Emil Žagar

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.

An approach to geometric interpolation by Pythagorean-hodograph 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.

Curvature variation minimizing cubic Hermite interpolants

In this paper, planar parametric Hermite cubic interpolants with small curvature variation are studied. By minimization of an appropriate approximate functional, it is shown that a unique solution of the interpolation problem exists, and has a nice geometric interpretation. The best solution of such a problem is a quadratic geometric interpolant. The optimal approximation order 4 of the solution is confirmed. The approach is combined with strain energy minimization in order to obtain G 1 cubic interpolatory spline.

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.

Some new G1 quartic parametric approximants of circular arcs

Abstract In this paper a method for approximation of circular arc by quartic Bezier curve is presented. Interpolation by geometrically continuous ( G 1 ) parametric polynomials is considered and necessary and sufficient conditions for the existence of the solution are established. Optimal asymptotic approximation order eight is also confirmed for the obtained approximant. Application of our new solution to approximation of conic sections is presented and some numerical examples provided.

Three-pencil lattices on triangulations

In this paper, three-pencil lattices on triangulations are studied. The explicit representation of a lattice, based upon barycentric coordinates, enables us to construct lattice points in a simple and numerically stable way. Further, this representation carries over to triangulations in a natural way. The construction is based upon group action of S3 on triangle vertices, and it is shown that the number of degrees of freedom is equal to the number of vertices of the triangulation.

Geometric Interpolation by Quartic Rational Spline Motions

We discuss piecewise rational motions with first order geometric continuity. In addition we describe an interpolation scheme generating rational spline motions of degree four matching given positions which are partially complemented by associated tangent information. As the main advantage of using geometric interpolation, it makes it possible to deal successfully with the unequal distribution of degrees of freedom between the trajectory of the origin and the rotation part of the motion.

CLOSED FORM FORMULA FOR THE NUMBER OF RESTRICTED COMPOSITIONS

In this paper, compositions of a natural number are studied. The number of restricted compositions is given in a closed form, and some applications are presented.

Barycentric coordinates for Lagrange interpolation over lattices on a simplex

In this paper, a (d + 1)-pencil lattice on a simplex in

Hermite Geometric Interpolation by Rational Bézier Spatial Curves

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