Euclidean space in the context of Orientation-reversing

In geometry, a three-dimensional space is a mathematical space in which three values (termed coordinates) are required to determine the position of a point. Alternatively, it can be referred to as 3D space, 3-space or, rarely, tri-dimensional space. Most commonly, it means the three-dimensional Euclidean space, that is, the Euclidean space of dimension three, which models physical space. More general three-dimensional spaces are called 3-manifolds. The term may refer colloquially to a subset of space, a three-dimensional region (or 3D domain), a solid figure.

Technically, a tuple of n numbers can be understood as the Cartesian coordinates of a location in a n-dimensional Euclidean space. The set of these n-tuples is commonly denoted ${\displaystyle \mathbb {R} ^{n},}$ and can be identified to the pair formed by a n-dimensional Euclidean space and a Cartesian coordinate system.When n = 3, this space is called the three-dimensional Euclidean space (or simply "Euclidean space" when the context is clear). In classical physics, it serves as a model of the physical universe, in which all known matter exists. When relativity theory is considered, it can be considered a local subspace of space-time. While this space remains the most compelling and useful way to model the world as it is experienced, it is only one example of a 3-manifold. In this classical example, when the three values refer to measurements in different directions (coordinates), any three directions can be chosen, provided that these directions do not lie in the same plane. Furthermore, if these directions are pairwise perpendicular, the three values are often labeled by the terms width/breadth, height/depth, and length.

Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum known as spacetime. The concept of space is considered to be of fundamental importance to an understanding of the physical universe. However, disagreement continues between philosophers over whether it is itself an entity, a relationship between entities, or part of a conceptual framework.

In the 19th and 20th centuries mathematicians began to examine geometries that are non-Euclidean, in which space is conceived as curved, rather than flat, as in the Euclidean space. According to Albert Einstein's theory of general relativity, space around gravitational fields deviates from Euclidean space. Experimental tests of general relativity have confirmed that non-Euclidean geometries provide a better model for the shape of space.

In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting (projective space) and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, for a given dimension, and that geometric transformations are permitted that transform the extra points (called "points at infinity") to Euclidean points, and vice versa.

Properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than can be expressed by a transformation matrix and translations (the affine transformations). The first issue for geometers is what kind of geometry is adequate for a novel situation. Unlike in Euclidean geometry, the concept of an angle does not apply in projective geometry, because no measure of angles is invariant with respect to projective transformations, as is seen in perspective drawing from a changing perspective. One source for projective geometry was indeed the theory of perspective. Another difference from elementary geometry is the way in which parallel lines can be said to meet in a point at infinity, once the concept is translated into projective geometry's terms. Again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a perspective drawing. See Projective plane for the basics of projective geometry in two dimensions.

In mathematics, a Euclidean plane is a Euclidean space of dimension two, denoted ${\displaystyle {\textbf {E}}^{2}}$ or ${\displaystyle \mathbb {E} ^{2}}$. It is a geometric space in which two real numbers are required to determine the position of each point. It is an affine space, which includes in particular the concept of parallel lines. It has also metrical properties induced by a distance, which allows to define circles, and angle measurement.

A Euclidean plane with a chosen Cartesian coordinate system is called a Cartesian plane.The set ${\displaystyle \mathbb {R} ^{2}}$ of the ordered pairs of real numbers (the real coordinate plane), equipped with the dot product, is often called the Euclidean plane or standard Euclidean plane, since every Euclidean plane is isomorphic to it.

In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is one less than that of the ambient space. Two lower-dimensional examples of hyperplanes are one-dimensional lines in a plane and zero-dimensional points on a line.

Most commonly, the ambient space is n-dimensional Euclidean space, in which case the hyperplanes are the (n − 1)-dimensional "flats", each of which separates the space into two half spaces. A reflection across a hyperplane is a kind of motion (geometric transformation preserving distance between points), and the group of all motions is generated by the reflections. A convex polytope is the intersection of half-spaces.

In mathematics, a ball is the solid figure bounded by a sphere; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them).

These concepts are defined not only in three-dimensional Euclidean space but also for lower and higher dimensions, and for metric spaces in general. A ball in n dimensions is called a hyperball or n-ball and is bounded by a hypersphere or (n−1)-sphere. Thus, for example, a ball in the Euclidean plane is the same thing as a disk, the planar region bounded by a circle. In Euclidean 3-space, a ball is taken to be the region of space bounded by a 2-dimensional sphere. In a one-dimensional space, a ball is a line segment.

