geometry

Points

Main article: Point (geometry)

Points are considered fundamental objects in Euclidean geometry. They have been defined in a variety of ways, including Euclid's definition as 'that which has no part'[31] and through the use of algebra or nested sets.[32] In many areas of geometry, such as analytic geometry, differential geometry, and topology, all objects are considered to be built up from points. However, there has been some study of geometry without reference to points.[33]

Lines

Main article: Line (geometry)

Euclid described a line as "breadthless length" which "lies equally with respect to the points on itself".[31] In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analytic geometry, a line in the plane is often defined as the set of points whose coordinates satisfy a given linear equation,[34] but in a more abstract setting, such as incidence geometry, a line may be an independent object, distinct from the set of points which lie on it.[35] In differential geometry, a geodesic is a generalization of the notion of a line to curved spaces.[36]

Planes

Main article: Plane (geometry)

A plane is a flat, two-dimensional surface that extends infinitely far.[31] Planes are used in every area of geometry. For instance, planes can be studied as a topological surface without reference to distances or angles;[37] it can be studied as an affine space, where collinearity and ratios can be studied but not distances;[38] it can be studied as the complex plane using techniques of complex analysis;[39] and so on.

Angles

Main article: Angle

Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other.[31] In modern terms, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.[40]