Related papers: Besant quadrilaterals
We consider 3-periodic orbits in an elliptic billiard. Numerical experiments conducted by Dan Reznik have shown that the locus of the centers of inscribed circles of the corresponding triangles is an ellipse. We prove this fact by the…
We consider triangle faced convex polyhedra inscribed in the unit sphere $S^2$ in ${\Bbb{R}}^3$. One way of measuring their deviation from regular polyhedra with triangular faces is to consider the quotient of the lengths of the longest and…
Let P be a point inside a convex quadrilateral ABCD. The lines from P to the vertices of the quadrilateral divide the quadrilateral into four triangles. If we locate a triangle center in each of these triangles, the four triangle centers…
In this paper, a novel technique for tight outer-approximation of the intersection region of a finite number of ellipses in 2-dimensional (2D) space is proposed. First, the vertices of a tight polygon that contains the convex intersection…
A parallelogram is conformally inscribed in four lines in the plane if it is inscribed in a scaled copy of the configuration of four lines. We describe the geometry of the three-dimensional Euclidean space whose points are the…
We establish a determinant formula for the bilinear form associated with the elliptic hypergeometric integrals of type $BC_n$ by studying the structure of $q$-difference equations to be satisfied by them. The determinant formula is proved…
We study a two-loop four-point function with one internal mass. This Feynman integral is one of the simplest Feynman integrals depending on two elliptic curves. We transform the associated differential equation into an $\varepsilon$-form.…
We classify all the relative positions between an ellipsoid and an elliptic paraboloid when the ellipsoid is small in comparison with the paraboloid ({\it small} meaning that the ellipsoid cannot be tangent to the paraboloid at two points…
The locus of centers of inscribed circles in triangles, the 3-periodic orbits of an elliptic billiard, is also an ellipse. In this work we obtain the canonical equation of this ellipse, complementing the previous results obtained by O.…
For an even set of points in the plane, choose a max-sum matching, that is, a perfect matching maximizing the sum of Euclidean distances of its edges. For each edge of the max-sum matching, consider the ellipse with foci at the edge's…
The goal of this paper is to determine the number of perpendicularly inscribed polygons that intersect a given side of a regular polygon with an odd number of sides. This is done using circular permutations with repetition, and some special…
The space of quadrilaterals with fixed side lengths is an elliptic curve. Darboux used this to prove a porism on foldings. In this article, the space of oriented quadrilaterals is studied on the base of biquadratic equations between their…
This note constructs sharp obstructions for stabilized symplectic embeddings of an ellipsoid into a ball, in the case when the initial four-dimensional ellipsoid has `eccentricity' of the form 3n-1 for some integer n.
In this note, we will consider two classical volume problems related to elliptic integrals. The first problem has a neat formula by means of elliptic integrals. We remade it with details. In the second problem, we found a messy formula. On…
We analyze the inverse problem of identifying the diffusivity coefficient of a scalar elliptic equation as a function of the resolvent operator. We prove that, within the class of measurable coefficients, bounded above and below by positive…
We strongly believe that in order to prove two important geometrical pro\-blems in convexity, namely, the G. Bianchi and P. Gruber's Conjecture \cite{bigru} and the J. A. Barker and D. G. Larman's Conjecture \cite{Barker}, it is necessary…
We present higher dimensional versions of the classical results of Euler and Fuss, both of which are special cases of the celebrated Poncelet porism. Our results concern polytopes, specifically simplices, parallelotopes and cross polytopes,…
Consider a random set of points on the unit sphere in $\mathbb{R}^d$, which can be either uniformly sampled or a Poisson point process. Its convex hull is a random inscribed polytope, whose boundary approximates the sphere. We focus on the…
An equidistant polytope is a special equidistant set in the space $\mathbb{R}^n$ all of whose boundary points have equal distances from two finite systems of points. Since one of the finite systems of the given points is required to be in…
We prove that over a Poncelet triangle family interscribed between two nested ellipses $\mathcal{E},\mathcal{E}_c$, (i) the locus of the orthocenter is not only a conic, but it is axis-aligned and homothetic to a $90^o$-rotated copy of…