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Euclidean plane

The Euclidean plane is a two-dimensional affine space equipped with an inner product on its vector space of translations, enabling the definition of distances, angles, and rigid motions that form the foundation of classical plane geometry.^[1] Analytically, it can be modeled as the set \mathbb{R}^2 paired with the Euclidean distance function d(\mathbf{x}, \mathbf{y}) = \|\mathbf{x} - \mathbf{y}\|_2 = \sqrt{(x_1 - y_1)^2 + (x_2 - y_2)^2}, where the norm arises from a positive definite quadratic form, ensuring properties like the Pythagorean theorem hold for orthogonal vectors.^[2] This structure satisfies key axioms, including the existence of unique lines between points, congruence of segments and angles, and the parallel postulate, which states that through a point not on a given line, exactly one parallel line can be drawn.^[1] In synthetic terms, the Euclidean plane consists of points and lines without reference to coordinates, where geometric figures are defined by incidence, order, congruence, and continuity relations, as formalized in modern axiomatizations like Hilbert's.^[1] It distinguishes itself from non-Euclidean planes by the symmetry of orthogonality and the congruence of all right angles, leading to characteristic theorems such as the sum of angles in a triangle equaling 180 degrees.^[2] The plane's isometries—translations, rotations, reflections, and glide reflections—preserve distances and orientations, making it a model for rigid body motions in physics and computer graphics.^[1] Historically rooted in Euclid's Elements (circa 300 BCE), the Euclidean plane provides the axiomatic basis for much of elementary mathematics, influencing fields from architecture to cartography, while serving as a benchmark for contrasting geometries like hyperbolic or elliptic spaces.^[2]

Fundamentals

Definition and axioms

The Euclidean plane is a two-dimensional flat space that satisfies the axioms of Euclidean geometry and serves as the ambient space for the study of plane geometry, where points, lines, and figures are defined without intrinsic curvature.^[3] It can be formalized as the set \mathbb{R}^2 of ordered pairs of real numbers, equipped with the Euclidean metric that measures distances between points. The distance d between two points (x_1, y_1) and (x_2, y_2) in this space is defined by the formula

d((x_1, y_1), (x_2, y_2)) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2},

which induces the standard topology and geometry on the plane.^[4] The initial axiomatic foundation for the Euclidean plane was provided by Euclid in his treatise Elements (circa 300 BCE), where five postulates specifically govern plane constructions and relations.^[5] These postulates are:

A straight line can be drawn between any two points.
Any terminated straight line can be extended indefinitely.
A circle can be drawn with any given center and radius.
All right angles are equal to each other.
If a straight line intersects two other straight lines such that the sum of the interior angles on one side is less than two right angles, then the two lines, if extended, will meet on that side.^[6]

The fifth postulate, known as the parallel postulate, asserts that given a line and a point not on it, exactly one line through the point is parallel to the given line; this ensures the plane's flatness by preventing hyperbolic or elliptic curvatures that arise in non-Euclidean geometries.^[7] To resolve ambiguities and gaps in Euclid's system—such as undefined terms like "between" and incomplete continuity assumptions—David Hilbert presented a rigorous axiomatization in his 1899 work Grundlagen der Geometrie (Foundations of Geometry), comprising 20 independent axioms that fully characterize plane Euclidean geometry.^[8] These are grouped as follows:

Incidence axioms: These define points and lines as primitive elements, with relations like: two distinct points determine a unique line; every line contains at least two points; there exist three points not all on the same line. They establish the basic combinatorial structure without order or measurement.^[8]
Order axioms (4 axioms): Introducing betweenness (II, 1–4), these specify that for any three collinear points, exactly one lies between the other two; they prevent cycles and ensure linear ordering on lines, foundational for defining segments and rays.^[8]
Congruence axioms (5 axioms): These define equality of segments and angles (III, 1–5), such as congruence being an equivalence relation for segments (III, 1) and the ability to superimpose congruent figures (III, 4–5); they enable the measurement and comparison central to Euclidean constructions.^[8]
Parallelism axiom (1 axiom): Stating that through a point not on a line, there exists one and only one parallel line (IV, 1), this is equivalent to Euclid's fifth postulate and ensures the plane's flatness.^[8]
Continuity axioms (2 axioms): The Archimedean axiom (V, 1) guarantees that the real numbers densely embed into the plane's lengths, while the completeness axiom (V, 2) ensures every bounded nonempty set of points has a least upper bound, providing the full real-line continuum for the plane.^[8]

