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Two-dimensional space
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A two-dimensional space is a mathematical space with two dimensions, meaning points have two degrees of freedom: their locations can be locally described with two coordinates or they can move in two independent directions. Common two-dimensional spaces are often called planes, or, more generally, surfaces. These include analogs to physical spaces, like flat planes, and curved surfaces like spheres, cylinders, and cones, which can be infinite or finite. Some two-dimensional mathematical spaces are not used to represent physical positions, like an affine plane or complex plane.
Flat
[edit]The most basic example is the flat Euclidean plane, an idealization of a flat surface in physical space such as a sheet of paper or a chalkboard. On the Euclidean plane, any two points can be joined by a unique straight line along which the distance can be measured. The space is flat because any two lines transversed by a third line perpendicular to both of them are parallel, meaning they never intersect and stay at uniform distance from each-other.
Curved
[edit]Two-dimensional spaces can also be curved, for example the sphere and hyperbolic plane, sufficiently small portions of which appear like the flat plane, but on which straight lines which are locally parallel do not stay equidistant from each-other but eventually converge or diverge, respectively. Two-dimensional spaces with a locally Euclidean concept of distance but which can have non-uniform curvature are called Riemannian surfaces. (Not to be confused with Riemann surfaces.) Some surfaces are embedded in three-dimensional Euclidean space or some other ambient space, and inherit their structure from it; for example, ruled surfaces such as the cylinder and cone contain a straight line through each point, and minimal surfaces locally minimize their area, as is done physically by soap films.
Relativistic
[edit]Lorentzian surfaces look locally like a two-dimensional slice of relativistic spacetime with one spatial and one time dimension; constant-curvature examples are the flat Lorentzian plane (a two-dimensional subspace of Minkowski space) and the curved de Sitter and anti-de Sitter planes.
Non-Euclidean
[edit]Other types of mathematical planes and surfaces modify or do away with the structures defining the Euclidean plane. For example, the affine plane has a notion of parallel lines but no notion of distance; however, signed areas can be meaningfully compared, as they can in a more general symplectic surface. The projective plane does away with both distance and parallelism. A two-dimensional metric space has some concept of distance but it need not match the Euclidean version. A topological surface can be stretched, twisted, or bent without changing its essential properties. An algebraic surface is a two-dimensional set of solutions of a system of polynomial equations.
Information-holding
[edit]Some mathematical spaces have additional arithmetical structure associated with their points. A vector plane is an affine plane whose points, called vectors, include a special designated origin or zero vector. Vectors can be added together or scaled by a number, and optionally have a Euclidean, Lorentzian, or Galilean concept of distance. The complex plane, hyperbolic number plane, and dual number plane each have points which are considered numbers themselves, and can be added and multiplied. A Riemann surface or Lorentz surface appear locally like the complex plane or hyperbolic number plane, respectively.
Definition and meaning
[edit]Mathematical spaces are often defined or represented using numbers rather than geometric axioms. One of the most fundamental two-dimensional spaces is the real coordinate space, denoted consisting of pairs of real-number coordinates. Sometimes the space represents arbitrary quantities rather than geometric positions, as in the parameter space of a mathematical model or the configuration space of a physical system.
Non-real numbers
[edit]More generally, other types of numbers can be used as coordinates. The complex plane is two-dimensional when considered to be formed from real-number coordinates, but one-dimensional in terms of complex-number coordinates. A two-dimensional complex space – such as the two-dimensional complex coordinate space, the complex projective plane, or a complex surface – has two complex dimensions, which can alternately be represented using four real dimensions. A two-dimensional lattice is an infinite grid of points which can be represented using integer coordinates. Some two-dimensional spaces, such as finite planes, have only a finite set of elements.
Further reading
[edit]- Hartshorne, Robin (2000). Geometry: Euclid and Beyond. Springer. doi:10.1007/978-0-387-22676-7. ISBN 0-387-98650-2.
- Kinsey, Laura Christine (1993). Topology of Surfaces. Springer. doi:10.1007/978-1-4612-0899-0. ISBN 0-387-94102-9.
