Geometry of circle equations in n dimensional space

The values for x, y, ... n that satisfy the circle equations translate to a single point in n dimensional space or nDspace. The geometry of the point when plotted as described earlier in the clear definition of n-dimensional spaces is as follows:  

nDspace	Equation	Geometry
1	x^2 = 1	Point on a number line at -1 or 1
2	x^2 + y^2 = 1	Point on the circumference of a circle whose radius is 1 unit
3	x^2 + y^2 + z^2 = 1	Point on the surface of a sphere whose radius is 1 unit
4	x^2 + y^2 + z^2 + w^2 = 1	Point on the surface of a sphere defined by x, y and z whose centre O1 is w units from the centre O. Effectively this can represent oscillation between -1 and 1
5	x^2 + y^2 + z^2 + w^2 + v^2 = 1	Point on the surface of a sphere defined by x, y and z whose centre O1 is w,v units from the centre O. Effectively this can represent a point on a sphere with a circular orbit
6	x^2 + y^2 + z^2 + w^2 + v^2 + u^2 = 1	Point on the surface of a sphere defined by x, y and z whose centre O1 is defined by w,v and u from the centre O. Effectively this can represent a point on a sphere with a spherical orbit
7	x^2 + y^2 + z^2 + w^2 + v^2 + u^2 + t^2 = 1	Point on the surface of a sphere defined by x, y and z whose centre O2 is defined by w,v and u from the centre O1, t units from centre O. Effectively this can represent a point on a sphere with a spherical orbit t units from a third vertex
8	x^2 + y^2 + z^2 + w^2 + v^2 + u^2 + t^2 + s^2 = 1	Point on the surface of a sphere defined by x, y and z whose centre O2 is defined by w,v and u from the centre O1, t,s units from centre O. Effectively this can represent a point on a sphere with a spherical orbit on a given plane
9	x^2 + y^2 + z^2 + w^2 + v^2 + u^2 + t^2 + s^2 + r^2 = 1	Point on the surface of a sphere defined by x, y and z whose centre O2 is defined by w,v and u from the centre O1, t,s,r units from centre O. Effectively this can represent a point on a sphere with a spherical orbit t units from a third vertex to depict the position of a point in relation to three bodies
n	x^2 + y^2 + z^2 + w^2 + v^2 + u^2 + t^2 + s^2 + r^2 + ... + n^2 = 1	Point depicting relationship between n spherical bodies

The ellipse can be plotted with the equation x^2/A^2 + y^2/B^2 = 1 and the ellipsoid with x^2/A^2 + y^2/B^2 + z^2/C^2 = 1, and n connected ellipses can be plotted with x^2/A^2 + y^2/B^2 + z^2/C^2 + ... + n^2/N^2 = 1 in the same manner.

If the vertices represent the centre of gravity of bodies, it may be seen that they are also ruled by these fundamental equations. If the distance between Sun and Earth is one unit, then the distance between Earth and Moon is 0.001 units approximately, and 0.001^2 = 1 x 10^-6. It would be interesting to scale and map actual distances in the above equations for any three bodies, or n bodies in general.

It may also be noted that all function variables could be nested and graphed elegantly in nDspace.