Hilbert's framework proves the consistency and completeness of Euclidean plane geometry relative to the real numbers, resolving foundational issues in Euclid's original postulates.^[8]

Basic properties

The Euclidean plane, founded on axioms of incidence, order, congruence, parallelism, and continuity, possesses core properties that follow directly from these foundations. Betweenness, defined via the order axioms, establishes a total order on points along any line: for distinct points A and B, any third point C on the line through A and B is either between A and B (lying in the open segment they determine), or A between B and C, or B between A and C.^[9] This relation ensures the plane's lines are Dedekind-complete chains, preventing gaps in linear order. Congruence of segments equates those of equal length, while angle congruence equates those superimposable by rigid motion; these are axiomatized such that if two segments are congruent and adjacent segments from their endpoints are congruent, the included angles are congruent (SAS criterion).^[10] The existence of midpoints follows from these axioms: for any segment AB, there exists a point M such that AM \cong MB, constructed via the circle axiom and congruence properties.^[11] A hallmark plane-specific consequence is the Pythagorean theorem, which holds due to the parallel postulate. In a right triangle with legs of lengths a and b and hypotenuse c,

a^2 + b^2 = c^2,

proved by dropping an altitude to the hypotenuse, creating similar triangles whose ratios yield the relation through area equivalence or similarity arguments.^[12] This theorem fails in non-Euclidean geometries without the parallelism axiom. The isometries of the Euclidean plane—distance-preserving transformations—comprise translations (rigid shifts along a vector), rotations (about a fixed point by an angle), reflections (over a line), and glide reflections (reflection followed by translation parallel to the line). Every non-identity isometry is exactly one of these types, and they generate the Euclidean group E(2), a semidirect product of the translation group \mathbb{R}^2 and the orthogonal group O(2) of rotations and reflections, with order 2 elements being orientation-reversing (reflections and glides) and others orientation-preserving.^[13] Underlying these structures, the Euclidean plane is coordinatized by the real numbers \mathbb{R}, an Archimedean ordered field where, for any positive x, y \in \mathbb{R}, there exists a natural number n such that nx > y. Completeness arises from the Dedekind cut construction: every partition of \mathbb{R} into non-empty lower and upper sets with no greatest lower element defines a unique real number as the cut's supremum, ensuring every bounded non-empty subset has a least upper bound, which underpins the plane's continuity.^[14] Topologically, the Euclidean plane \mathbb{R}^2 is simply connected, as every closed curve can be continuously shrunk to a point without leaving the space, reflecting its contractibility. It exhibits zero Gaussian curvature everywhere, enabling the parallel postulate—through a point not on a line, exactly one parallel exists—unlike positively curved elliptic planes (where parallels intersect) or negatively curved hyperbolic planes (with multiple parallels).^[15]

