Ordered 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 ternary 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. There exist at least two points. If A and B are distinct points, there exists a C such that [ABC]. If [ABC], then A and C are distinct (A ≠ C). If [ABC], then [CBA] but not [CAB]. If C and D are distinct points on the line AB, then A is on the line CD. If AB is a line, there is a point C not on the line AB.