Approximating rational triangular Bézier surfaces by polynomial triangular Bézier surfaces

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.

As is known, differential geometry studies the properties of curve lines (tangent, curvature, torsion), surfaces (bending, first and second basic quadratic forms) and their families in small, that is, in the neighborhood of the point by means of differential calculus. Algebraic geometry studies properties of algebraic curves, surfaces, and algebraic varieties in general [1; 17]: order, class, genre, existence of singular points and lines, curves and surfaces family intersections (sheaves, bundles, congruences, complexes and their characteristics). Rational curves and surfaces occupy a special place among them: • their design by bi-rational (Cremona) transformations [10; 21]; • investigation of their properties by mapping to lines and planes [9; 21; 22]; • construction of smooth contours from arcs of rational curves belonging to surfaces [10]. It seems that the main results obtained in this direction by mathematicians in the second half of the 19th century by structural and geometric methods should be the theoretical support for the design of technical forms that meet a number of pre-set requirements using modern computational tools and information technologies. It is obvious that application of Cremona transformations’ powerful apparatus is useful when designing, for example, pipes of complex geometry according to set of streamlines, thin-walled shells for a given mesh manifold of curvature lines etc. Apparently, this stage should precede computer graphics’ calculation procedures. However, in Russian publications on applied (engineering) geometry, only a little attention is paid to the study of surfaces in general. The author knows nothing about the use of this approach for solving of these applied problems. In this regard, the aims of this paper are: • illustration of method for mapping a surface to a plane to study its properties in general by the example of construction a flat model for a hyperboloid of one sheet; • constructive approach to the construction of smooth one-dimensional contours on rational surfaces.

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.

A quaternion rational surface is a rational surface generated by two rational space curves via quaternion multiplication. In general, the structure of the graded minimal free resolution of a rational surface is unknown. The goal of this paper is to construct the graded minimal free resolution of a quaternion rational surface generated by two rational space curves. We will provide the explicit formulas for the maps of these graded minimal free resolutions. The approach we take is to utilize the information of the μ -bases of the generating rational curves, and create the generating sets for the first and second syzygy modules in the graded minimal free resolutions. In addition, we show that the ideal generated by the first syzygy module expressed in terms of moving planes is exactly the same as the ideal generated by the parametrization in the affine ring.

This work deals with the problem of practical mesh generation for surface normal approximation. Part of its contribution is in presenting previous work in a unified framework. A new algorithm for surface normal approximation is then introduced which improves upon existing ones in a number of aspects. In particular, it produces better approximations of surfaces both in practice and in the theoretical limit regime. Additionally, it resolves in a simple way some of the problems that previous methods for surface approximation suffered from.

This is a short paper announcing several new results. The best known algorithm for solving a Toeplitz system of linear equations requires <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(n^2)</tex> operations. We present a new algorithm that reduces this computation to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(n\log^{2}n)</tex> . Furthermore the new algorithm handles all degenerate cases. We show that the solution process of a Toeplitz system is intimately related to the computation of certain Padé approximants. Actually the Padé computation is a special case of rational Hermite interpolation. Our primary result shows that all rational Hermite interpolants along an antidiagonal of the Rational Interpolation Table can be computed by the extended Euclidean algorithm. We present new fast algorithms for such computations and indicate extensions and improvements. Several application areas can make use of these results. We include a brief discussion on coding but leave digital filters and circuit synthesis to the full paper.

The μ-bases of rational curves and surfaces are newly developed tools which play an important role in connecting parametric forms and implicit forms of curves and surfaces. However, exact μ-bases may have high degree with complicated rational coefficients and are often hard to compute (especially for surfaces), and sometimes they are not easy to use in geometric modeling and processing applications. In this paper, we introduce approximate μ-bases for rational curves and surfaces, and present an algorithm to compute approximate μ-bases. The algorithm amounts to solving a generalized eigenvalue problem and some quadratic programming problems with linear constraints. As applications, approximate implicitization and degree reduction of rational curves and surfaces with approximate μ-bases are discussed. Both the parametric equations and the implicit equations of the approximate curves/surfaces are easily obtained by using the approximate μ-bases. As indicated by the examples, the proposed algorithm may be a useful alternative to other methods for approximate implicitization.Keywordsapproximate μ-basesapproximate implicitization