Historical development

Ancient origins

The concepts underlying the Euclidean plane emerged from practical needs in ancient civilizations, particularly for land measurement and construction. In ancient Egypt around 2000 BCE, geometry developed empirically to survey fields after Nile floods, using tools like the merkhet for alignment and ropes for right angles.^[16] The Rhind Mathematical Papyrus, dating to circa 1650 BCE, contains problems on calculating areas of triangles and circles, reflecting these surveying techniques without formal proofs.^[17] Similarly, in Babylon around the same period, cuneiform tablets demonstrate knowledge of Pythagorean triples and approximations for circular areas, applied to architecture and astronomy.^[18] Greek thinkers in the 6th century BCE built on these influences, shifting toward deductive reasoning. Thales of Miletus (c. 624–546 BCE), often credited as the founder of Greek geometry, introduced theorems such as the equality of base angles in isosceles triangles and the intercept theorem for parallel lines, likely inspired by Egyptian and Babylonian practices during his travels.^[19] His work emphasized proofs, marking a transition from empirical to logical methods.^[20] The Pythagorean school, founded by Pythagoras (c. 570–495 BCE) in Croton, Italy, advanced studies of right triangles, formalizing the relationship now known as 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.^[21] This school integrated geometry with philosophy, viewing numbers and shapes as fundamental to the cosmos.^[22] Euclid of Alexandria (c. 300 BCE) synthesized these developments in his seminal work, Elements, providing the first comprehensive axiomatic treatment of plane geometry in Books I–IV. Book I covers basic constructions like equilateral triangles and congruence; Book II addresses geometric algebra; Book III deals with circles and inscribed angles; and Book IV constructs regular polygons.^[5] This compilation drew from prior Greek sources while establishing a rigorous framework that influenced mathematics for over two millennia.^[23] The preservation and extension of these ideas occurred in the Islamic world during the 8th–9th centuries CE. Al-Khwarizmi (c. 780–850 CE), working in Baghdad's House of Wisdom, authored The Compendious Book on Calculation by Completion and Balancing, incorporating geometric methods inspired by Euclid into algebraic solutions for plane figures and applying them to inheritance and surveying problems.^[24]^[25] His works, along with translations of Greek texts by other scholars, bridged ancient traditions, ensuring the transmission of Euclidean plane concepts to medieval Europe.^[25]

Modern axiomatization

In the 19th century, mathematicians identified several gaps in Euclid's ancient axiomatic system for plane geometry, including the lack of explicit axioms for order (betweenness) on lines, incidence relations between points and lines, and continuity to guarantee the existence of intersections in constructions such as drawing circles.^[26] These deficiencies allowed for ambiguities in proofs and potential inconsistencies, prompting efforts to develop rigorous, complete, and independent sets of axioms that could derive all Euclidean theorems without hidden assumptions.^[8] A key early contribution came from Moritz Pasch in 1882, who introduced axioms for betweenness to formalize the ordering of points on a line and the separation properties of lines in the plane.^[26] Pasch's axiom states that if a line intersects one side of a triangle, it must also intersect exactly one of the other two sides, ensuring that lines divide the plane into distinct half-planes and preventing pathological configurations absent in Euclidean geometry.^[26] This addressed Euclid's implicit reliance on intuitive notions of "inside" and "outside" without proof, providing a foundation for ordered geometry that influenced subsequent axiomatizations.^[26] Building on such work, Mario Pieri advanced the axiomatization of incidence in 1895, proposing a system for projective geometry that treated points and lines as primitive concepts with minimal assumptions about their mutual relations. Pieri's axioms emphasized the independence of incidence from metric properties, allowing Euclidean plane geometry to be derived as a special case by adding order and congruence conditions, thus clarifying the foundational structure detached from coordinate or measurement assumptions. David Hilbert's seminal 1899 monograph Grundlagen der Geometrie provided a comprehensive modern foundation by organizing axioms into five groups: incidence, order (incorporating Pasch's betweenness), congruence, parallelism, and continuity.^[8] Hilbert demonstrated the independence of each axiom by constructing models where specific ones fail while others hold, and ensured completeness through Archimedean and completeness axioms that model the real numbers, resolving Euclid's gaps such as the continuity required for circle constructions to intersect properly.^[8] For instance, his continuity axioms guarantee that any line through a point inside a circle intersects the circle at two points, enabling rigorous proofs of existence in geometric constructions.^[27] In 1932, Garrett Birkhoff proposed a streamlined metric-based axiomatization with just four postulates, integrating real numbers directly to define distance and angle measures via ruler and protractor operations.^[28] Birkhoff's system posits points as pairs of real coordinates, lines by equations, a positive definite distance function, and a congruence axiom for angles modulo $2\pi, while assuming the ruler-compass constructibility aligns with real arithmetic, thus deriving Euclidean properties from analytic foundations without separate incidence or order primitives.^[28] These developments profoundly influenced the foundations of mathematics by establishing geometry as a formal system amenable to logical analysis, with Hilbert's framework linking to set theory through models over the reals and to model theory via categoricity and completeness results.^[29] For the Euclidean plane, this axiomatization confirms unique up-to-isometry realization in \mathbb{R}^2, underpinning applications in algebra and analysis while highlighting geometry's dependence on the continuum.^[29]

