Typical curve with G1 constraints for curve completion

This paper presents a novel algorithm for planar G1 interpolation using typical curves with monotonic curvature. The G1 interpolation problem is converted into a system of nonlinear equations and sufficient conditions are provided to check whether there is a solution. The proposed algorithm was applied to a curve completion task. The main advantages of the proposed method are its simple construction, compatibility with NURBS, and monotonic curvature.

Iterations and Groups of Formal Transformations

In this paper, we consider the problem of formal iteration. We construct an area preserving mapping which does not have any square root. This leads to a counterexample to Moser’s existence theorem for an interpolation problem. We give examples of formal transformation groups such that the iteration problem has a solution for every element of the groups

Planar Typical Bézier Curves with a Single Curvature Extremum

This paper focuses on planar typical Bézier curves with a single curvature extremum, which is a supplement of typical curves with monotonic curvature by Y. Mineur et al. We have proven that the typical curve has at most one curvature extremum and given a fast calculation formula of the parameter at the curvature extremum. This will allow designers to execute a subdivision at the curvature extremum to obtain two pieces of typical curves with monotonic curvature. In addition, we put forward a sufficient condition for typical curve solutions under arbitrary degrees for the G1 interpolation problem. Some numerical experiments are provided to demonstrate the effectiveness and efficiency of our approach.

Certain non-homogeneous matricial domains and Pick–Nevanlinna interpolation problem

In this paper, we consider certain matricial domains that are naturally associated to a given domain of the complex plane. A particular example of such domains is the spectral unit ball. We present several results for these matricial domains. Our first result shows — generalizing a result of Ransford–White for the spectral unit ball — that the holomorphic automorphism group of these matricial domains does not act transitively. We also consider [Formula: see text]-point and [Formula: see text]-point Pick–Nevanlinna interpolation problem from the unit disc to these matricial domains. We present results providing necessary conditions for the existence of a holomorphic interpolant for these problems. In particular, we shall observe that these results are generalizations of the results provided by Bharali and Chandel related to these problems.

Rational Interpolation: Jacobi’s Approach Reminiscence

We treat the interpolation problem {f(xj)=yj}j=1N for polynomial and rational functions. Developing the approach originated by C. Jacobi, we represent the interpolants by virtue of the Hankel polynomials generated by the sequences of special symmetric functions of the data set like {∑j=1Nxjkyj/W′(xj)}k∈N and {∑j=1Nxjk/(yjW′(xj))}k∈N; here, W(x)=∏j=1N(x−xj). We also review the results by Jacobi, Joachimsthal, Kronecker and Frobenius on the recursive procedure for computation of the sequence of Hankel polynomials. The problem of evaluation of the resultant of polynomials p(x) and q(x) given a set of values {p(xj)/q(xj)}j=1N is also tackled within the framework of this approach. An effective procedure is suggested for recomputation of rational interpolants in case of extension of the data set by an extra point.

Lidstone–Euler Second-Type Boundary Value Problems: Theoretical and Computational Tools

General nonlinear high odd-order differential equations with Lidstone–Euler boundary conditions of second type are treated both theoretically and computationally. First, the associated interpolation problem is considered. Then, a theorem of existence and uniqueness of the solution to the Lidstone–Euler second-type boundary value problem is given. Finally, for a numerical solution, two different approaches are illustrated and some numerical examples are included to demonstrate the validity and applicability of the proposed algorithms.

Interpolation by Differences In H∞

We pose an interpolation problem for the space of bounded analytic functions in the disk. The interpolation is performed by a function and its di˛erence of values in points whose subscripts are related by an increasing application. We impose that the data values satisfy certain conditions related to the pseudohyperbolic distance, and characterize interpolating sequences in terms of uniformly separated subsequences.

Data Interpolation by Near-Optimal Splines with Free Knots Using Linear Programming

The problem of obtaining an optimal spline with free knots is tantamount to minimizing derivatives of a nonlinear differentiable function over a Banach space on a compact set. While the problem of data interpolation by quadratic splines has been accomplished, interpolation by splines of higher orders is far more challenging. In this paper, to overcome difficulties associated with the complexity of the interpolation problem, the interval over which data points are defined is discretized and continuous derivatives are replaced by their discrete counterparts. The l∞-norm used for maximum rth order curvature (a derivative of order r) is then linearized, and the problem to obtain a near-optimal spline becomes a linear programming (LP) problem, which is solved in polynomial time by using LP methods, e.g., by using the Simplex method implemented in modern software such as CPLEX. It is shown that, as the mesh of the discretization approaches zero, a resulting near-optimal spline approaches an optimal spline. Splines with the desired accuracy can be obtained by choosing an appropriately fine mesh of the discretization. By using cubic splines as an example, numerical results demonstrate that the linear programming (LP) formulation, resulting from the discretization of the interpolation problem, can be solved by linear solvers with high computational efficiency and the resulting spline provides a good approximation to the sought-for optimal spline.