We all have in mind Einstein's famous thought experiment in the elevator where we observe the free fall of a body and then the trajectory of a light ray. Simply here, in addition to the qualitative aspect, we carry out the exact calculation. We consider a uniformly accelerated reference frame in rectilinear translation and we show that the trajectories of the particles are ellipses centered on the horizon of the events. The frame of reference is non-inertial, the space-time is flat, the metric is non-Minkowskian and the computations are performed within the framework of special relativity. The deviation, compared to the classical case, is important close to the horizon, but small in the box, and the interest is above all theoretical and pedagogical. The study helps the student to become familiar with the concepts of metric, coordinate velocity, horizon, and, to do the analogy with the black hole.

This article deals with special curves similar to those known in mathematics as Lamé curves. Their equations have a certain similarity with the equations of Lamé curves. But these curves are more like ellipses in appearance. Like ellipses, they have major and minor axes. These are such curves that their major and minor semi-axis coincide with an ellipse but are different than an ellipse. The main feature of these curves is that for any triangle whose base coincides with the axis of a given curve and the vertex is on a given curve, the condition aⁿ+bⁿ=cⁿ is obtained. The number “n” is called the degree or rank of this curve. To distinguish this group of curves, we called them Elbe curves. The equations of these lines are also provided in the paper.

This paper presents a new approach to a different proof of "Fermat's Last Theorem.” For the proof of the theorem, it is proposed to use a more straightforward geometrical approach. A special family of curves, the Elba curves, is introduced to facilitate the proof. This approach makes the proof easier to comprehend by mathematicians and those interested in the subject.

This paper presents a new approach to a different proof of "Fermat's Last Theorem.” For the proof of the theorem, it is proposed to use a more straightforward geometrical approach. A special family of curves, the Elba curves, is introduced to facilitate the proof. This approach makes the proof easier to comprehend by mathematicians and those interested in the subject.

Autonomous aerial refueling technology is an effective solution to extend flight duration of unmanned aerial vehicles, and also a great challenge due to its high risk. For autonomous probe-and-drogue refueling tasks, relative navigation to provide relative position between the receiver aircraft and the refueling drogue is the first and essential step, and vision-based method is the most frequently used. A new monocular vision navigation sensor with image processing strategy consisting of the drogue detection method and the tracking method is developed for autonomous aerial refueling in this paper. In the drogue detection method, thresholding and mathematical morphology method are adopted to eliminate image interference, and contours extraction method is applied to obtain all contours, which are then subsequently checked to achieve target contour of drogue. In the tracking method, a rectangle of interest (ROI) of current frame image is determined by positioning results of last frame, and then processed by the previous drogue detection method. Finally, the proposed image processing strategy in monocular vision navigation sensor is validated using real flight images, which are captured from an autonomous aerial refueling testbed using a micro six-rotor aircraft as receiver aircraft.

Based on measured astronomical position data of heavenly objects in the Solar System and other planetary systems, all bodies in space seem to move in some kind of elliptical motion with respect to each other. According to Kepler’s 1^st Law, “orbit of a planet with respect to the Sun is an ellipse, with the Sun at one of the two foci.” Orbit of the Moon with respect to Earth is also distinctly elliptical, but this ellipse has a varying eccentricity as the Moon comes closer to and goes farther away from the Earth in a harmonic style along a full cycle of this ellipse. In this paper, our research results are summarized, where it is first mathematically shown that the “distance between points around any two different circles in three dimensional space” is equivalent to the “distance of points around a vector ellipse to another fixed or moving point, as in two dimensional space”. What is done is equivalent to showing that bodies moving on two different circular orbits in space vector wise behave as if moving on an elliptical path with respect to each other, and virtually seeing each other as positioned at an instantaneously stationary point in space on their relative ecliptic plane, whether they are moving with the same angular velocity, or different but fixed angular velocities, or even with different and changing angular velocities with respect to their own centers of revolution. This mathematical revelation has the potential to lead to far reaching discoveries in physics, enabling more insight into forces of nature, with a formulation of a new fundamental model regarding the motions of bodies in the Universe, including the Sun, Planets, and Satellites in the Solar System and elsewhere, as well as at particle and subatomic level. Based on the demonstrated mathematical analysis, as they exhibit almost fixed elliptic orbits relative to one another over time, the assertion is made that the Sun, the Earth, and the Moon must each be revolving in their individual circular orbits of revolution in space. With this expectation, individual orbital parameters of the Sun, the Earth, and the Moon are calculated based on observed Earth to Sun and Earth to Moon distance data, also using analytical methods developed as part of this research to an approximation. This calculation and analysis process have revealed additional results aligned with observation, and this also supports our assertion that the Sun, the Earth, and the Moon must actually be revolving in individual circular orbits.

Medical research has suggested that the anterior-posterior (AP)-diameter of the inferior vena cava (IVC) and its associated temporal variation as imaged by bedside ultrasound is useful in guiding fluid resuscitation of the critically-ill patients. This paper develops semi-automatic active ellipse and rectangle algorithms for measurement and tracking of the AP-diameter. The proposed algorithms are compared with an expert manual measurement and the previous work based on active circle model. It is shown that regardless of the shape of the IVC, the rectangle model always outperforms the two other models and performs very close to manual measurement.

unlike the volume, the expression for the surface area of a regular conoid has not yet been obtained by means of direct integration or a differential geometry procedure. As this non-developable form is relatively used in engineering, the difficulty to determine its surface, represents a serious shortcoming for several problems which arise in radiative transfer, lighting and construction, to cite just a few. In order to solve the problem, I conceived the surface as a set of linearly dwindling ellipses which remain parallel to a circular directrix, a typical problem appears when searching the length of such ellipses. I employed a new procedure which, in principle, consists in dividing the surface into infinitesimal elliptic strips to which we have successively applied Ramanujan’s second approximation for the longitude of the ellipse. In this manner, we can obtain the perimeter of any transversal curve pertaining to the said form as a function of the radius of the directrix and the position of the ellipse’s center on the X-axis. Integrating the so-found perimeters of the differential strips for the whole span of the conoid, an unexpected solution emerges through the newly found number psi (ψ) which seems like a refined approximation to the third decimal of Pi but derived from a definite integral equation. As the strips are in truth slanted in the symmetry axis, their width is not uniform and we need to perform some adjustments in order to complete the problem with sufficient precision, but this is discussed separately in the annexes. Relevant implications for mathematical symmetry applied to multifarious architectural and engineering forms can be derived from this finding

Based on measured astronomical position data of heavenly objects in the Solar System and other planetary systems, all bodies in space seem to move in some kind of elliptical motion with respect to each other, whereas objects follow parabolic escape orbits while moving away from Earth and bodies asserting a gravitational pull, and some comets move in near-hyperbolic orbits when they approach the Sun. In this article, it is first mathematically proven that the “distance between points on any two different circles in three-dimensional space” is equivalent to the “distance of points on a vector ellipse from another fixed or moving point, as in two-dimensional space.” Then, it is further mathematically demonstrated that “distance between points on any two different circles in any number of multiple dimensions” is equivalent to “distance of points on a vector ellipse from another fixed or moving point”. Finally, two special cases when the “distance between points on two different circles in multi-dimensional space” become mathematically equivalent to distances in “parabolic” or “near-hyperbolic” trajectories are investigated. Concepts of “vector ellipse”, “vector hyperbola”, and “vector parabola” are also mathematically defined. The mathematical basis derived in this Article is utilized in the book “Everyhing Is A Circle: A New Model For Orbits Of Bodies In The Universe” in asserting a new Circular Orbital Model for moving bodies in the Universe, leading to further insights in Astrophysics.