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The Absolute Geometry

It is thus worth our while studying the evolution of the image of a material particle inside an Absolute from the point of view of a material particle outside Absolute and moving towards that Absolute or away from it. That evolution is an indication of the structure which a material point inside the Absolute exposes to the Universe around it at any moment. Let therefore consider a particle outside the Absolute. What this particle would see from the Absolute itself is a disk: the base of the cone of tangent lines to the Absolute, drawn from the point indicated by the particle in space. The equation of the plane containing the disk, called polar plane of that particle, can be easily obtained by polarizing the equation of the Absolute sphere (For the algebra involved in this section one of the classical treatises of Non Euclidean Geometry should be consulted. See e.g. Coolidge, 1909). Now, if the external particle moves towards the Absolute, its polar plane moves towards the particle. When they are incident, the plane is tangent to Absolute in the point where the particle touches the Absolute. As the particle proceeds inside the Absolute, its polar plane becomes external, being actually an imaginary plane.

Now, assume a material particle inside the Absolute, part of the structure of a material point, moving towards an external particle. This internal particle will have an apparent motion in a straight line along the direction of relative motion. If initially this direction meets the Absolute in two points, say P and Q, then as the two particles approach to, or depart from, each other, the line PQ – let’s call it ray from now on, for reasons that will become clear soon – changes its position, thus describing a congruence of lines, while the internal particle, which indicates a point on the generic ray, describes a certain trajectory inside the Absolute. The amount of motion out of the line PQ is a measure of both closeness of the two particles and their relative velocity.

This measure will be given by the component of motion of the internal particle perpendicular to the line PQ. Of course, this condition of perpendicularity is understood here in the sense of Absolute Geometry: two directions are perpendicular if they are conjugated with respect to the Absolute of space. Knowing the equations of the lines, this condition is easily expressed algebraically by polarization of the equation of Absolute. Another way to express the perpendicularity is by incidence: the condition that a line (L) incident to PQ is perpendicular to PQ, can be expressed by the condition that (L) is incident to the line conjugated to PQ with respect to Absolute.

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