Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Distance Between Two Circles In Any Number Of Dimensions Is A Vector Ellipse

Version 1 : Received: 22 April 2021 / Approved: 23 April 2021 / Online: 23 April 2021 (11:36:05 CEST)

How to cite: Tan, A.P. Distance Between Two Circles In Any Number Of Dimensions Is A Vector Ellipse. Preprints 2021, 2021040632. https://doi.org/10.20944/preprints202104.0632.v1 Tan, A.P. Distance Between Two Circles In Any Number Of Dimensions Is A Vector Ellipse. Preprints 2021, 2021040632. https://doi.org/10.20944/preprints202104.0632.v1

Abstract

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.

Keywords

Conic Sections, Topology, Circle, Ellipse, Hyperbola, Parabola, Orbit, Trajectory, Orbital Mechanics, Solar System, Planetary System, Planet, Satellite, Comet, Sun, Earth, Moon

Subject

Physical Sciences, Mathematical Physics