Embedding in the context of "Line (mathematics)"

In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.

Surfaces have been extensively studied from various perspectives: extrinsically, relating to their embedding in Euclidean space and intrinsically, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.

In geometry, a straight line, usually abbreviated line, is an infinitely long object with no width, depth, or curvature. It is a special case of a curve and an idealization of such physical objects as a straightedge, a taut string, or a ray of light. Lines are spaces of dimension one, which may be embedded in spaces of dimension two, three, or higher. The word line may also refer, in everyday life, to a line segment, which is a part of a line delimited by two points (its endpoints).

Euclid's Elements defines a straight line as a "breadthless length" that "lies evenly with respect to the points on itself", and introduced several postulates as basic unprovable properties on which the rest of geometry was established. Euclidean line and Euclidean geometry are terms introduced to avoid confusion with generalizations introduced since the end of the 19th century, such as non-Euclidean, projective, and affine geometry.

In Euclidean geometry, a plane is a flat two-dimensional surface that extends indefinitely. Euclidean planes often arise as subspaces of three-dimensional space ${\displaystyle \mathbb {R} ^{3}}$.A prototypical example is one of a room's walls, infinitely extended and assumed infinitesimally thin.While a pair of real numbers ${\displaystyle \mathbb {R} ^{2}}$ suffices to describe points on a plane, the relationship with out-of-plane points requires special consideration for their embedding in the ambient space ${\displaystyle \mathbb {R} ^{3}}$.

In mathematics, an n-sphere or hypersphere is an ⁠${\displaystyle n}$⁠-dimensional generalization of the ⁠${\displaystyle 1}$⁠-dimensional circle and ⁠${\displaystyle 2}$⁠-dimensional sphere to any non-negative integer ⁠${\displaystyle n}$⁠.

The circle is considered 1-dimensional and the sphere 2-dimensional because a point within them has one and two degrees of freedom respectively. However, the typical embedding of the 1-dimensional circle is in 2-dimensional space, the 2-dimensional sphere is usually depicted embedded in 3-dimensional space, and a general ⁠${\displaystyle n}$⁠-sphere is embedded in an ⁠${\displaystyle n+1}$⁠-dimensional space. The term hypersphere is commonly used to distinguish spheres of dimension ⁠${\displaystyle n\geq 3}$⁠ which are thus embedded in a space of dimension ⁠${\displaystyle n+1\geq 4}$⁠, which means that they cannot be easily visualized. The ⁠${\displaystyle n}$⁠-sphere is the setting for ⁠${\displaystyle n}$⁠-dimensional spherical geometry.

In graph theory, a lattice graph, mesh graph, or grid graph is a graph whose drawing, embedded in some Euclidean space ⁠${\displaystyle \mathbb {R} ^{n}}$⁠, forms a regular tiling. This implies that the group of bijective transformations that send the graph to itself is a lattice in the group-theoretical sense.

Typically, no clear distinction is made between such a graph in the more abstract sense of graph theory, and its drawing in space (often the plane or 3D space). This type of graph may more shortly be called just a lattice, mesh, or grid. Moreover, these terms are also commonly used for a finite section of the infinite graph, as in "an 8 × 8 square grid".

In topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solid figures; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions of two variables; see the figure at right. However, surfaces can also be defined abstractly, without reference to any ambient space. For example, the Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space.

Topological surfaces are sometimes equipped with additional information, such as a Riemannian metric or a complex structure, that connects them to other disciplines within mathematics, such as differential geometry and complex analysis. The various mathematical notions of surface can be used to model surfaces in the physical world.