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In geometry, a point is an abstract idealization of an exact position, without size, in physical space, [1] or its generalization to other kinds of mathematical spaces.As zero-dimensional objects, points are usually taken to be the fundamental indivisible elements comprising the space, of which one-dimensional curves, two-dimensional surfaces, and higher-dimensional objects consist; conversely ...
Primitives (undefined terms) are the most basic ideas. Typically they include objects and relationships. In geometry, the objects are things like points, lines and planes while a fundamental relationship is that of incidence – of one object meeting or joining with another. The terms themselves are undefined.
With these modern definitions, every geometric shape is defined as a set of points; this is not the case in synthetic geometry, where a line is another fundamental object that is not viewed as the set of the points through which it passes. However, there are modern geometries in which points are not primitive objects, or even without points.
Although these terms are not further defined, Euclid uses them to construct more complex geometric concepts. [5] Whether a particular function or value is undefined, depends on the rules of the formal system in which it is used. For example, the imaginary number is undefined within the set of real numbers.
So plane geometry with points as the fundamental elements and plane geometry with lines as the fundamental elements are equivalent except for interpretation. This leads to the concept of duality in projective geometry, the principle that the roles of points and lines can be interchanged in a theorem in projective geometry and the result will ...
If a point B lies between points A and C, B is also between C and A, and there exists a line containing the distinct points A, B, C. If A and C are two points, then there exists at least one point B on the line AC such that C lies between A and B. [7] Of any three points situated on a line, there is no more than one which lies between the other ...