- Needham, Tristan (2021). Visual Differential Geometry and Forms. Princeton. ISBN 0-691-20370-9.
- Stillwell, John (1992). Geometry of Surfaces. Springer. doi:10.1007/978-1-4612-0929-4. ISBN 0-387-97743-0.
- Yaglom, Isaak Moiseevich (1968) [1963]. Complex Numbers in Geometry. Translated by Primrose, Eric J. F. Academic Press. LCCN 66-26269.
Two-dimensional space
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Fundamentals
Definition
Two-dimensional space, or 2D space, is formally defined in topology as a two-dimensional manifold: a Hausdorff, second-countable topological space that is locally homeomorphic to the Euclidean plane .[4] This means every point in the space has a neighborhood that can be continuously mapped onto an open subset of via a homeomorphism, allowing the space to be "flattened" locally without distortion.[4] Unlike one-dimensional space, which requires only a single coordinate to specify points (as in a line), or three-dimensional space needing three (as in ordinary volume), two-dimensional space demands exactly two independent coordinates to uniquely determine any point, parameterizing its extent in terms of area rather than length or volume.[5] The concept originates from the axiomatic foundations laid by Euclid in his Elements, composed circa 300 BCE in Alexandria, where plane geometry is developed deductively from 23 definitions, five postulates, and five common notions, establishing rigorous rules for points, lines, and surfaces without reliance on empirical measurement.[6] Euclid's system treats the plane as an infinite, flat expanse where constructions are limited to straightedge and compass, forming the basis for later mathematical developments.[6] Two-dimensional spaces can be viewed intrinsically, through properties measurable solely within the space itself (such as distances along geodesics or angle sums in triangles), or extrinsically, as subsets embedded in a higher-dimensional ambient space where curvature arises from the embedding.[7] The standard model is the Euclidean plane , an infinite flat surface where points are identified by pairs of real numbers , embodying the simplest, zero-curvature realization of 2D space.[8]Properties
Two-dimensional spaces, or surfaces, possess key topological invariants that classify them up to homeomorphism. The genus of a closed orientable surface is a topological invariant that, for such surfaces, satisfies the relation with the Euler characteristic , intuitively representing the number of 'handles' or 'holes' (e.g., a sphere has genus 0, while a torus has genus 1).[9] Another fundamental invariant is the Euler characteristic , computed for a polyhedral decomposition of the surface as , where is the number of vertices, the number of edges, and the number of faces.[10] For closed orientable surfaces, this simplifies to .[10] These invariants remain unchanged under continuous deformations, providing a complete classification for closed orientable two-dimensional manifolds.[10] Orientability is another intrinsic property distinguishing two-dimensional spaces. A surface is orientable if it admits a consistent choice of orientation, meaning a continuous selection of a normal vector field that does not reverse direction along any closed path.[11] Equivalently, an orientable surface does not contain a subset homeomorphic to a Möbius strip.[12] The Möbius strip, formed by twisting and joining the ends of a rectangular strip, exemplifies a nonorientable two-dimensional embedding, as traversing its central loop reverses the local orientation.[13] Nonorientable surfaces like the Möbius strip lack a global "inside" and "outside," impacting applications in geometry and physics.[13] Connectivity describes how paths behave within two-dimensional spaces. A space is path-connected if any two points can be joined by a continuous path.[14] Among path-connected spaces, a domain is simply connected if every closed curve can be continuously contracted to a point within the domain.[14] In two dimensions, such as subsets of the Euclidean plane, a bounded region is simply connected if both it and its complement in the plane are connected.[14] Multiply connected spaces, by contrast, contain "holes" where certain closed curves cannot be contracted, as in an annulus.[14] The Euclidean plane itself serves as the prototypical simply connected two-dimensional space.[14] A hallmark implication of two-dimensionality is the Jordan curve theorem, which underscores how curves partition the plane. This theorem states that any simple closed curve in the plane divides it into exactly two connected components: a bounded interior region and an unbounded exterior region, with the curve serving as the boundary of each.[15][16] This separation property fails in higher dimensions but defines the topological behavior unique to two-dimensional spaces.[15]Euclidean Geometry
Coordinate Systems
