Approximation of curves by polygons

Let A be a smooth curve in a Euclidean space E given by an arc length parametrization f: [0, 1] → E. Let π_n = {0 = t₀ ≤ t₁ ≤ … ≤ t_n = 1} be a partition of [0, 1] and let P_n be the polygon with vertices f(t₀), f(t₁),…, f(t_n). Let L(A) and L(P_n) denote the lengths of A and P_n, respectively. The paper investigates the behavior of n² |L(A) ? L(P_n)| when the partition π_n is induced by the sequence nθ(mod 1) for some irrational number θ. It turns out that this behavior depends on the partial quotients of the continued fraction expansion of θ.     

Approximation of planar BV homeomorphisms by diffeomorphisms

We show that a planar BV homeomorphism can be approximated in the area strict sense, together with its inverse, with smooth or piecewise affine homeomorphisms.     

Approximation of circular arcs by parametric polynomial curves

In this paper the approximation of circular arcs by parametric polynomial curves is studied. If the angular length of the circular arc is h, a parametric polynomial curve of arbitrary degree \(n \in {\mathbb{N}}\) , which interpolates given arc at a particular point, can be constructed with radial distance bounded by h ²ⁿ . This is a generalization of the result obtained by Lyche and Mørken for odd n.     

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 can interpolate given points in is confirmed for under certain natural restrictions. This conclusion also implies the optimal asymptotic approximation order. More generally, the optimal order can be achieved as soon as the interpolating curve exists.

     

Approximation of curves by fairness splines with tangent conditions

We present in this paper an approximation method of curves from sets of Lagrangian data and vectorial tangent subspaces. We define a discrete smoothing fairness spline with tangent conditions by minimizing certain quadratic functional on finite element spaces. Convergence theorem is established and some numerical and graphical examples are analyzed in order to show the validity and the effectiveness of this paper.     

Inhomogeneous Diophantine approximation on planar curves

In this paper we develop the inhomogeneous metric theory of simultaneous Diophantine approximation on planar curves. Our results naturally extend the homogeneous Khintchine and Jarník type theorems established in Beresnevich et al. (Ann Math 166(2):367–426, 2007) and Vaughan and Velani (Invent Math 166:103–124, 2006) and are the first of their kind. The key lies in obtaining essentially the best possible results regarding the distribution of ‘shifted’ rational points near planar curves.     

Generalized planar curves and quaternionic geometry

Motivated by the analogies between the projective and the almost quaternionic geometries, we first study the generalized planar curves and mappings. We follow, recover, and extend the classical approach, see e.g., (Sov. Math. 27(1) 63–70 (1983), Rediconti del circolo matematico di Palermo, Serie II, Suppl. 54 75–81) (1998), Then we exploit the impact of the general results in the almost quaternionic geometry. In particular we show, that the natural class of ℍ-planar curves coincides with the class of all geodesics of the so called Weyl connections and preserving this class turns out to be the necessary and sufficient condition on diffeomorphisms to become morphisms of almost quaternionic geometries.     

On evolutoids of planar convex curves

There are many interesting curves which we can associate with a given convex curve, however, in this work we are especially interested in studying the relations between the given curve and its evolutoids: that is, the curve obtained as the envelope of lines making a fixed angle with the normal line at every point of the curve. The first result is an inequality between the area enclosed by the given curve and the area enclosed by its evolutoid. Also, we proved that a convex curve is of constant width (centrally symmetric) if and only if its evolutoid for a fixed angle is of constant width (centrally symmetric).     

Analytic invariant curves for a planar mapping

This paper is concerned with the existence of analytic invariant curves for a planar mapping. By constructing a convergent power series solution of an auxiliary equation, analytic solutions of the original equation are obtained. In this paper, we discuss not only the general case, but also the critical cases as well, in particular, the case where β is a unit root is discussed.     

Remark on invariant curves for planar mappings

Bounded solutions for a functional equation were investigated by Ng and Zhang [C. T. Ng, W. Zhang, Invariant curves for planar mapping, J. Difference Eqn. Appl. 3 (1997) 147-168]. In this paper, we consider its solutions which may be unbounded. We give not only existence and uniqueness of its solutions but also the convex and concave solutions. Moreover, some examples are given.     

Autoduality for treelike curves with planar singularities

Let C be a treelike curve, that is, a reduced curve whose irreducible components meet transversally at disconnecting points ofC. AssumeC has planar singularities. In this note we prove an Autoduality Theorem for C. More precisely, in [6] a natural Abel map A: C → $\bar J$ to a certain compactification $\bar J$ of the generalized Jacobian of C has been constructed. In this note we prove that the pullback map $A^* :Pic^0 (\bar J) \to Pic^0 (C)$ is an isomorphism.     

Analytic invariant curves for a planar map

In this paper, we study the existence of analytic invariant curves for two-dimensional maps of the form F(x,y) = (x + y, y + G(x) + H(x + y)).     

Quadratic invariant curves for a planar mapping

A planar mapping was derived from a second order delay differential equation with a piecewise constant argument. Invariant curves for the planar mapping reflects on the dynamics of the differential equation. Results were reported on a planar mapping admitting quadratic invariant curves y=x ²+C, except for the case -3/4≥C≤0. This remaining case is now resolved, and we describe the solutions of the functional equation K(x ²+C)+k(x)=x by iterations of y.     

Asymptotics of tangent points for planar curves

In this paper, we prove a universal inequality described the asymptotic behavior of tangent points for differentiable planar curves. As corollaries, we obtain some (partially known) assertions on the asymptotics of mean value points for a number of the classical theorems in analysis, We formulate some unsolved problems.