Piecewise Algebraic Curves Research Articles

Least Squares Fitting of Piecewise Algebraic Curves

A piecewise algebraic curve is defined as the zero contour of a bivariate spline. In this paper, we present a new method for fittingC1piecewise algebraic curves of degree 2 over type-2 triangulation to the given scattered data. By simultaneously approximating points, associated normals and tangents, and points constraints, the energy term is also considered in the method. Moreover, some examples are presented.

Lagrange interpolation by bivariate splines on cross-cut partitions

The piecewise algebraic curve is a generalization of the classical algebraic curve. In this paper, we give the Nöther-type theorem of piecewise algebraic curves on cross-cut partitions. By using interpolation along a piecewise algebraic curve, we also present a method to construct Lagrange interpolation sets for bivariate spline spaces on cross-cut partitions.

Nöther-type theorem of piecewise algebraic curves*

Abstract The piecewise algebraic curve is a generalization of the classical algebraic curve. This paper describes the improvement of the Nother-type theorem of piecewise algebraic curves on the star region. Moreover, the Nother-type theorem of piecewise algebraic curves on the cross-cut partition is discussed.

Cayley–Bacharach theorem of piecewise algebraic curves

The piecewise algebraic curve, determined by a bivariate spline function, is a generalization of the classical algebraic curve. In this paper, by using Bezout's theorem and Nöther-type theorem of piecewise algebraic curves, the Cayley–Bacharach theorem and Hilbert function of C 0 piecewise algebraic curves are presented.

Estimation of the Bezout number for piecewise algebraic curve

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. In this paper, a conjecture on triangulation is confirmed. The relation between the piecewise linear algebraic curve and four-color conjecture is also presented. By Morgan-Scott triangulation, we will show the instability of Bezout number of piecewise algebraic curves. By using the combinatorial optimization method, an upper bound of the Bezout number defined as the maximum finite number of intersection points of two piecewise algebraic curves is presented.

Piecewise algebraic curve

A piecewise algebraic curve is defined by a bivariate spline function. Using the techniques of the B- net form of bivariate splines function, discriminant sequence of polynomial (cf. Yang Lu et al. (Sci. China Ser. E 39(6) (1996) 628) and Yang Lu et al. (Nonlinear Algebraic Equation System and Automated Theorem Proving, Shanghai Scientific and Technological Education Publishing House, Shanghai, 1996)) and the number of sign changes in the sequence of coefficients of the highest degree terms of sturm sequence, we determine the number of real intersection points of two piecewise algebraic curves whose common points are finite. A lower bound of the number of real intersection points is given in terms of the method of rotation degree of vector field.

Multivariate spline and algebraic geometry

The purpose of this survey is to emphasize the special relationship between multivariate spline and algebraic geometry. We will not only point out the algebraic–geometric method of multivariate spline, but also the algebraic–geometric background and essence of multivariate spline. Especially, we have made an introduction to the so-called piecewise algebraic curve, piecewise algebraic variety, and some of their properties.

Applications of symmetry methods in basic problems of orthotropic

We discuss basic problems of orthotropic elasticity in a plane domain whose boundary is a piecewise-algebraic curve. First, by means of bi-analytic functions, a basic problem is reduced to a boundary value problem for analytic functions. Then, by use of the generalized symmetry principle for algebraic curves, a boundary value problem for analytic functions is converted to a problem on a Riemann surface; then the solution to the original problem is obtained in closed form for a domain with algebraic boundaries having genusρ≥0

THE BEZOUT NUMBER FOR PIECEWISE ALGEBRAIC CURVES

The computation of the Bezout number, the maximum number of intersection points between two piecewise algebraic curves whose common points are finite, is considered. A piecewise algebraic curve is a curve determined by a bivariate spline function. It is found that the maximum number of intersections depends not only on the degrees and the differentiability of the spline functions, but also on the structure of the partition on which the spline functions are defined.

Geometric control of G2-cubic A-splines

A-splines are piecewise algebraic curves, constructed in a control polygon that is a sequence of triangles meeting at the vertices. The arc in a given triangle is a cubic segment that joins these vertices and interpolates also the slope and curvature at each endpoint. This arc in each triangle is controlled by three other shape handles: an interior point that is interpolated, the slope at the interior point, and a tension parameter. The individual arcs can never be singular and are always convex. An inflection point of an A-spline must be placed at a junction point.

Planar piecewise algebraic curves

An approach is described for piecing together segments of planar algebraic curves with derivative continuity. The application of piecewise algebraic curves to area modelling (the two-dimensional analogue of solid modelling) is discussed. A technique is presented for expressing a planar rational parametric curve as an algebraic curve segment. An upper bound is derived for the farthest distance between two algebraic curves (one of which may also be a parametric curve) within a specified region.