geometrypoints

A geometrypoint, or simply a point in geometry, is the fundamental object that represents a location in space with no size, extent, or shape. In formal treatments, a point is often treated as a primitive concept not defined in terms of other objects; it is something that exists because a geometric framework requires a notion of position. In practical terms, a point is usually represented by a coordinate description in a chosen space.

In two-dimensional geometry, a point is typically given by Cartesian coordinates (x, y); in three dimensions, by (x, y, z). More generally, a point in n-dimensional space is an element of R^n. In projective or homogeneous coordinate systems, a point can be represented by a tuple (x:y:z:w), allowing affine transformations to be handled more conveniently. Points encode positions that can be used to define other geometric objects.

A key property of a point is that it has no length, area, or volume. Distances between points are defined by a metric on the space, with the distance between two distinct points being positive. Two points are identical if and only if their coordinates agree.

Points serve as the building blocks for many geometric constructions: lines, planes, curves, and polygons are defined as sets of points satisfying certain conditions. In computational settings, points are stored as coordinate tuples and subjected to transformations such as translation, rotation, scaling, or projection.

Across geometries, the basic idea of a point remains: a precise location in a space. In non-Euclidean and topological contexts, points still denote positions, but their relationships (distance, incidence, neighborhood) are defined by the underlying geometric or topological structure.