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The physical principle of constant velocity of light is expressed by the requirement that the change from one inertial frame to another is determined by a motion of Minkowski space, i.e. by a transformation
preserving space-time intervals. This means that
for each pair of points x and y in R1,3.
An early appreciation of the role of motion in geometry was given by Alhazen (965 to 1039). His work "Space and its Nature" uses comparisons of the dimensions of a mobile body to quantify the vacuum of imaginary space.
In the 19th century Felix Klein became a proponent of group theory as a means to classify geometries according to their "groups of motions". He proposed using symmetry groups in his Erlangen program, a suggestion that was widely adopted. He noted that every Euclidean congruence is an affine mapping, and each of these is a projective transformation; therefore the group of projectivities contains the group of affine maps, which in turn contains the group of Euclidean congruencies. The term motion, shorter than transformation, puts more emphasis on the adjectives: projective, affine, Euclidean. The context was thus expanded, so much that "In topology, the allowed movements are continuous invertible deformations that might be called elastic motions."
In 1914 D. M. Y. Sommerville used the idea of a geometric motion to establish the idea of distance in hyperbolic geometry when he wrote Elements of Non-Euclidean Geometry. He explains:
By a motion or displacement in the general sense is not meant a change of position of a single point or any bounded figure, but a displacement of the whole space, or, if we are dealing with only two dimensions, of the whole plane. A motion is a transformation which changes each point P into another point P ? in such a way that distances and angles are unchanged.
Any motion is a one-to-one mapping of space R onto itself such that every three points on a line will be transformed into (three) points on a line.
The identical mapping of space R is a motion.
The product of two motions is a motion.
The inverse mapping of a motion is a motion.
If we have two planes A, A' two lines g, g' and two points P, P' such that P is on g, g is on A, P' is on g' and g' is on A' then there exist a motion mapping A to A', g to g' and P to P'
There is a plane A, a line g, and a point P such that P is on g and g is on A then there exist four motions mapping A, g and P onto themselves, respectively, and not more than two of these motions may have every point of g as a fixed point, while there is one of them (i.e. the identity) for which every point of A is fixed.
There exists three points A, B, P on line g such that P is between A and B and for every point C (unequal P) between A and B there is a point D between C and P for which no motion with P as fixed point can be found that will map C onto a point lying between D and P.