Curvature-dependent parameterization of curves and surfaces

A curve D on a compact analytic surface Y is called an anti-canonical cycle if Y is smooth near D,-D is a canonical divisor, and if either D is an irreducible rational curve with an ordinary node or D is reducible and the distinct irreducible components Dow ..., D-I of D are smooth rational curves forming a polygon. This paper deals with rational surfaces Y (with at worst rational double points) endowed with an anti-canonical cycle D such that D contains no exceptional curve and the intersection matrix (Di.Dj) is negative. This amounts to D. D ? 0 if D is irreducible and Di. Di -2 for all i if D is reducible. The Picard group of these surfaces is studied in Chapter I. One of our main findings is that for s ? 5 the Picard group of such a surface contains a naturally defined infinite root system which enables us to give a precise description of the classes which represent exceptional curves. The restriction to s Md for rational surfaces Y endowed with an anti-canonical divisor D = Do + + D, with a given cycle of self-intersection numbers d = (-Do.Do,..., -Ds-1.Ds-1) of length s ? 5. In order to get a reasonable space we must discard some pairs (Y, D), however. The base Md is entirely given in terms of the (abstract) infinite root system defined in Chapter I. In fact, it is nothing but the orbit space M introduced and investigated in [19]. In that paper we proved that the global holomorphic functions separate the points of Md and we gave a precise description of the holomorphic hull Md of Md. This holomorphic hull is a Stein manifold and the added material Md Md is naturally stratified.

Rational Curves and Surfaces

Curves with rational Frenet-Serret motion

• NP-Completeness and APX-Hardness • We prove that the problems MinFixMask OPT and MinFixMask BOUND are NP-complete. • We establish that FixMasks + is NP-complete and APX-hard. • We show NP-completeness of DFixMasks + . • Exact, FPT, and Approximation Algorithms for DFixMasks + • We design a fixed-parameter tractable (FPT) algorithm running in O *(2 M ) time, parameterized by the number of shield configurations M . • We present a polynomial-time 1 M -approximation algorithm. • We propose an exponential k M -approximation method (for a treatment plan length k ). • We provide a tight ( 1 − 1 e ) -approximation algorithm by exploiting monotonicity and submodularity of the objective function. • Quasi-Polynomial Algorithms and Logarithmic Approximations • We develop quasi-polynomial-time algorithms for MinFixMask OPT , MinFixMask OPT + , MinFixMask BOUND , and MinFixMask BOUND + , assuming that the maximum prescribed dose d ^ max is polynomially bounded. • Under the same assumption, we achieve polynomial-time approximation algorithms with a logarithmic factor log ( d ^ max ) for MinFixMask OPT and MinFixMask OPT + . • Polynomial-Time Solvable Special Cases • We identify polynomial-time solvable cases: FixMask and FixMasks + when M = 1 . • We show that the problems MinFixMask BOUND and MinFixMask BOUND + can be solved in polynomial time when T max 1, N ;. In this paper, we study a combinatorial problem which arises in the development of innovative treatment strategies and equipment using tunable shields in internal radiotherapy. From an algorithmic point of view, the problem is related to circular integer word decomposition into circular binary words under constraints. We consider several variants of the problem, depending on constraints and parameters and present exact, approximation, fixed parameter tractable algorithms and NP-hardness and APX-hardness results.

The rational Bernstein bases and the multirational blossoms

A new surface error calculation method for layered manufacturing processes is proposed in this paper. The developed method is used to generate the layers by adaptively varying the thickness of the layers based on the surface approximation errors. Traditionally, the surface errors are calculated using local approximation techniques. In this paper, the surface approximation errors are calculated more accurately by marching through the surface points and determining the distances between layers and the surface points. Using the calculated distances, the adaptive layers are generated for both traditional two‐dimensional layer and ruled‐layer approximation methods. It has been shown that layered manufacturing (rapid prototyping) processes can achieve better accuracy and efficiency using the proposed surface error calculation and the adaptive ruled layer approximation methods. Computer implementation and illustrative examples are also presented in this paper.

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.