Related papers: Ford Circles and Spheres
The Farey sequence is a natural exhaustion of the set of rational numbers between 0 and 1 by finite lists. Ford Circles are a natural family of mutually tangent circles associated to Farey fractions: they are an important object of study in…
We define Ford Spheres $\mathcal{P}$ in hyperbolic $n$-space associated to Clifford-Bianchi groups $PSL_2(O)$ for $O$ orders in rational Clifford algebras associated to positive definite, integral, primitive quadratic forms. For…
We show how the size of the Galois groups of iterates of a quadratic polynomial $f(x)$ can be parametrized by certain rational points on the curves $C_n:y^2=f^n(x)$ and their quadratic twists. To that end, we study the arithmetic of such…
We give an elementary geometric proof using Ford circles that the convergents of the continued fraction expansion of a real number $\alpha$ coincide with the rationals that are best approximations of the second kind of $\alpha$.
One distinguishing feature of rational curves is that they have algebraic parameterizations. Arc spaces are a way of describing approximations to parameterizations of all curves in some fixed space. Playing on these descriptions, this paper…
We offer some elementary characterisations of group and round quadratic forms. These characterisations are applied to establish new (and recover existing) characterisations of Pfister forms. We establish "going-up" results for group and…
We propose a new method for constructing rational spatial Pythagorean Hodograph (PH) curves based on determining a suitable rational framing motion. While the spherical component of the framing motion is arbitrary, the translation part is…
The ruled surfaces, i.e., surfaces generated by one parametric set of lines, are widely used in the~field of applied geometry. An~isophote on a surface is a curve consisting of surface points whose normals form a constant angle with some…
Elliptic functions are largely studied and standardized mathematical objects. The two usual approaches are due to Jacobi and Weierstrass. From a contour integral which allowed us to unify many summation formulae (Euler-MacLaurin, Poisson,…
Most genuine multi-sided surface representations depend on a 2D domain that enables a mapping between local parameters and global coordinates. The shape of this domain ranges from regular polygons to curved configurations, but the simple…
Diophantine approximation is the problem of approximating a real number by rational numbers. We propose a version of this in which the numerators are approximately related to the denominators by a Laurent polynomial. Our definition is…
The Gaussian formula and spherical aberrations of the static and relativistic curved mirrors are analyzed using the optical path length (OPL) and Fermat's principle. The geometrical figures generated by the rotation of conic sections about…
The mathematical formalism necessary for the diagramatic evaluation of quantum corrections to a conformally invariant field theory for a self-interacting scalar field on a curved manifold with boundary is considered. The evaluation of…
We treat the circular and elliptic restricted three-body problems in inertial frames as periodically forced Kepler problems with additional singularities and explain that in this setting the main result of [4] is applicable. This guarantees…
A parameterized surface can be represented as a projection from a certain toric surface. This generalizes the classical homogeneous and bihomogeneous parameterizations. We extend to the toric case two methods for computing the implicit…
We study the possibility that galactic rotation curves can be explained by a gravitational potential that contains a linear term as well as a Newtonian one. This hypothesis, suggested by conformal gravity, does allow good fits to the…
Convex regularization techniques are now widespread tools for solving inverse problems in a variety of different frameworks. In some cases, the functions to be reconstructed are naturally viewed as realizations from random processes; an…
A rational spherical triangle is a triangle on the unit sphere such that the lengths of its three sides and its area are rational multiples of $\pi$. Little and Coxeter have given examples of rational spherical triangles in 1980s. In this…
Generalized cycles can be thought of as the extension of form-cycle duality between holomorphic forms and cycles, to meromorphic forms and generalized cycles. They appeared as an ubiquitous tool in the study of spectral curves and…
We consider the space of all configurations of finitely many (potentially nested) circles in the plane. We prove that this space is aspherical, and compute the fundamental group of each of its connected components. It turns out these…