Ordered geometry is a form of geometry featuring the concept of intermediacy (or "betweenness") but, like projective geometry, omitting the basic notion...

other geometries. Absolute geometry is an extension of ordered geometry, and thus, all theorems in ordered geometry hold in absolute geometry. The converse...

In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting") the metric notions of distance...

Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry, Elements...

Foundations of geometry is the study of geometries as axiomatic systems. There are several sets of axioms which give rise to Euclidean geometry or to non-Euclidean...

affine and Euclidean geometry. Projective geometry is not "ordered" and so it is a distinct foundation for geometry. Projective geometry is less restrictive...

algebraic geometry Noncommutative geometry Ordered geometry Parabolic geometry Plane geometry Projective geometry Quantum geometry Riemannian geometry Ruppeiner...

In geometry, a point is an abstract idealization of an exact position, without size, in physical space, or its generalization to other kinds of mathematical...

generally than above, the concept of a line segment can be defined in an ordered geometry. A pair of line segments can be any one of the following: intersecting...

orderings. Every subfield of an ordered field is also an ordered field in the inherited order. Every ordered field contains an ordered subfield that is isomorphic...

theory. Contributors to ordered geometry were listed in a 1961 textbook: It was Pasch in 1882, who first pointed out that a geometry of order could be developed...

In abstract algebra, an ordered ring is a (usually commutative) ring R with a total order ≤ such that for all a, b, and c in R: if a ≤ b then a + c ≤...

Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric...

In geometry, a straight line, usually abbreviated line, is an infinitely long object with no width, depth, or curvature, an idealization of such physical...

In mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts...

name of Ricci calculus Absolute geometry Also called neutral geometry, a synthetic geometry similar to Euclidean geometry but without the parallel postulate...

(1969, p. 192) axiomatizes the special case of affine geometry over the reals as ordered geometry together with an affine form of Desargues's theorem and...

systems of geometry include ordered geometry, absolute geometry, affine geometry, Euclidean geometry, projective geometry, and hyperbolic geometry. For each...

mathematics such as David Hilbert's axioms for geometry, and the theories of ordered groups, ordered fields, and local fields. An algebraic structure...

statement, is also included and remains an axiom in Hilbert's treatment. Ordered geometry Pasch's axiom Pasch 1912 Coxeter (1969, p. 179) states the result in...

should not be confused with discrete geometry (combinatorial geometry). Order theory is the study of partially ordered sets, both finite and infinite. It...

unnecessarily powerful, instead proving the theorem using only the axioms of ordered geometry. This proof is by Leroy Milton Kelly. Aigner & Ziegler (2018) call...

In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the position of the points...

non-Archimedean ordered field based on the field of rational functions. In this geometry, there are significant differences from Euclidean geometry; in particular...

In geometry, Pasch's axiom is a statement in plane geometry, used implicitly by Euclid, which cannot be derived from the postulates as Euclid gave them...

the Dehn field, an example of a non-Archimedean ordered field, to construct non-Euclidean geometries in which the parallel postulate fails to be true...

reflexive, antisymmetric, and transitive. A partially ordered set (poset for short) is an ordered pair P = ( X , ≤ ) {\displaystyle P=(X,\leq )} consisting...

green and blue) in an image sensor or display can be ordered in different patterns, called pixel geometry. The geometric arrangement of the primary colors...

Relativitätstheorie (Springer Verlag, 1933) Hughes plane Finite geometry Ordered geometry Hall plane of order 9 Mac Lane, Saunders (1964). "Oswald Veblen...

Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems...