Implicit Algebraic Surfaces Research Articles

Detecting isometries and symmetries of implicit algebraic surfaces

<abstract><p>We presented a new and complete algorithm for detecting isometries and symmetries of implicit algebraic surfaces. First, our method reduced the problem to the case of isometries fixing the origin. Second, using tools from elimination theory and polynomial factoring, we determined the desired isometries between the surfaces. We have implemented the algorithm in Maple to provide evidences of the efficiency of the method.</p></abstract>

Computing symmetries of implicit algebraic surfaces

We present a complete algorithm to compute the rotational, axial, reflectional and central symmetries of an algebraic surface defined by means of its implicit equation. The algorithms rely on the fact that the symmetries of the surface satisfy an algebraic condition verified by the polynomial defining the surface, and use a variety of ideas, some of them extensions to space of ideas already used to compute the symmetries of planar implicit curves, and some of them new. We flesh out our algorithms with several examples.

Offset approximation of polygons on an ellipsoid

We present an approximation method for offset curves of polygons on an oblate ellipsoid using implicit algebraic surfaces. The polygon on the ellipsoid is given by a set of vertices, i.e. points on the ellipsoid. The edges are the shortest geodesic paths connecting two consecutive points. The offset curve of the polygon consists of two parts. The offset curve of an edge is the set of points that are at the same distance from each point on the edge, in the normal direction of the edge. The offset curve of a vertex is the set of points that are at the same geodesic distance from the vertex. Our offset approximation method uses plane section curves for offset curves of edges and prolate ellipsoids for offset curves of vertices. Since our offset approximation curve is constructed from implicit algebraic surfaces, it is easy to check whether a given point on the oblate ellipsoid has intruded into the inside of the offset curve. Moreover our method achieves extremely small approximation errors. We apply our method to numerical examples on the Earth.

Recognizing algebraic affine rotation surfaces

We investigate the problem of recognizing a generalization of surfaces of revolution appearing in the field of affine differential geometry, namely affine rotation surfaces. By using some notions from affine differential geometry, we determine how to detect whether or not a given implicit algebraic surface is an affine rotation surface. These results generalize some previous results of the authors on surfaces of revolution. As a by-product, we also provide an algorithm to detect whether or not an algebraic surface is an affine sphere, a generalization of Euclidean spheres of interest not only in geometry, but also in other fields.

Blending Implicit Algebraic Surfaces by Combining Division with Filling with S-Patches

Based on the method of approximating boundary curves with splines,we use a spline surface of degree 3 to blend two arbitrary implicit algebraic surfaces.Smooth transition is achieved between the spline surface and two base surfaces.Furthermore,we use GB-splines to blend two algebraic surfaces which can exactly represent Lissajous curves,conics,trigonometric function curves,hyperbolic function curves,catenary curves and helixes etc.For several implicit algebraic surfaces,we present the blending method that combines splitting and filling holes with S-patches.G1-continiuty is persisted between splitting blending patches and S-patches.All blending patches are control point patches and the shape of each patch can be adjusted by shape parameters intuitively.

On the isotopic meshing of an algebraic implicit surface

We present a new and complete algorithm for computing the topology of an algebraic surface S given by a square free polynomial in Q [ X , Y , Z ] . Our algorithm involves only subresultant computations and entirely relies on rational manipulation, which makes it direct to implement. We extend the work in Diatta et al. (2008) , on the topology of non-reduced algebraic space curves, and apply it to the polar curve or apparent contour of the surface S . We exploit a simple algebraic criterion to certify the pseudo-genericity and genericity position of the surface. This gives us rational parametrizations of the components of the polar curve, which are used to lift the topology of the projection of the polar curve. We deduce the connection of the two-dimensional components above the cell defined by the projection of the polar curve. A complexity analysis of the algorithm is provided leading to a bound in O ˜ B ( d 21 τ ) for the complexity of the computation of the topology of an implicit algebraic surface defined by integer coefficient polynomial of degree d and coefficient size τ . Examples illustrate the implementation in Mathemagix of this first complete code for certified topology of algebraic surfaces.

Computing minimum distance between two implicit algebraic surfaces

The minimum distance computation problem between two surfaces is very important in many applications such as robotics, CAD/CAM and computer graphics. Given two implicit algebraic surfaces, a new method based on the offset technique is presented to compute the minimum distance and a pair of points where the minimum distance occurs. The new method also works where there are an implicit algebraic surface and a parametric surface. Quadric surfaces, tori and canal surfaces are used to demonstrate our new method. When the two surfaces are a general quadric surface and a surface which is a cylinder, a cone or an elliptic paraboloid, the new method can produce two bivariate equations where the degrees are lower than those of any existing method.

Basic lines, axes and geometric modeling on implicit algebraic surfaces

We introduce some new concepts such as basic lines and divide the blending problems into two classes, Class I blending and Class II blending. We further develop the results [Li, A software system on blending of pipe surfaces, Master Thesis, Jilin University, June, 2000; Wu et al., Blending of implicit algebraic surfaces, Proceedings of the ASCM 1995, Beijing, China, August 18–20, 1995, pp. 125–131; Wu and Han, Newton form formulae for the n-way blending of quadratic surfaces, Proceedings of IJCC Workshop on Digital Engineering, August 21–22, 2003; Wu and Zhou, On blending of several quadratic algebraic surfaces, Comput. Aided Geom. Design 17(9) (2000) 759–766; Wu, Zhou and Feng, Blending two quadratic algebraic surfaces with cubic surfaces, Proceedings of ASCM, 1996, Kobe, Japan, August 20–22, 1996, pp. 73–79] and analyze the geometric condition for the Class I blending problem by using characteristic roots. We obtain the lowest degree of Class II blending according to the different position of the axes of the primary surfaces. And we derive the efficient parameterization method for a kind of cubic blending surfaces on the basis of the results obtained by Berry and Patterson [Implicitization and parameterization of nonsingular cubic surfaces, Comput. Aided Geom. Design 19 (2001) 723–738].

Formula omitted] modeling with A-patches from rational trivariate functions

We approximate a manifold triangulation in R 3 using smooth implicit algebraic surface patches, which we call A-patches. Here each A-patch is a real iso-contour of a trivariate rational function defined within a tetrahedron. The rational trivariate function provides increased degrees of freedom so that the number of surface patches needed for free-form shape modeling is significantly reduced compared to earlier similar approaches. Furthermore, the surface patches have quadratic precision, that is they exactly recover quadratic surfaces. We give conditions under which a C 1 smooth and single sheeted surface patch is isolated from the multiple sheets.

Hyperfinite interpolation, Wu’s Method and blending of implicit algebraic surfaces

One of the central questions in CAGD[1] is blending surfaces which provides the theoretical basis for the design technology of space surfaces. We will discuss the general theories and algorithms for multivariate hyperfinite interpolation and their application to the blending of implicit algebraic surfaces, and investigate the existence conditions of hyperfinite interpolation. Based on Wu’s theory on blending implicit algebraic surfaces, the problem of blending two quadric surfaces is studied. The conditions for the coefficient ofg i under which there exists the cubic blending surfaceS(f) (the lowest degree) are obtained and the concrete expressions off are presented if they exist. These results can be applied directly to CAGD.

Spline Approximations of Real Algebraic Surfaces

We use a combination of both symbolic and numerical techniques to construct several degree boundedG0andG1continuous, piecewise spline approximations of real implicit algebraic surfaces for both computer graphics and geometric modeling. These approximations are based upon an adaptive triangulation (aG0planar approximation) of the real components of the algebraic surface, and include both singular points and singular curves on the surface. A curvilinear wireframe is also constructed using minimum bending energy, parametric curves with additionally normals varying along them. The spline approximations over the triangulation or curvilinear wireframe could be one of several forms: either low degree, implicit algebraic splines (triangular A-patches) or multivariate functional B-splines (B-patches) or standardized rational Bernstein–Bézier patches (RBB), or triangular rational B-Splines. The adaptive triangulation is additionally useful for a rapid display and animation of the implicit surface.

Higher-order interpolation and least-squares approximation using implicit algebraic surfaces

In this article, we characterize the solution space of low-degree, implicitly defined, algebraic surfaces which interpolate and/or least-squares approximate a collection of scattered point and curve data in three-dimensional space. The problem of higher-order interpolation and least-squares approximation with algebraic surfaces under a proper normalization reduces to a quadratic minimization problem with elegant and easily expressible solutions. We have implemented our algebraic surface-fitting algorithms, and included them in the distributed and collaborative geometric environment SHASTRA. Several examples are given to illustrate how our algorithms are applied to algebraic surface design.

Implicit curves and surfaces in CAGD

The role of implicit curves and surfaces in computer-aided geometric design (CAGD) are described. The ways in which the study of implicit algebraic curves and surfaces draws on algebraic geometry are reviewed. The implicitization of parametric curves and surfaces, parameterization of implicits, and techniques used to circumvent conversions between implicit and parametric representations are discussed. >

Algebraic surface design with Hermite interpolation

This paper presents an efficient algorithm called Hermite interpolation, for constructing low-degree algebraic surfaces, which contain, with C 1 or tangent plane continuity, any given collection of points and algebraic space curves having derivative information. Positional as well as derivative constraints on an implicitly defined algebraic surface are translated into a homogeneous linear system, where the unknowns are the coefficients of the polynomial defining the algebraic surface. Computaional details of the Hermite interpolation algorithm are presented along with several illustrative applications of the interpolation technique to construction of joining or blending surfaces for solid models as well as fleshing surfaces for curved wire frame models. A heuristic approach to interactive shape control of implicit algebraic surfaces is also given, and open problems in algebraic surface design are discussed.