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.     

Efficient topology determination of implicitly defined algebraic plane curves

This paper is devoted to present a new algorithm computing in a very efficient way the topology of a real algebraic plane curve defined implicitly. This algorithm proceeds in a seminumerical way by performing a symbolic preprocessing which allows later to accomplish the numerical computations in a very accurate way.     

Optimal affine reparametrization of rational curves

Let K be a characteristic zero field, let ?(t) be a birational parametrization ?(t) of a K-definable curve C with coefficients in an algebraic extension K(α) over K. We propose an algorithm to solve the optimization problem of computing the affine reparametrization t→at+b such that ?(at+b) has coefficients over an extension of K with algebraic degree as small as possible.     

Triangular Bernstein moment-based identification of algebraic curves

Extending our previous results, in this paper we present a theoretical improvement of a strategy for the identification of binary images with algebraic boundaries. Such identification is obtained from few samples and it is based on a representation of the image shape in terms of non-separable bivariate Bernstein polynomials piecewisely defined over triangular domains.     

Local shape of generalized offsets to algebraic curves

Offsetting is an important operation in computer aided design, with applications also in other contexts like robot path planning or tolerance analysis. In this paper we study the local behavior of an algebraic curve under a variation of the usual offsetting construction, namely the generalized offsetting process (Sendra and Sendra, 2000a). More precisely, here we discuss when and how this geometric construction may cause local changes in the shape of an algebraic curve, and we compare our results with those obtained for the case of classical offsets (Alcazar and Sendra, 2007). For these purposes, we use well-known notions of Differential Geometry, and also the notion of local shape introduced in Alcazar and Sendra (2007). Our analysis shows important differences between the topological properties of classical and generalized offsets, both at regular and singular points.     

Radical parametrizations of algebraic curves by adjoint curves

We present algorithms for parametrizing by radicals an irreducible curve, not necessarily plane, when the genus is less than or equal to 4 and the curve is defined over an algebraically closed field of characteristic zero. In addition, we also present an algorithm for parametrizing by radicals any irreducible plane curve of degree d having at least a point of multiplicity d−r, with 1≤r≤4 and, as a consequence, every irreducible plane curve of degree d≤5 and every irreducible singular plane curve of degree 6.     

Degenerate parametric curves

This paper discusses two degenerate cases of polynomial parametric curves for which the degrees of the defining polynomials can be reduced without altering the curve. The first case is the improperly parametrized curve for which each point on the curve corresponds to several parameter values. The second case, which can only occur for rational polynomial parametric curves, exists when the defining polynomials all have a common factor.

This paper describes how to detect and correct each type of degeneracy. Examples are given which demonstrate that seemingly innocuous Bézier curves may suffer from either of these degeneracies.     

Topology of real algebraic space curves

In this paper we give a new projection-based algorithm for computing the topology of a real algebraic space curve given implicitly by a set of equations. Under some genericity conditions, which may be reached through a linear change of coordinates, we show that a plane projection of the given curve, together with a special polynomial in the ideal of the curve contains all the information needed to compute its topological shape. Our method is also designed in such a way to exploit important particular cases such as complete intersection curves or curves contained in nonsingular surfaces.     

Convex hulls of objects bounded by algebraic curves

We present an0(n ·d ^o(1)) algorithm to compute the convex hull of a curved object bounded by0(n) algebraic curve segments of maximum degreed.Research supported in part by NSF Grant MIP-85 21356, ARO Contract DAA G29-85-C0018 under Cornell MSI, and ONR Contract N00014-88-K-0402. This paper is an updated version of a part of [6].     

Rational Pythagorean-hodograph space curves

A method for constructing rational Pythagorean-hodograph (PH) curves in R³ is proposed, based on prescribing a field of rational unit tangent vectors. This tangent field, together with its first derivative, defines the orientation of the curve osculating planes. Augmenting this orientation information with a rational support function, that specifies the distance of each osculating plane from the origin, then completely defines a one-parameter family of osculating planes, whose envelope is a developable ruled surface. The rational PH space curve is identified as the edge of regression (or cuspidal edge) of this developable surface. Such curves have rational parametric speed, and also rational adapted frames that satisfy the same conditions as polynomial PH curves in order to be rotation-minimizing with respect to the tangent. The key properties of such rational PH space curves are derived and illustrated by examples, and simple algorithms for their practical construction by geometric Hermite interpolation are also proposed.     

