Gašper Jaklič

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

One direction in exploring similarities among biological sequences (such as DNA, RNA, and proteins), is to associate with such systems ordered sets of sequence invariants. These invariants represent selected properties of mathematical objects, such as matrices, that one can associate with biological sequences. In this article, we are exploring properties of recently introduced Line Distance matrices, and in particular we consider properties of their eigenvalues. We prove that Line Distance matrices of size n have one positive and n - 1 negative eigenvalues. Visual representation of Cauchys interlacing property for Line Distance matrices is considered. Matlab programs for line distance matrices and examples are available on the following website: www.fmf.uni-lj.si/ approximately jaklicg/ldmatrix.html.

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.

Characterization of complex biological systems by matrix invariants.

Let G be a simple, connected graph with n vertices and eigenvalues λ 1 > λ 2 ≥ … ≥ λ n . If n is even, define H = n /2 and L = H + 1. If n is odd, define H = L = ( n + 1)/2. Define the HL-index of G to be R ( G ) = max(| λ H |, | λ L |). The eigenvalues λ H and λ L appear in chemical graph theory in the study of molecular stability. In this paper, bounds on HL-index for chemical and general graphs are studied. It is shown that there exist graphs with arbitrarily large HL-index.

An approach to geometric interpolation by Pythagorean-hodograph curves

A technique for determining the permeability of a phospholipid membrane on a single giant unilamellar vesicle (GUV) is described, which complements the existing methods utilizing either a planar black lipid membrane or sub-micrometre-sized liposomes. A single GUV is transferred using a micropipette from a solution of a nonpermeable solute into an iso-osmolar solution of a solute with a higher membrane permeability. Osmotical swelling of the vesicle is monitored with a CCD camera mounted on a phase contrast microscope, and a sequence of images is obtained. On each image, the points on the vesicle contour are determined using Sobel filtering with adaptive binarization threshold, and from these, the vesicle radius is calculated with great accuracy. From the time dependence of the vesicle radius, the membrane permeability is obtained. Using a test set of data, the method provided a consistent estimate of the POPC membrane permeability for glycerol, P = 1.7 × 10−8 m s−1, with individual samples ranging from P = 1.61 × 10−8 m s−1 to P = 1.98 × 10−8 m s−1. This value is ≈40% lower than the one obtained on similar systems. Possible causes for this discrepancy are discussed.

HL-index of a graph

In this paper, a classical problem of the construction of a cubic G^1 continuous interpolatory spline curve is considered. The only data prescribed are interpolation points, while tangent directions are unknown. They are constructed automatically in such a way that a particular minimization of the strain energy of the spline curve is applied. The resulting spline curve is constructed locally and is regular, cusp-, loop- and fold-free. Numerical examples demonstrate that it is satisfactory as far as the shape of the curve is concerned.

Determining membrane permeability of giant phospholipid vesicles from a series of videomicroscopy images

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.

Planar cubic G1 interpolatory splines with small strain energy

Interpolation by rational spline motions is an important issue in robotics and related fields. In this paper a new approach to rational spline motion design is described by using techniques of geometric interpolation. This enables us to reduce the discrepancy in the number of degrees of freedom of the trajectory of the origin and of the rotational part of the motion. A general approach to geometric interpolation by rational spline motions is presented and two particularly important cases are analyzed, i.e., geometrically continuous quartic rational motions and second order geometrically continuous rational spline motions of degree six. In both cases sufficient conditions on the given Hermite data are found which guarantee the uniqueness of the solution. If the given data do not fulfill the solvability conditions, a method to perturb them slightly is described. Numerical examples are presented which confirm the theoretical results and provide evidence that the obtained motions have nice shapes.