Coordinate systems

Cartesian coordinates

The Cartesian coordinate system provides an algebraic framework for representing points and geometric objects in the Euclidean plane, transforming geometric problems into equations solvable by algebraic methods. Invented by René Descartes in his 1637 treatise La Géométrie, this system marked a pivotal advancement by systematically linking algebraic notation to geometric constructions, enabling the resolution of complex figures through coordinate assignments rather than purely synthetic proofs.^[30] In La Géométrie, Descartes assigned coordinates to points using intersecting lines as references, laying the groundwork for analytic geometry and influencing subsequent mathematical developments.^[30] In this system, the Euclidean plane is modeled as the set \mathbb{R}^2, where each point is uniquely specified by an ordered pair of real numbers (x, y), corresponding to distances along two perpendicular axes: the horizontal x-axis and vertical y-axis, intersecting at the origin (0, 0).^[31] The axes form a right-handed orthogonal frame, with positive directions extending rightward and upward from the origin, allowing precise location of any point via signed distances from these references. Straight lines in the plane are represented by linear equations of the form ax + by + c = 0, where a, b, and c are real constants (with a and b not both zero), encapsulating all points satisfying the relation.^[32] Common transformations within the Cartesian system include translations and scalings, which preserve the plane's structure while shifting or resizing coordinates. A translation by vector (h, k) maps a point (x, y) to (x + h, y + k), effectively relocating the origin without altering orientations or relative distances.^[33] Scaling by factors s_x and s_y transforms (x, y) to (s_x x, s_y y), stretching or compressing along the axes, though non-uniform scalings may distort angles unless s_x = s_y. Curves beyond straight lines are often parameterized as x = f(t), y = g(t), where t varies over an interval, tracing paths like parabolas or ellipses through functional dependence on a single parameter.^[34] The primary advantage of Cartesian coordinates lies in converting geometric inquiries into algebraic manipulations, such as finding line intersections by solving simultaneous linear equations, which yields exact coordinates without ruler-and-compass constructions.^[30] For instance, the intersection of lines a_1 x + b_1 y + c_1 = 0 and a_2 x + b_2 y + c_2 = 0 is determined via Cramer's rule or substitution, providing a systematic algebraic resolution to what was previously a synthetic challenge. This approach not only simplifies computations but also extends to higher-degree equations for conics and beyond, underpinning much of modern geometry and its applications.^[32]

Polar and other systems

In the Euclidean plane, polar coordinates provide an alternative representation to Cartesian coordinates, specifying each point by a radial distance r \geq 0 from a fixed origin called the pole and an angle \theta measured counterclockwise from the positive x-axis.^[35] This system leverages rotational symmetry, making it particularly useful for problems involving circles, rotations, or radial patterns.^[36] The angle \theta is typically expressed in radians and can take any real value, though representations are not unique since adding $2\pi k to \theta (for integer k) or negating r with \theta + \pi yields equivalent points.^[35] Conversions between polar and Cartesian coordinates are given by the equations

x = r \cos \theta, \quad y = r \sin \theta

and inversely,

r = \sqrt{x^2 + y^2}, \quad \theta = \atantwo(y, x),

where \atantwo accounts for the correct quadrant.^[35] These relations derive from the unit circle definitions of sine and cosine. In polar form, geometric objects often simplify: a circle centered at the origin has equation r = a for radius a > 0, while the Archimedean spiral, which winds outward at a constant rate, follows r = a \theta for \theta \geq 0 and scaling constant a > 0.^[37]^[38] Other coordinate systems build on or extend polar coordinates for specific purposes. Cylindrical coordinates (r, \theta, z) generalize polar to three dimensions, but restricted to the plane (where z = 0), they coincide exactly with polar coordinates, emphasizing the radial and angular components without height.^[39] Homogeneous coordinates, used in projective geometry, represent points in the Euclidean plane as triples (x : y : w) with w \neq 0, where finite points correspond to (x/w, y/w) in affine (Euclidean) space; scaling the triple leaves the point unchanged, allowing compact handling of lines, intersections, and points at infinity.^[40] Polar coordinates prove advantageous in applications exploiting symmetry, such as describing rotations: a counterclockwise rotation by angle \phi transforms (r, \theta) simply to (r, \theta + \phi), preserving radial distance.^[41] This facilitates analysis in contexts like orbital mechanics or periodic phenomena. Additionally, polar forms simplify certain calculus operations, such as integrating over circular regions, by aligning with the natural geometry and reducing computational complexity through symmetry.^[41]

