How did Felix Klein connect Euclidean length and angle to the cross ratio?
The sight lines drawn from the image in the reality plane (RP) to the artist's eye intersect the picture plane (PP) to form a projective, or perspective, drawing. The horizontal line drawn parallel to PP corresponds to the horizon. Early perspective experimenters sometimes used translucent paper or glass for the picture plane, which they drew on while looking through a small hole to keep their focus steady.
In his 1872 Erlanger Programm, Felix Klein proposed a unifying framework where Euclidean geometry and other geometries are treated as special cases of projective geometry.1
While early projective geometers viewed the cross ratio as an unsatisfactory concept because its definition relied on Euclidean length, Karl G.C. von Staudt later defined the cross ratio without any reference to length. Building on this, Klein demonstrated in 1873 that the fundamental Euclidean properties of length and angle magnitude could be defined entirely in terms of this abstract, non-metric cross ratio.2
By doing so, Klein successfully inverted the traditional hierarchy: rather than projective geometry being a derivative of Euclidean concepts, he established projective geometry as the more basic foundation from which Euclidean measurements could be derived. This approach allowed for a systematic classification of geometries based on the groups of transformations that leave specific properties invariant.21