Jernej Kozak

On geometric interpolation by planar parametric polynomial curves

In this paper the problem of geometric interpolation of planar data by parametric polynomial curves is revisited. The conjecture that a parametric polynomial curve of degree < n can interpolate 2n given points in R 2 is confirmed for n < 5 under certain natural restrictions. This conclusion also implies the optimal asymptotic approximation order. More generally, the optimal order 2n can be achieved as soon as the interpolating curve exists.

On geometric interpolation of circle-like curves

In this paper, geometric interpolation of certain circle-like curves by parametric polynomial curves is studied. It is shown that such an interpolating curve of degree n achieves the optimal approximation order 2n, the fact already known for particular small values of n. Furthermore, numerical experiments suggest that the error decreases exponentially with growing n.

On interpolation by Planar cubic

In this paper, the geometric interpolation of planar data points and boundary tangent directions by a cubic G 2 Pythagorean-hodograph (PH) spline curve is studied. It is shown that such an interpolant exists under some natural assumptions on the data. The construction of the spline is based upon the solution of a tridiagonal system of nonlinear equations. The asymptotic approximation order 4 is confirmed.

G^2

The purpose of this paper is to provide sufficient geometric conditions that imply the existence of a cubic parametric polynomial curve which interpolates six points in the plane. The conditions turn out to be quite simple and depend only on certain determinants derived from the data points.

pythagorean-hodograph spline curves

The problem of geometric interpolation by Pythagorean-hodograph (PH) curves of general degree n is studied independently of the dimension d ≥ 2. In contrast to classical approaches, where special structures that depend on the dimension are considered (complex numbers, quaternions, etc.), the basic algebraic definition of a PH property together with geometric interpolation conditions is used. The analysis of the resulting system of nonlinear equations exploits techniques such as the cylindrical algebraic decomposition and relies heavily on a computer algebra system. The nonlinear equations are written entirely in terms of geometric data parameters and are independent of the dimension. The analysis of the boundary regions, construction of solutions for particular data and homotopy theory are used to establish the existence and (in some cases) the number of admissible solutions. The general approach is applied to the cubic Hermite and Lagrange type of interpolation. Some known results are extended and numerical examples provided.

Geometric interpolation by planar cubic polynomial curves

In this paper, geometric interpolation by parametric polynomial curves is considered. Discussion is focused on the case where the number of interpolated points is equal to r + 2, and n=r denotes the degree of the interpolating polynomial curve. The interpolation takes place in

An approach to geometric interpolation by Pythagorean-hodograph curves

\mathbb R^d

On Geometric Interpolation by Polynomial Curves

with d=n. Even though the problem is nonlinear, simple necessary and sufficient conditions for existence of the solution are stated. These conditions are entirely geometric and do not depend on the asymptotic analysis. Furthermore, they provide an efficient and stable way to the numeric solution of the problem.

Shape preserving approximation

Abstract An algorithm that produces a shape preserving interpolatory hyperbolic splines is discussed here. It is based upon Schweikerts spline in tension, but with tension parameters determined in advance. These are chosen in such a way that the approximating function differs as little as possible from a cubic polynomial interpolant. By this approach the approximation power of the cubic spline is preserved wherever possible. At the end, a shape preserving approximation of a planar curve is discussed.

Geometric Lagrange interpolation by planar cubic Pythagorean-hodograph curves

In this paper, the geometric Lagrange interpolation of four points by planar cubic Pythagorean-hodograph (PH) curves is studied. It is shown that such an interpolatory curve exists provided that the data polygon, formed by the interpolation points, is convex, and satisfies an additional restriction on its angles. The approximation order is 4. This gives rise to a conjecture that a PH curve of degree n can, under some natural restrictions on data points, interpolate up to n+1 points.