Geometric structures

Lines, angles, and distances

In the Euclidean plane, the distance between two points P_1(x_1, y_1) and P_2(x_2, y_2) is given by the [formula d](/page/Formula_D) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}, which follows directly from the Pythagorean theorem applied to the right triangle formed by the horizontal and vertical segments connecting the points.^[42] This metric defines the straight-line distance and underpins the plane's uniformity. A line in the Euclidean plane can be characterized by its slope m = \frac{y_2 - y_1}{x_2 - x_1}, where the points (x_1, y_1) and (x_2, y_2) lie on the line, provided the line is not vertical.^[43] Two lines with slopes m_1 and m_2 are perpendicular if m_1 \cdot m_2 = -1, a condition arising from the fact that the angles they form with the horizontal differ by 90 degrees. The acute angle \phi between two non-perpendicular lines satisfies \tan \phi = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right|, derived from the tangent subtraction formula using the slopes as tangents of inclination angles. Angles in the Euclidean plane are measured in degrees or radians, with one full rotation corresponding to 360 degrees or $2\pi radians. In any triangle, the sum of the interior angles equals \pi radians (or 180 degrees), a consequence of the exterior angle theorem, which states that an exterior angle equals the sum of the two remote interior angles.^[44] Triangle congruence criteria include side-angle-side (SAS), where two sides and the included angle determine congruence, and angle-side-angle (ASA), where two angles and the included side suffice. The midpoint of a line segment joining points (x_1, y_1) and (x_2, y_2) is \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right), which bisects the segment into two equal parts. The midpoint theorem states that the segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length, provable using properties of similar triangles or vector averages in the plane.

Circles and conics

In the Euclidean plane, a circle is defined as the locus of all points equidistant from a fixed point called the center, with this distance denoted as the radius r.^[45] In Cartesian coordinates, with the center at (h, k), the equation of the circle is given by

(x - h)^2 + (y - k)^2 = r^2.

This equation represents the set of points (x, y) satisfying the constant distance r from the center, derived from the distance formula in the plane.^[45] The circumference of the circle, or the length of its boundary, is $2\pi r, while its area is \pi r^2; these measures follow from integrating the arc length and applying the fundamental theorem of calculus, though they can also be established geometrically using limits of polygonal approximations in Euclidean constructions.^[45] Key properties of the circle include the perpendicularity of the tangent line to the radius at the point of tangency, ensuring that any line touching the circle at exactly one point forms a right angle with the line from the center to that point. Additionally, an inscribed angle subtending a given arc measures half the central angle subtending the same arc, a relation that holds because angles in the same segment are equal and the angle in a semicircle is a right angle.^[46] The power of a point theorem further characterizes intersections: for a point P outside the circle, if two secants from P intersect the circle at points A, B and C, D respectively, then PA \cdot PB = PC \cdot PD, reflecting an invariant product of segment lengths.^[47] Conic sections encompass a family of curves in the Euclidean plane, including ellipses, parabolas, and hyperbolas, obtained as intersections of a plane with a right circular cone; the circle is a special limiting case.^[48] Their general equation in Cartesian coordinates is