In cosmology, a static universe (also referred to as stationary, infinite, static infinite or static eternal) is a cosmological model in which the universe is both spatially and temporally infinite, and space is neither expanding nor contracting. Such a universe does not have so-called spatial curvature; that is to say that it is 'flat' or Euclidean. A static infinite universe was first proposed by English astronomer Thomas Digges (1546–1595).

In contrast to this model, Albert Einstein proposed a temporally infinite but spatially finite model - static eternal universe - as his preferred cosmology during 1917, in his paper Cosmological Considerations in the General Theory of Relativity.

In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets.

A topological space is the most general type of a mathematical space that allows for the definition of limits, continuity, and connectedness. Common types of topological spaces include Euclidean spaces, metric spaces and manifolds.

In mathematics, a space is a set (sometimes known as a universe) endowed with a structure defining the relationships among the elements of the set.A subspace is a subset of the parent space which retains the same structure.While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, or probability spaces, it does not define the notion of "space" itself.

A space consists of selected mathematical objects that are treated as points, and selected relationships between these points. The nature of the points can vary widely: for example, the points can represent numbers, functions on another space, or subspaces of another space. It is the relationships that define the nature of the space. More precisely, isomorphic spaces are considered identical, where an isomorphism between two spaces is a one-to-one correspondence between their points that preserves the relationships. For example, the relationships between the points of a three-dimensional Euclidean space are uniquely determined by Euclid's axioms, and all three-dimensional Euclidean spaces are considered identical.

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 or other geometric elements on a manifold such as Euclidean space. The coordinates are not interchangeable; they are commonly distinguished by their position in an ordered tuple, or by a label, such as in "the x-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa; this is the basis of analytic geometry.

In mathematics, a plane is a two-dimensional space or flat surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. When working exclusively in two-dimensional Euclidean space, the definite article is used, so the Euclidean plane refers to the whole space.

Several notions of a plane may be defined. The Euclidean plane follows Euclidean geometry, and in particular the parallel postulate. A projective plane may be constructed by adding "points at infinity" where two otherwise parallel lines would intersect, so that every pair of lines intersects in exactly one point. The elliptic plane may be further defined by adding a metric to the real projective plane. One may also conceive of a hyperbolic plane, which obeys hyperbolic geometry and has a negative curvature.

In 3D human-computer interaction, positional tracking, also called pose tracking, is a process that tracks the position and/or orientation of head-mounted displays, controllers, or other input devices within Euclidean space. Pose tracking is often referred to as 6DOF tracking, for the six degrees of freedom in which the objects are often tracked.

In some consumer GPS systems, orientation data is added additionally using magnetometers, which give partial orientation information, but not the full orientation that pose tracking provides.

In geometry, a Cartesian coordinate system (UK: /kɑːrˈtiːzjən/, US: /kɑːrˈtiːʒən/) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, called coordinate lines, coordinate axes or just axes (plural of axis) of the system. The point where the axes meet is called the origin and has (0, 0) as coordinates. The axes directions represent an orthogonal basis. The combination of origin and basis forms a coordinate frame called the Cartesian frame.

Similarly, the position of any point in three-dimensional space can be specified by three Cartesian coordinates, which are the signed distances from the point to three mutually perpendicular planes. More generally, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n. These coordinates are the signed distances from the point to n mutually perpendicular fixed hyperplanes.

In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents a point P in space. Its length represents the distance in relation to an arbitrary reference origin O, and its direction represents the angular orientation with respect to given reference axes. Usually denoted x, r, or s, it corresponds to the straight line segment from O to P.In other words, it is the displacement or translation that maps the origin to P:

The term position vector is used mostly in the fields of differential geometry, mechanics and occasionally vector calculus.Frequently this is used in two-dimensional or three-dimensional space, but can be easily generalized to Euclidean spaces and affine spaces of any dimension.

Solid geometry or stereometry is the geometry of three-dimensional Euclidean space (3D space).A solid figure is the region of 3D space bounded by a two-dimensional closed surface; for example, a solid ball consists of a sphere and its interior.

Solid geometry deals with the measurements of volumes of various solids, including pyramids, prisms, cubes (and other polyhedrons), cylinders, cones (including truncated) and other solids of revolution.