Ordering (geometry)

Ordered geometry is a form of geometry featuring the concept of intermediacy (or "betweenness") but, like projective geometry, omitting the basic notion of measurement. Ordered geometry is a fundamental geometry forming a common framework for affine, Euclidean, absolute, and hyperbolic geometry (but not for projective geometry).

Moritz Pasch first defined a geometry without reference to measurement in 1882. His axioms were improved upon by Peano (1889), Hilbert (1899), and Veblen (1904). Euclid anticipated Pasch's approach in definition 4 of The Elements: "a straight line is a line which lies evenly with the points on itself".

The only primitive notions in ordered geometry are points A, B, C, ... and the relation of intermediacy [ABC] which can be read as "B is between A and C".

The segment AB is the set of points P such that [APB].

The interval AB is the segment AB and its end points A and B.

The ray A/B (read as "the ray from A away from B") is the set of points P such that [PAB].

The line AB is the interval AB and the two rays A/B and B/A. Points on the line AB are said to be collinear.

An angle consists of a point O (the vertex) and two non-collinear rays out from O (the sides).

A triangle is given by three non-collinear points (called vertices) and their three segments AB, BC, and CA.

If three points A, B, and C are non-collinear, then a plane ABC is the set of all points collinear with pairs of points on one or two of the sides of triangle ABC.

If four points A, B, C, and D are non-coplanar, then a space (3-space) ABCD is the set of all points collinear with pairs of points selected from any of the four faces (planar regions) of the tetrahedron ABCD.

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