(iv) A compass on [mathematical expression not reproducible] with a moving origin anchored on the projective point [mathematical expression not reproducible] of [mathematical expression not reproducible] associated with [S.sup.*].

Also, if [bar.E] is always associated by convention with the canonical projective point [mathematical expression not reproducible] on the celestial circle of E, then we deduce that [[tau].sup.[??]] corresponds by a projective transformation to 0.

Then, from the three echoing causal structures Ech, [bar.E]ch, and [??]ch, we have e = [E.sub.p] = E* = E' where E* is associated with the projective point [mathematical expression not reproducible] and E' is associated with the projective point [mathematical expression not reproducible].

Then, the localization protocol at [E.sub.p] gives the formula [mathematical expression not reproducible] because [[bar.E].sup.*] is associated with the projective point [mathematical expression not reproducible].

(iv) Two compasses on the specific hemisphere of [mathematical expression not reproducible] defined above with a moving origin anchored on the projective point [mathematical expression not reproducible] associated with [S.sup.*].

(v) Two compasses on each celestial hemisphere [mathematical expression not reproducible], [mathematical expression not reproducible], [mathematical expression not reproducible], and [mathematical expression not reproducible] with a common moving origin for angle measurements anchored on the projective point [1, 1].

Caption: Figure 15: The projective disk on the celestial hemisphere [mathematical expression not reproducible] centered at [E.sub.p] and the four canonical projective points and the corresponding projective point for e.

If a point is considered as an outlier, its projective point is used to replace this outlier point.

Then, we use these extrinsic parameters to project the 3D points [D.sub.q] of these features onto the AR frame, obtaining a set of 2D projective points [q.sub.proj], which has the number of [N.sub.q] points.

The sampling points produced by SLE are evenly distributed in the design space and projective points in lower dimensions are almost uniform [20].

The points produced by this method are evenly distributed in the design space and projective points in lower dimensions are almost uniform.

The distributions of the produced sampling points are even in the design space and the projective points in lower dimensions are almost uniform, especially for projecting to each coordinate axis.