In this paper we examine bicycle curves from the point of view of Fourier series. We analyze necessary and sufficient conditions for a circle to admit an infinitesimal deformation as a bicycle curve with rotation number ρ and derive conditions equivalent to those obtained by Tabachnikov. We then show that whenever ρ ∈ Q is not 1 2 , that there is no such deformation. Introduction Two closed cur...

In this paper, we discuss rotation number on the invariant curve of a one parameter family of outer billiard tables. Given a convex polygon η, we can construct an outer billiard table T by cutting out a fixed area A from the interior of η. T is piece-wise hyperbolic and the polygon η is an invariant curve of T under the billiard map φ. We will show that, if β ∈ η is a periodic point under φ wit...

A plane algebraic curve can be represented as the zero-set of a polynomial in two or if one takes homogenous coordinates: three variables. The coefficients of the polynomial determine the curve uniquely. Thus features of the curve, like for instance rotation symmetry, must find their correspondence in the algebraic structure of the coefficients of the polynomial. In this article we will investi...

objective: to investigate the friction of rabbit’s knee joint with designed pendulum friction tester to diagnose and evaluate different pathological and therapeutic conditions following haemarthrosis, osteoarthritis, and other joint disorders. materials and methods: friction coefficient of 10 adult male albino rabbit’s joint (5 right and 5 left knees) weighted 1.59 ± 0.27 kg, after designing a...

In this paper, a class of rational spatial curves that have a rational binormal is introduced. Such curves (called PB curves) play an important role in the derivation of rational rotation-minimizing conformal frames. The PB curve construction proposed is based upon the dual curve representation and the Euler-Rodrigues frame obtained from quaternion polynomials. The construction significantly si...

Tabachnikov’s conjecture is proved: for any closed curve Γ lying inside a convex closed curve Γ1 the mean absolute curvature T (Γ) exceeds T (Γ1) if Γ = kΓ1. §1. The problem setting and main ideas Let Γ(s), s ∈ [0, L(Γ)], be a naturally parametrized closed curve on a plane. We say that Γ(s) belongs to the class BV 1 if the velocity Γ′(s) exists and is continuous everywhere on [0, L(Γ)] except f...

In Euclidean 3-space, the Frenet and Darboux instantaneous rotation vectors for a curve (c) on the surface which the parameter curves are perpendicular to each other are known. Let us consider an arbitrary curve (c) and the parameter curves (Cl) , (cz) passing through a point P on surface. If the Darboux instantaneous rotation vectors of these curves are shown by W, WI and Wz respectively, then...

We extend the MOND analysis to a sample of 17 high surface brightness, early-type disc galaxies with rotation curves derived from a combination of 21cm Hi line observations and optical spectroscopic data. A number of these galaxies have asymptotic rotation velocities between 250 and 350 km/s making them among the most massive systems (in terms of baryonic mass) considered in the context of MOND...