Related papers: Besant quadrilaterals

We describe invariants of centers of ellipse-inscribed triangle families with two vertices fixed to the ellipse boundary and a third one which sweeps it. We prove that: (i) if a triangle center is a fixed affine combination of barycenter…

We propose a simple derivation of an upper bound for the perimeter of an ellipse. The procedure, which relies on the use of elliptic integrals, consists in introducing, via inequalities and convexity properties, specific integrals which can…

We give a characterization of all three points in $\mathbb R^4$ with integer coordinates which are at the same Euclidean distance apart. In three dimension the problem is characterized in terms of solutions of the Diophantine equations…

This paper, motivated by problems in Diophantine analysis which can be formulated as problems of finding rational points on the intersection of two quadrics, presents an explicit construction of a rationally defined isomorphism (biregular…

Suppose that we are given two distinct points, $P_1$ and $P_2$, in the interior of a triangle, $T$. Is there always an ellipse inscribed in $T$ which also passes through $P_1$ and $P_2$ ? If yes, how many such ellipses ? We answer those…

We consider second-order elliptic equations in a half space with leading coefficients measurable in a tangential direction. We prove the $W^2_p$-estimate and solvability for the Dirichlet problem when $p\in (1,2]$, and for the Neumann…

In this note the following is shown. Consider the quadratic form on (complex) matrices Q(A):=tr(A^2). Let A be such a matrix. Then an ellipse can be found, with the vector from center to focus determined by the value of Q at the traceless…

We present a general algorithm for finding the overlap area between two ellipses. The algorithm is based on finding a segment area (the area between an ellipse and a secant line) given two points on the ellipse. The Gauss-Green formula is…

We investigate the sign patterns of coefficients in the Ehrhart polynomial of the Cartesian product between the $r$-th pyramid over the Reeve tetrahedron and the hypercube $[0, n]^n$. This investigation yields partial results on the sign…

If $ABC$ is a given triangle in the plane, $P$ is any point not on the extended sides of $ABC$ or its anticomplementary triangle, $Q$ is the complement of the isotomic conjugate of $P$ with respect to $ABC$, $DEF$ is the cevian triangle of…

We introduce a new approach to the anisotropic Calder\'on problem, based on a map called Poisson embedding that identifies the points of a Riemannian manifold with distributions on its boundary. We give a new uniqueness result for a large…

Many questions about triangles and quadrilaterals with rational sides, diagonals and areas can be reduced to solving certain Diophantine equations. We look at a number of such questions including the question of approximating arbitrary…

The B\^{o}cher-Grace Theorem can be stated as follows: Let $p$ be a third degree complex polynomial. Then there is a unique inscribed ellipse interpolating the midpoints of the triangle formed from the roots of $p$, and the foci of the…

We study the inverse problem of recovering the order and the diffusion coefficient of an elliptic fractional partial differential equation from a finite number of noisy observations of the solution. We work in a Bayesian framework and show…

Witt (1996) has shown that for an elliptical potential, the four images of a quadruply lensed quasar lie on a rectangular hyperbola that passes through the unlensed quasar position and the center of the potential as well. Wynne and…

We compute an analogue of Pascal's triangle enriched in bilinear forms over a finite field. This gives an arithmetically meaningful count of the ways to choose $j$ ring homomorphisms into an algebraic closure from an \'etale extension of…

We analyze the characteristic polynomial associated to an ellipsoid and another quadric in the context of the contact detection problem. We obtain a necessary and sufficient condition for an efficient method to detect contact. This…

We give a fast, exact algorithm for solving Dirichlet problems with polynomial boundary functions on quadratic surfaces in R^n such as ellipsoids, elliptic cylinders, and paraboloids. To produce this algorithm, first we show that every…

We introduce the tetrahedron trinomial coefficient transform which takes a Pascal-like arithmetical triangle to a sequence. We define a Pascal-like infinite tetrahedron H, and prove that the application of the tetrahedron trinomial…

The Neumann problem on an ellipsoid in R^n asks for a function harmonic inside the ellipsoid whose normal derivative is some specified function on the ellipsoid. We solve this problem when the specified function on the ellipsoid is a…