Point (geometry)

A point is a fundamental element of the geometry. Clearly, one imagines an object without including any extension before. When access to the axiomatic geometry (synthetic geometry) exist on equal footing with the points other classes of geometric objects, such as the straight line. In analytical geometry and the differential geometry, however, all other geometric objects are defined as sets of points. Functions can be considered as points in a function space in the functional analysis. In high geometry are considered as points of the associated dual space, for example, levels of a three-dimensional projective space.

The point counts as a special circuit with a radius of zero to the cone sections.

Ancient geometry, to the synthetic geometry

According to Proclus Pythagoras was the first who offered a definition of a point as a unit ( monas ), which has a position. The Greek mathematician Euclid defined by 300 BC, the point in its elements as something that has no parts and uses the term semeion ( characters), an abstract term to understand the most extensively discussed in response to the Platonic school difficulties is to detect the relationship between the points which do not have the expansion and from them presented composed lines having an extension; For example, in Aristotle's De generatione et corruptione.

For rates and their evidence in synthetic geometry, however, the true nature of points and lines does not matter, only the specified by axioms relationship among these objects. David Hilbert is attributed the saying that one could take " points, lines and planes " at any time say " tables, chairs, and beer mugs "; it 'll only ensure that the axioms are satisfied.

A point is in this case a term to take the individual axioms reference. An example is the first axiom of Hilbert's axiom system:

The meaning of point arises from the totality of the axiom system. An interpretation as an object without extension is not mandatory.

In the projective plane, the terms point and line even completely interchangeable. Thus, it is possible here, a straight line is infinitely imagined as infinitely small point, and a long and infinitely thin as.

Analytic Geometry

In analytic geometry, the geometric space is represented as a - dimensional vector space over a field. Each element of this vector space is called a point. A base defines a co-ordinate system and the components of a vector with respect to this basis are referred to as the co-ordinates of the point. A point has this dimension zero.

All other geometric objects are defined as sets of points. Such as an affinity Just as one-dimensional sub-space, and a plane is defined as a two-dimensional affine subspace. A sphere is defined as the set of points having a specific distance from the center.

Differential Geometry

In the differential geometry of the elements of a manifold are called points. These are in this case no vectors, a point can also be provided with a local map with coordinates.

Quotes

From Oskar Perron, the following remark comes from:

" A point is exactly what the intelligent, but harmless, unsophisticated reader imagines underneath. "