBS patches. Due to the discontinuity of the parameter between the patches, it is more complicated to compute geodesic paths on several connected patches than on one single patch. Thus, a creeping ray-tracing algorithm is presented in this paper to compute the geodesic paths of creeping rays on the complex objects that are modeled as the combination of several NURBS surface patches. In the algorithm, the creeping ray tracing on each surface patch is performed by solving the geodesic equations with a Runge-Kutta method. When the creeping ray propagates from one patch to another, a transition method is developed to handle the transition of the creeping ray tracing across the border between the patches. This creeping ray-tracing algorithm can meet practical requirements because it can be applied to the objects with complex shapes. The algorithm can also extend the applicability of NURBS for electromagnetic and optical applications. The validity and usefulness of the algorithm can be verified from the numerical results.

A rational B-spline curve or surface is a collection of points associated with a mass (weight) distribution. These mass distributions can be used to exert local control over the morph between two rational B-spline curves or surfaces. Here we propose a technique for designing customized morphs by attaching appropriate mass distributions to target B-spline curves and surfaces. We also develop a user interface for this morphing method that is easy to use and requires no knowledge of B-splines on the part of the designer.

Techniques for computing intersections of algebraic surfaces with piecewise rational polynomial parametric surface patches and intersections of two piecewise rational polynomial parametric surface patches are discussed. The techniques are classified using four categories-lattice evolution methods, marching methods, subdivision methods, and analytic methods-and their principal features are discussed. It is shown that some of these methods also apply to the general parametric surface-intersection problem.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Rational B-spline surfaces make it possible to merge the concepts of freeform surfaces and that of surfaces described by rational polynomials especially conic sections. For ray tracing it is crucial to determine the intersection between ray and object. Therefore an algorithm is developed that is suitable for a VLSI implementation. Some alternatives for the implementation of this algorithm are presented and discussed. The paper concludes with a discussion of some problems and possible further developments.

Approximating rational triangular Bézier surfaces by polynomial triangular Bézier surfaces

Rational-Ruled Surfaces: Implicitization and Section Curves

In this paper has been considered the main content and distinctive features of the “Geometric Modeling” training course for the “Mathematics and Computer Science” training program 02.03.01 (“Mathematical and Computer Modeling” specialization).&#x0D; The goal of the “Geometric Modeling” course study is the assimilation of mathematical methods for construction of geometric objects with complex curved shapes, and techniques for their computer visualization by using polygons of curves and surfaces. Methods for construction of structures’ curved shapes using spline representations, as well as techniques for construction of surfaces and volumetric geometries using motion operations and basic logical operations on geometric objects are considered. The spline representations include linear and bilinear splines, Hermite cubic splines and Hermite surfaces, natural cubic and bicubic interpolation splines, Bezier curves and surfaces, rational Bezier splines, B-splines and B-spline surfaces, NURBS-curves and NURBS-surfaces, transfinite interpolation methods, and splines of surfaces with triangular form. Logical operations for intersection of two spline curves, and intersection of two parametric surfaces are considered. The principles of scientific visualization and computer animation are considered in this course as well.&#x0D; Some examples for visualization of initial data and results of curves and surfaces construction in two- and three-dimensional spaces through the software shell developed by authors and used by students while doing tests have been demonstrated. The software shell has a web interface with the WebGL library graphic support. Tasks for four practical studies in a computer classroom, as well as several variations of homework are represented.&#x0D; The problems occurring in preparation materials for some course sections are discussed, as well as the practical importance of acquired knowledge for the further progress of students.&#x0D; The paper may be interesting for teachers of “Geometric Modeling” and “Computer Graphics” courses aimed to students with a specialization in mathematics and information, as well as to those who independently develop software interfaces for algorithms of geometric modeling.

This is a 2-part introduction to the dynamics of rational surface automorphisms. Such maps can be written in coordinates as rational functions or polynomials. The first part concerns polynomial automorphisms of complex 2-space and includes the complex Henon family.The second part concerns compact (complex) rational surfaces. The basic properties of automorphisms of positive entropy are given, as well as the construction of invariant currents and measures. This is illustrated by a number of examples.KeywordsCohomology ClassBlow DownRational SurfaceFatou ComponentExceptional CurveThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The realization of the data exchanges between the reverse engineering system and CAD / CAM system is by Graphics Interchange standards, of which the initial graphics exchanges specification IGES is one of the important ways that data exchanges in systems. At first, the paper introduces the structures of the five parts that composes the IGES file. Secondly, it also introduces the meaning of the parameters that the rational B-spline surface in IGES and gives the method to identify and extract IGES surface data in the reverse engineering system. At last, it uses the OpenGL graphics system to show the surface data. Then, the exchanges of the surface data between the systems can realize easily by the method.

Topological and differential-equation methods for surface intersections

This paper focuses on the design of the extended branch structure for rapidly and simultaneously interpolating non-uniform rational B-spline (NURBS) surfaces and their partial derivatives on CNC machines. NURBS surfaces are concerned in the applications of molding process. However, the copious and complicated operations usually limit the applications of NURBS surfaces in CNC machining systems. In this paper, by considering the derivatives of basis functions, an extended branch structure is derived such that the computation time can be significantly reduced in computing NURBS surfaces and their partial derivatives. According to the machining results on a 3-axis vertical machining center, the extended branch structure is feasible to implement on CNC machines and is more efficient than existing approaches in computation.

The construction of offset curves is an important problem encountered often in design processes and in interrogation of geometric models. In this paper the problem of construction of offsets of curves lying on the same parametric surface is addressed. A novel algorithm is introduced, whose main feature is the use of geodesic paths to determine points of the offset. The offset is then approximated in the underlying surface parameter space by B-splines interpolating data points obtained by traveling a known distance along the geodesics departing from corresponding points of the progenitor in a direction perpendicular to the latter. A comprehensive error checking scheme has been devised allowing adaptive improvement of the approximation of the offset. The applicability of the algorithm is demonstrated by number of numerical examples.

Nonuniform rational B-spline (NURBS) curves and surfaces, which are based on rational and B-splines, are defined. The important characteristics of NURBS that have contributed to their wide acceptance as standard tools for geometry representation and design are summarized. Their application to representing conic sections and commonly used surfaces, designing curves and surfaces, and modifying shapes is examined. >

Although rational polynomial curves and surfaces have become standard forms in computer-aided design, they have many problems. For example, a Newton-Raphson algorithm for dealing with rational polynomial curves tends to be non-robust. This is a fatal problem. In the past, we proposed a new Newton-Raphson method for dealing with the rational polynomial curves. In the method, we homogenize the coordinates of a rational curve when it is applied to the Newton-Raphson algorithm. Then it becomes very robust.This paper proposes a method of adapting the algorithm to the Newton-Raphson method for dealing with the surface-surface intersection problem. Like the case of a curve, the Newton-Raphson algorithm was found to be very robust when the coordinates of a curved surface are homogenized. In this paper, we discuss the reason why the conventional method tends to be non-robust while the new method is robust. The conventional method deals with a rational function that always has a horizontal asymptote as well as vertical ones while the new method deals with a non-rational function and thus has no asymptote. The result of our analysis points out that the cause of the non-robustness lies in the horizontal asymptote of the rational function inherent in the conventional Newton-Raphson method. Our numerical experiments show that our method is very robust.