Fig. 1
a Line from the neck of the cylindrical implant to the apex, and symmetrical from left to right, is the direction \(\overrightarrow{\mathrm{OA}}\); the direction \(\overrightarrow{\mathrm{OA}}\) touches the centre of the implant shoulder in the implant reference point, i.e., the point O. Line from the point O to the external reference point, i.e., the point R, is the direction \(\overrightarrow{\mathrm{OR}}\); in present example the point R is the centre of the implant shoulder of a further dental implant. \({\overrightarrow{\mathrm{OA}}}_{0}\) = (0 0 l) were the lengthwise starting vector (red arrow), where l is the implant length, and RY and RZ are the components of the crosswise vector (blue arrow), \({\overrightarrow{\mathrm{OR}}}_{0}\) = (0 RY RZ), along plane AOR, represented by the trapezoid indicated by the thin black lines, and passing through point A = (a1, a2, a3), O = (o1, o2, o3) and R = (r1, r2, r3). b Drawing of the two vectors, \(\overrightarrow{\mathrm{OA}}\) and \(\overrightarrow{\mathrm{OR}}\) on mega pixel simulated two-dimensional image of a dental implant with measured reference points on another implant: mesial point, distal point, point A, point O and point R with (c) the list of variables and that of their values as obtained from the free standalone software Osiris 4.19 which was used to acquire the two-dimensional coordinates of the points. d Three-dimensional renderings of the cylindrical implant using the three consecutive rotational angles φ, θ and ψ along the three direction of the main implant axes Xi, Yi and Zi