ax^2 + bxy + cy^2 + dx + ey + f = 0,

where the coefficients determine the type and orientation of the curve, with the discriminant b^2 - 4ac distinguishing ellipses (<0), parabolas (=0), and hyperbolas (>0) for non-degenerate cases.^[48] Conics are classified by their eccentricity e, a dimensionless parameter measuring deviation from circularity: e = 0 for a circle, $0 < e < 1 for an ellipse, e = 1 for a parabola, and e > 1 for a hyperbola.^[49] A unifying focus-directrix definition characterizes each conic as the locus of points P such that the ratio of the distance from P to a fixed point (focus F) to the distance from P to a fixed line (directrix l) equals the constant eccentricity e; this property, originally explored by Apollonius of Perga, provides an intrinsic geometric description independent of the cone intersection.^[50] For the parabola (e=1), this yields equal distances to focus and directrix, leading to a notable reflective property: incoming rays parallel to the axis of symmetry reflect off the curve and pass through the focus, as the tangent at any point bisects the angle between the axis-parallel ray and the line to the focus. Similar reflective behaviors hold for ellipses and hyperbolas, where rays from one focus reflect toward the other focus.^[51]^[52]

Polygons and polytopes

A polygon in the Euclidean plane is a closed figure bounded by a finite chain of line segments connected end-to-end, forming an n-sided shape for n ≥ 3. A regular polygon has all sides of equal length and all interior angles equal, with each interior angle measuring \frac{(n-2)\pi}{n} radians. Any simple polygon with n vertices admits a triangulation, partitioning its interior into n-2 non-overlapping triangles using n-3 non-intersecting diagonals, and for the resulting planar graph (including the outer face), Euler's formula holds: V - E + F = 2, where V is the number of vertices, E the number of edges, and F the number of faces.^[53]^[54]^[55] Convex polygons are those where all interior angles are less than \pi radians and any line segment connecting two points inside the polygon lies entirely within it. The intersection of any collection of convex sets, including polygons, is itself convex, preserving this property under intersection operations. In two dimensions, every boundary point of a convex polygon has a supporting line (the 2D analog of a supporting hyperplane) that touches the polygon at that point and leaves the entire polygon on one side of the line, as guaranteed by the supporting hyperplane theorem for convex sets.^[56]^[57] In the context of polytopes, a 2-polytope is precisely a convex polygon in the Euclidean plane, defined as the convex hull of finitely many points in \mathbb{R}^2 or the bounded intersection of finitely many half-planes.^[58] Non-convex polygons include those with interior angles exceeding \pi radians or self-intersecting edges, such as star polygons like the pentagram, a regular five-pointed figure denoted by the Schläfli symbol {5/2}, formed by connecting every second vertex of a regular pentagon. These self-intersections create intersecting line segments within the boundary. Regarding tilings, only three types of regular convex polygons—equilateral triangles, squares, and regular hexagons—can tile the Euclidean plane without gaps or overlaps in a monohedral fashion, due to the requirement that interior angles sum to $2\pi radians at each vertex.^[59]^[60]

Vector and algebraic aspects

Vectors and operations

In the Euclidean plane, vectors can be conceptualized as directed line segments, or arrows, originating from a point, or more formally as ordered pairs (a, b) where a, b \in \mathbb{R}, representing elements of the vector space \mathbb{R}^2.^[61] Position vectors specifically denote those arrows starting from the origin, providing a coordinate-based representation of displacement in the plane.^[62] Vector addition follows the parallelogram law: for vectors \mathbf{u} = (u_1, u_2) and \mathbf{v} = (v_1, v_2), the sum is \mathbf{u} + \mathbf{v} = (u_1 + v_1, u_2 + v_2), geometrically obtained as the diagonal of the parallelogram formed by \mathbf{u} and \mathbf{v} sharing a common initial point.^[63] Scalar multiplication by a real number k scales the vector: k\mathbf{u} = (k u_1, k u_2), which stretches or compresses the arrow while preserving direction if k > 0, or reversing it if k < 0.^[62] These operations enable linear combinations, such as \alpha \mathbf{u} + \beta \mathbf{v} for scalars \alpha, \beta \in \mathbb{R}, and the span of a set of vectors is the collection of all such combinations, forming a subspace of \mathbb{R}^2./04%3A_R/4.10%3A_Spanning_Linear_Independence_and_Basis_in_R) The standard basis vectors \mathbf{i} = (1, [0](/page/0)) and \mathbf{j} = ([0](/page/0), 1) provide a fundamental framework for \mathbb{R}^2, as any vector (a, [b](/page/List_of_punk_rap_artists)) can be uniquely expressed as a\mathbf{i} + [b](/page/List_of_punk_rap_artists)\mathbf{j}.^[64] These vectors are linearly independent, meaning the equation \alpha \mathbf{i} + \beta \mathbf{j} = ([0](/page/0), [0](/page/0)) holds only for \alpha = \beta = [0](/page/0), ensuring they form a basis that spans the entire plane without redundancy./04%3A_R/4.10%3A_Spanning_Linear_Independence_and_Basis_in_R) Affine combinations extend linear combinations by restricting the scalars \lambda_1, \lambda_2, \dots, \lambda_n such that \sum \lambda_i = 1, allowing points in the plane to be expressed as weighted averages of other points without shifting the origin.^[65] In particular, barycentric coordinates arise from affine combinations of three non-collinear points forming a triangle, where the coefficients represent areal weights summing to 1, facilitating the location of any interior point as a convex combination.^[66]