Constraint-based LN curves

We consider the design of parametric curves from geometric constraints such as distance from lines or points and tangency to lines or circles. We solve the Hermite problem with such additional geometric constraints. We use a family of curves with linearly varying normals, LN curves. The nonlinear equations that arise can be of algebraic degree 60. We solve them using the GPU on commodity graphics cards and achieve interactive performance. The family of curves considered has the additional property that the convolution of two curves in the family is again a curve in the family, assuming common Gauss maps, making the class more useful to applications. Further, we consider valid ranges in which the line tangency constraint can be imposed without the curve segment becoming singular. Finally, we remark on the larger class of LN curves and how it relates to Bézier curves.     

Local control of interpolating rational cubic spline curves

A rational spline based on function values only was constructed in the authors’ earlier works. This paper deals with the properties of the interpolation and the local control of the interpolant curves. The methods of value control, convex control and inflection-point control of the interpolation at a point are developed. Some numerical examples are given to illustrate these methods.     

Bisector curves of planar rational curves

This paper presents a simple and robust method for computing the bisector of two planar rational curves. We represent the correspondence between the foot points on two planar rational curves C₁(t) and C₂(r) as an implicit curve (t,r)=0, where (t,r) is a bivariate polynomial B-spline function. Given two rational curves of degree m in the xy-plane, the curve (t,r)=0 has degree 4m−2, which is considerably lower than that of the corresponding bisector curve in the xy-plane.     

Representation of piecewise continuous algebraic surfaces in terms of B-splines

A method for representing shape using portions of algebraic surfaces bounded by rectangular boxes defined in terms of triple product Bernstein polynomials is described and some of its properties are outlined. The method is extended to handle piecewise continuous algebraic surfaces within rectangular boxes defined in terms of triple products of B-spline basis functions. Next, two techniques for sculptured shape creation are studied. The first is based on geometric manipulation of existing primitives and the second on approximation/interpolation of lower dimensional entities using least-squares techniques based on singular value decomposition. In addition, several interrogation techniques, such as contouring, ray tracing and curvature evaluation, used in the design and analysis of piecewise continuous algebraic surfaces are discussed.     

Suboptimal Markovian smoothing estimates based on continuous curves of solutions of the algebraic Riccati inequality

Based on a result on continuous dependence of solutions of an algebraic Riccati equation on the data matrices, we construct continuous curves of solutions of an algebraic Riccati inequality, and derive suboptimal Markovian estimates for the steady-state smoothing problem.     

Detecting real singularities of a space curve from a real rational parametrization

In this paper we give an algorithm that detects real singularities, including singularities at infinity, and counts local branches and multiplicities of real rational curves in the affine n$n$-space without knowing an implicitization. The main idea behind this is a generalization of the D$D$-resultant (see [van den Essen, A., Yu, J.-T., 1997. The D$D$-resultant, singularities and the degree of unfaithfulness. Proc. Amer. Math. Soc. 25 (3), 689–695]) to n$n$ rational functions. This allows us to find all real parameters corresponding to the real singularities between the solutions of a system of polynomials in one variable.     

Harmonic rational Bézier curves, p-Bézier curves and trigonometric polynomials

In a recent article, Ge et al. (1997) identify a special class of rational curves (Harmonic Rational Bézier (HRB) curves) that can be reparameterized in sinusoidal form. Here we show how this family of curves strongly relates to the class of p-Bézier curves, curves easily expressible as single-valued in polar coordinates. Although both subsets do not coincide, the reparameterization needed in both cases is exactly the same, and the weights of a HRB curve are those corresponding to the representation of a circular arc as a p-Bézier curve. We also prove that a HRB curve can be written as a combination of its control points and certain Bernstein-like trigonometric basis functions. These functions form a normalized totally positive B-basis (that is, the basis with optimal shape preserving properties) of the space of trigonometric polynomials {1, sint, cost, …. sinmt, cosmt} defined on an interval of length < π.     

Smoothing rational B-spline curves using the weights in an optimization procedure

Freeform shape design is typically accomplished in an interactive manner and shapes generated by a computer are rarely immediately acceptable. The available techniques for any subsequent modifications depend on the chosen representation for the geometry. In many computer aided styling and design systems which use nonuniform rational B-splines (NURBS) for representation of geometry, the use of the weights as a shape control tool is very inadequately supported. In fact they are often hidden from the user and therefore remain unused. This paper investigates the possibilities of entering the weights in an automatic fairing process. In order to produce a curve with a more gradual change in curvature and the smallest deviation from its initial shape the perturbation of the weights is stated as an optimization problem. Examples of applications to automotive shape design are presented and discussed.