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

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.

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.

ARTICLE
Consider a system of two circles in three-dimensional space, with the geometry of the system demonstrated as in Figure 1 in Cartesian     1 2 3ˆˆˆ, , , ,  x y z u u u coordinates. The two circles have vector radii 1  r (1) -(2) and 2  r (3), with constant magnitudes 1 r (4) and 2 r (4), respectively, which are radius vectors at point 1 P phased at   0    and 2 P phased at    on the two circles, respectively, phased apart by a constant or time   t -dependent angle 0  (1).
The scalar magnitude 1 r (4) of the radius vector 1  r (1) -(2) is calculated as the square root of 2 1 r (4), which in turn is calculated in terms of the Dot Product 1   1 1    r r (4) of the 1  r (1) -(2) vector with itself. In the same way, the scalar magnitude 2 r (4) of the radius vector 2  r (3) is calculated as the square root of 2 2 r (4), which in turn is calculated in terms of the Dot based on (1) - (2) and (5), and the location of 2 P phased at    is defined by the vector   2   r (3). Note that the inclination angle  (1) between the planes of these two circles is also taken to be constant.  ; The vector distance     d (7) -(9) between any of these two points 1 P phased at   0    and 2 P phased at    on the two respective circles, and its magnitude   d  (10) which is also the scalar distance between 1 P and 2 P , is as demonstrated in Figure 1 and expressed in the following vector equations in (7) -(10), based on (1) - (6). It is important to note that the scalar distance   d  (10) between 1 P and 2 P is calculated as the square root of which in turn is calculated in terms of the Dot Product 1 itself. When the phase difference 0  (1) is constant for all  , magnitudes a (16) and b (17) are constant for all  , and when the phase difference 0    We now continue to analyze the situation of the distance between points 1 P and 2 P on two circles in multiple dimensions.
The centers of these two circles in four-dimensions are displaced by a constant or variable r Cos Cos r r Cos Sin r Sin Cos r Sin Sin their magnitude pair Consider a system of two circles in five dimensions defined in terms of perpendicular coordinates    , , , u u u u u .
Based on the sequence of reasoning and mathematical derivation through (1) -(53), we can reach the general conclusion in (56) -(73) for the case of two circles in multiple dimensions.
For the case of two circles in multiple dimensions, in matrix notation, for one of the circles we can define the vector radius as 1  r (56) in odd   In other words, the vector distance  