In Euclidean two-dimensional space, the Cartesian coordinate system provides a fundamental method for locating points using ordered pairs of real numbers , where the -axis extends horizontally from an origin point and the -axis extends vertically, forming perpendicular lines that divide the plane into four quadrants.[17] This system, named after René Descartes, allows for straightforward representation of points, lines, and shapes through algebraic equations, enabling precise calculations in vector spaces and geometry.[18] To transform points under rotation in the Cartesian system, the coordinates after a counterclockwise rotation by angle around the origin are given by the equations: These formulas derive from the properties of orthogonal transformations preserving distances and angles in the plane.[19] Polar coordinates offer an alternative representation using a radial distance from the origin and an angle measured counterclockwise from the positive -axis, denoted as . Conversion to Cartesian coordinates uses the relations and , while the reverse employs and with quadrant adjustments./11%3A_Parametric_Equations_and_Polar_Coordinates/11.03%3A_Polar_Coordinates) This system excels in problems exhibiting circular or rotational symmetry, such as describing orbits or waves, where equations simplify due to natural alignment with radial and angular variations.[18] Other coordinate systems extend these representations for specific needs. Parametric curves describe paths in the plane via functions and , where is a parameter tracing the curve, useful for modeling trajectories like ellipses without explicit functional relations between and ./10%3A_Parametric_Equations_And_Polar_Coordinates/10.01%3A_Curves_Defined_by_Parametric_Equations) Homogeneous coordinates, represented as with , embed the Euclidean plane into projective space by identifying points where coordinates are scalar multiples, facilitating transformations like perspective projections and handling points at infinity.[20] Cartesian coordinates are particularly advantageous for linear algebra operations, such as matrix multiplications and solving systems of equations, due to their orthogonal basis aligning with vector additions and projections.[18] In contrast, polar coordinates prove superior for radial problems, reducing complexity in integrals or differential equations involving angular dependence, as seen in applications like fluid dynamics around circular objects.[21]Distance and Area
In Euclidean two-dimensional space, the distance between two points is measured using the Euclidean distance formula, which quantifies the straight-line separation. For points and , the distance is given byThis formula arises directly from the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. Euclid proved this in Book I, Proposition 47 of his Elements, using geometric constructions involving similar triangles and areas of squares built on the sides. The theorem's derivation involves constructing squares on each side of the right triangle and showing equality through congruence and area addition, establishing the foundation for distance metrics in flat space.[22] Angles in two-dimensional Euclidean space are measured in either degrees or radians, with radians preferred in advanced mathematics for their natural relation to arc lengths. Degrees divide a full circle into 360 equal parts, a convention tracing back to ancient Babylonian astronomy around 2000 BCE, where 360 approximated the days in a year. Radians, defined as the ratio of arc length to radius, were formalized by James Thomson in 1871, though the concept appeared earlier in works by Roger Cotes in 1714; one full circle equals radians. The angle between two vectors and is calculated using the dot product:
where and magnitudes are Euclidean distances from the origin. This formula derives from the law of cosines in vector geometry.[23][24]/12:_Vectors_in_Space/12.03:_The_Dot_Product) Area in two-dimensional Euclidean space measures the enclosed surface, with calculations varying by shape. For a circle of radius , the area is , a result Archimedes established around 250 BCE by approximating the circle with inscribed and circumscribed polygons and taking limits, linking it to the constant as the ratio of circumference to diameter. For polygons, the shoelace formula provides an efficient coordinate-based method: for vertices listed in order, the area is
with ; this derives from summing trapezoidal areas under coordinate lines, equivalent to Green's theorem in vector calculus. For irregular shapes, area is computed via integration, such as for regions bounded by a curve from to , generalizing polygonal approximations through limits of Riemann sums.[25][26][27] The underlying structure for these measurements is the Euclidean metric tensor, which in Cartesian coordinates defines the infinitesimal distance element as
This diagonal tensor (Kronecker delta) encodes the flat geometry, where distances and areas follow from integrating along paths or over regions, consistent with Pythagoras at each point.[28][29]