Quaternion rational surfaces

<p>In this paper, we provided, first, a general symbolic algorithm for computing the symmetries of a given rational surface, based on the classical differential invariants of surfaces, i.e., <italic>Gauss curvature</italic> and <italic>mean curvature</italic>. In practice, the algorithm works well for sparse parametrizations (e.g., toric surfaces) and PN surfaces. Additionally, we provided a specific, and symbolic, algorithm for computing the symmetries of ruled surfaces. This algorithm works extremely well in practice, since the problem is reduced to that of rational space curves, which can be efficiently solved by using existing methods. The algorithm for ruled surfaces is based on the fact, proven in the paper, that every symmetry of a rational surface must also be a symmetry of its <italic>line of striction</italic>, which is a rational space curve. The algorithms have been implemented in the computer algebra system Maple, and the implementations have been made public. Evidence of their performance is given in the paper.</p>

Syzygies for translational 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.

Rational Curves and Surfaces

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

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.

Curves with rational Frenet-Serret motion

Quantum cohomology via [formula omitted]-modules

It is often necessary to represent a hand-drawn shape accurately. Modeling such shapes manually is both cumbersome and commercially expensive. The user's concern for curves that are easy to manipulate has been a major influence on the development of free-form curves. Rational parametric curves have been receiving considerable attention in the area of geometric modeling because any parametric polynomial curve can be expressed as a rational curve and most polynomial splines have rational extensions. Rational parametric curves can be used to model any object, like ships and airplanes, and even in the medical field for modeling the heart or other parts of the body. In object modeling, the rational cubic spline is most popular because it is the lowest degree that can define space curves and curves with points of zero curvature. The foremost objective of this work is to show how conic sections can adequately be used to represent curves and objects that were previously thought to require rational cubic splines. A single rational cubic curve is represented by two conics by splitting the rational cubic at its mid-point. The degree of smoothness has been taken into account to have a visually pleasing effect. It has been realized that the conic representation is advantageous over the rational cubic one in terms of computational requirements and shape control. All of this work is carried out on both 2D curves and 3D objects.

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.

We describe the relation between quasi-minuscule representations, polytopes and Weyl group orbits in Picard lattices of rational surfaces. As an application, to each quasi-minuscule representation we attach a class of rational surfaces, and realize such a representation as an associated vector bundle of a principal bundle over these surfaces. Moreover, any quasi-minuscule representation can be defined by rational curves, or their disjoint unions in a rational surface, satisfying certain natural numerical conditions.

Parameterizing rational offset canal surfaces via rational contour curves

In this paper, we discuss the singularities of rational space curves. Two methods are provided to compute the singularities of arbitrary degree curves. These methods are a generalization of the paper (Chen, Wang and Liu. Computing singular points of plane rational curves. Journal of Symbolic Computation 43, 92--117, 2008), which are based on the μ-basis of the rational space curve and on random technique. The μ-basis induces a matrix M which contains all the information about the singularities including the parameter values corresponding to the singularities, multiplicities and infinitely near singularities. These information can be obtained by computing the Smith form of the matrix M. We compare our methods with previous approaches such as generalized resultants, and provide some examples to illustrate the effectiveness of our methods.

Curvature-dependent parameterization of curves and surfaces