Two-dimensional space
Two-dimensional space, also known as 2D space, is a geometric structure consisting of an infinite flat plane where every point can be uniquely specified by two real-valued coordinates, typically denoted as (x, y), using a Cartesian coordinate system with perpendicular axes.[1] This space serves as the foundational setting for Euclidean plane geometry, where distances between points are measured via the Euclidean metric, defined as the square root of the sum of squared differences in coordinates: for points (x₁, y₁) and (x₂, y₂), the distance is √[(x₂ - x₁)² + (y₂ - y₁)²].[2] Unlike higher-dimensional spaces, two-dimensional space lacks depth or a third coordinate, making it ideal for modeling flat surfaces and shapes such as lines, circles, and polygons.[3] In this space, vectors are represented as ordered pairs (a₁, a₂), which can be visualized as directed line segments from the origin, and they obey vector addition via component-wise operations: (a₁, a₂) + (b₁, b₂) = (a₁ + b₁, a₂ + b₂).[3] The length (or norm) of a vector (a₁, a₂) is given by √(a₁² + a₂²), satisfying properties like positivity, homogeneity, and the triangle inequality, which underpin the Pythagorean theorem central to Euclidean geometry.[2] Angles between vectors are determined through the inner product: for vectors a = (a₁, a₂) and b = (b₁, b₂), a · b = a₁b₁ + a₂b₂, enabling concepts like perpendicularity and orthogonality.[3] Two-dimensional space distinguishes itself from affine spaces like ℝ² by incorporating a metric structure that preserves distances and angles under transformations such as rotations, reflections, and translations, forming an inner product space that supports rigorous geometric analysis.[2] It finds applications in fields ranging from computer graphics, where it models pixel grids and transformations, to physics, such as describing motion in a plane, and extends to non-Euclidean variants like hyperbolic or spherical geometry when curvature is introduced.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 \mathbb{R}^2.[4] This means every point in the space has a neighborhood that can be continuously mapped onto an open subset of \mathbb{R}^2 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 \mathbb{R}^2, an infinite flat surface where points are identified by pairs of real numbers (x, y), 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 g of a closed orientable surface is a topological invariant that, for such surfaces, satisfies the relation \chi = 2 - 2g with the Euler characteristic \chi, 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 \chi, computed for a polyhedral decomposition of the surface as \chi = V - E + F, where V is the number of vertices, E the number of edges, and F the number of faces.[10] For closed orientable surfaces, this simplifies to \chi = 2 - 2g.[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 (x, y), where the x-axis extends horizontally from an origin point (0, 0) and the y-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 (x', y') after a counterclockwise rotation by angle \theta around the origin are given by the equations: \begin{align*} x' &= x \cos \theta - y \sin \theta, \\ y' &= x \sin \theta + y \cos \theta. \end{align*} 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 r \geq 0 from the origin and an angle \theta measured counterclockwise from the positive x-axis, denoted as (r, \theta). Conversion to Cartesian coordinates uses the relations x = r \cos \theta and y = r \sin \theta, while the reverse employs r = \sqrt{x^2 + y^2} and \theta = \tan^{-1}(y/x) 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 x = x(t) and y = y(t), where t is a parameter tracing the curve, useful for modeling trajectories like ellipses without explicit functional relations between x and y./10%3A_Parametric_Equations_And_Polar_Coordinates/10.01%3A_Curves_Defined_by_Parametric_Equations) Homogeneous coordinates, represented as (x : y : w) with w \neq 0, 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 (x_1, y_1) and (x_2, y_2), the distance d is given byd = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.
This 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 $2\pi radians. The angle \theta between two vectors \mathbf{u} and \mathbf{v} is calculated using the dot product:
\cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}| |\mathbf{v}|},
where \mathbf{u} \cdot \mathbf{v} = u_x v_x + u_y v_y 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 r, the area is \pi r^2, a result Archimedes established around 250 BCE by approximating the circle with inscribed and circumscribed polygons and taking limits, linking it to the constant \pi as the ratio of circumference to diameter. For polygons, the shoelace formula provides an efficient coordinate-based method: for vertices (x_1, y_1), \dots, (x_n, y_n) listed in order, the area A is
A = \frac{1}{2} \left| \sum_{i=1}^{n} (x_i y_{i+1} - x_{i+1} y_i) \right|,
with (x_{n+1}, y_{n+1}) = (x_1, y_1); 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 \int_a^b f(x) \, dx for regions bounded by a curve y = f(x) from x = a to x = b, 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
ds^2 = dx^2 + dy^2.
This diagonal tensor g_{ij} = \delta_{ij} (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]