Primitive Relation for Elliptic Geometry

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This Demonstration shows that the binary relation (the distance of from equals ) is a suitable primitive notion for elliptic geometry. M. Pieri showed that the ternary relation of a point being equally distant from two other points (in symbols, ) can be used as the only primitive notion of Euclidean geometry of two or more dimensions [1]. Pieri's relation can also be used as a primitive defining relation for non-Euclidean geometries.

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In the usual model of the elliptic plane, a point is a line through the origin in a three-dimensional Euclidean space. Such a line is determined by a point on a unit sphere. Two antipodal points on the sphere represent the same point of the elliptic geometry. The distance of two points is measured by the angle between their radius vectors and is always less than or equal to . Points at a distance from a given point lie on a great circle. Lines of the model are planes through two points and the center of the sphere; they are shown as great circles on the unit sphere.

Ignoring anything outside the sphere, an alternative view of the model is that points in elliptic 2D geometry are antipodal pairs of points of a unit sphere and lines are of great circles.

So the collinearity of three points means that they are on the same great circle. Perpendicularity of two segments means perpendicularity of the great circles that contain the segments. Distance is obtained by the existence of points and such that distances , and are , and by the existence of points and that form a special equilateral triangle so that is the "midpoint" of (but the definition of midpoint is not yet available). Thus we have two definitions of midpoint, depending on which antipodal point is used. Points and are symmetric relative to if is the midpoint of or Finally, and are equidistant from if there exists a point such that and are symmetric relative to and is perpendicular to .

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Contributed by: Izidor Hafner (April 2018)
Open content licensed under CC BY-NC-SA


Snapshots


Details

(collinearity)

(perpendicularity)

(distance )

(midpoint)

(external midpoint)

(symmetric)

(Robinson's definition of Pieri's relation [2, pp. 72–73])

References

[1] M. Pieri, "La Geometria Elementare istituita sulle nozioni di punto e sfera," Memorie di matematica e di fisica della Società italiana delle Scienze, ser. 3(15), 1908 pp. 345–450.

[2] R. M. Robinson, "Binary Relations as Primitive Notions in Elementary Geometry: The Axiomatic Method with Special Reference to Geometry and Physics," in Proceedings of an International Symposium Held at the University of California, Berkeley, December 26, 1957–January 4, 1958, Amsterdam: North-Holland Publishing Company, 1959. doi:10.1017/S0022481200092690.



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