Starting with 2 Arbitrary points i may conceive of 2 mutual spheres, sharing a common radius. Where these spherical surfaces meet in a mutual circle i may pick an arbitrary point on that circle thus defining a plane in which the three radii lie. At that point i may conceive a third mutual sphere sharing a common radius, each with 2 others
USING THESE 3 SPHERES, THE PLANE AND THE COMMON RADII I MAY GIVE A REFERENCE TO ANY POINT IN SPACE BY AT LEAST 3 rotations of line from the common radii.
Any point in space will be the meet of at least three lines or 3 planes. the 3 lines will be from the centres of the spheres, the three planes will be through the point and any 2 centres of the three spheres.
This is a very flexible Reference frame which has a counterpart in the line of sight triangulation methods of land survey, but which is entirely self referencing. The radii as metron and the sector arcs are all that is necessary to fix a point. This reduces to a quaternion system in which the radius and 3 angle /arc play the roles.
To fit in with the Cartesian coordinates the point can be projected onto the 3 radii and from there onto the 3 orthogonal axes. Since the 3 radii lie in a plane this requires some fancy footwork! I would proceed by a rational parametrisation of the sphere.
Because the positioning is based on angles of line of sight I cannot specify a unique point only a unque pair of points which lie on a connecting good line. Norman deals with uch a reference system in the flatter videos in Universal hyperbolic geometry, particularly those dealing with elliptical geometries..