Related papers: Classification of the relative positions between a…
We characterize all possible relative positions between a hyperboloid of one sheet and a sphere through the roots of a characteristic polynomial associated to these quadrics. The classification is also suitable for a hyperboloid and a…
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…
Efficient methods to determine the relative position of two conics are of great interest for applications in robotics, computer animation, CAGD, computational physics, and other areas. We present a method to obtain the relative position of…
We obtain the distance of closest approach of the surfaces of two arbitrary ellipsoids valid at any orientation and separation, measured along their inter-center vector. This directional distance is derived from the Elliptic Contact…
We classify proper holomorphic mappings between generalized pseudoellipsoids of different dimensions. Those domains are parametrized by the exponents. The relations among them are also obtained. Main tool is the orthogonal decomposition of…
We classify the solutions to an overdetermined elliptic problem in the plane in the finite connectivity case. This is achieved by establishing a one-to-one correspondence between the solutions to this problem and a certain type of minimal…
An isometry is a geometric transformation that preserves distances between pairs of points. We present methods to classify isometries in the Euclidean plane, and extend these methods to spherical, single elliptical, and hyperbolic geometry.…
We establish some characterizations of elliptic hyperboloids (resp., ellipsoids) in the $(n+1)$-dimensional Euclidean space ${\Bbb E}^{n+1}$, using the $n$-dimensional area of the sections cut off by hyperplanes and the $(n+1)$-dimensional…
Both the ellipse and the hyperbola are geometric places that can be defined by establishing a relationship between points $P$ of the plane and two fixed points $A$ and $B$ (which are its foci $F'=A$ and $F=B$). Given two points $A$ and $B$…
We propose a new contact relation between polytopes. Intuitively, we say that two polytopes are in strong contact if a small enough object can pass from one of them to the other while remaining in their union. In the first half of the paper…
We consider both divergence and non-divergence parabolic equations on a half space in weighted Sobolev spaces. All the leading coefficients are assumed to be only measurable in the time and one spatial variable except one coefficient, which…
We propose a novel iterative process to establish the minimum separation between two ellipsoids. The method maintains one point on each surface and updates their locations in the theta-phi parametric space. The tension along the connecting…
Main Theorem. Two parabols have four common points. There exists a circle tangent to the sides of the obtained parabolic quadrilateral if and only if the diagonals of this quadrilateral are orthogonal. The proof of the Main Theorem is…
Various packing problems and simulations of hard and soft interacting particles, such as microscopic models of nematic liquid crystals, reduce to calculations of intersections and pair interactions between ellipsoids. When constrained to a…
A complete treatment of the intersections of two geodesics on the surface of an ellipsoid of revolution is given. With a suitable metric for the distances between intersections, bounds are placed on their spacing. This leads to fast and…
We describe a method by which the number of intersections one ellipse makes inside the plane of another can be determined. The method is based on applying a transformation that reverts one ellipse to the unit circle, and examining the…
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…
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…
Given an arbitrary convex symmetric n-dimensional body, we construct a natural and non-trivial continuous map which associates ellipsoids to ellipsoids, such that the Lowner-John ellipsoid of the body is its unique fixed point. A new…
It is well-known that in any codimension a simply connected Euclidean minimal surface has an associated one-parameter family of minimal isometric deformations. In this paper, we show that this is just a special case of the associated family…