Dot product, norms, and angles

In the Euclidean plane, the dot product provides a fundamental way to measure the interaction between two vectors, imposing a metric structure that allows for the computation of lengths and angles. For vectors \mathbf{u} = (u_1, u_2) and \mathbf{v} = (v_1, v_2), the dot product is defined algebraically as

\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2.

This operation, introduced in the context of modern vector analysis, is bilinear, meaning it is linear in each argument separately: (\alpha \mathbf{u} + \beta \mathbf{w}) \cdot \mathbf{v} = \alpha (\mathbf{u} \cdot \mathbf{v}) + \beta (\mathbf{w} \cdot \mathbf{v}) and similarly for the second argument, and symmetric such that \mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u}. Geometrically, the dot product equals the product of the vectors' magnitudes times the cosine of the angle \theta between them: \mathbf{u} \cdot \mathbf{v} = \|\mathbf{u}\| \|\mathbf{v}\| \cos \theta. These properties make the dot product the standard inner product on the Euclidean plane, enabling projections and other geometric interpretations. The norm, or length, of a vector \mathbf{u} is derived from the dot product as

\|\mathbf{u}\| = \sqrt{\mathbf{u} \cdot \mathbf{u}}.

This Euclidean norm satisfies the properties of a vector space norm, including positivity (\|\mathbf{u}\| \geq 0 with equality only for the zero vector), scalability (\|\alpha \mathbf{u}\| = |\alpha| \|\mathbf{u}\|), and the triangle inequality. A unit vector, or vector of length 1, has \|\mathbf{u}\| = 1; any nonzero vector can be normalized to a unit vector by \hat{\mathbf{u}} = \mathbf{u} / \|\mathbf{u}\|, which preserves direction while standardizing magnitude for applications like basis constructions. The angle \theta between two nonzero vectors \mathbf{u} and \mathbf{v} is determined via the dot product by

\cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \|\mathbf{v}\|},

where \theta ranges from 0 to \pi radians. Vectors are orthogonal if \mathbf{u} \cdot \mathbf{v} = 0, corresponding to \theta = \pi/2 and \cos \theta = 0; this condition defines perpendicularity in the Euclidean sense, crucial for coordinate systems and decompositions. A key inequality arising from the dot product is the Cauchy-Schwarz inequality, which states that

|\mathbf{u} \cdot \mathbf{v}| \leq \|\mathbf{u}\| \|\mathbf{v}\|,

with equality if and only if \mathbf{u} and \mathbf{v} are linearly dependent (parallel or anti-parallel). This bound, first proved for sums in the context of real analysis, limits how aligned vectors can be and underpins many theorems in geometry and analysis.

Analytic and calculus applications

Functions and gradients

In the Euclidean plane, a scalar function f: \mathbb{R}^2 \to \mathbb{R} assigns a real value to each point (x, y), enabling the representation of quantities such as height, temperature, or potential energy across the plane.^[67] The level sets of f are the curves where f(x, y) = c for a constant c, forming contours that illustrate regions of equal value and revealing the function's topological structure, such as hills and valleys in a topographic map.^[68] These contours are particularly useful for visualizing how the function varies spatially, with denser spacing indicating steeper changes. Partial derivatives measure the rate of change of f along the coordinate axes: the partial derivative with respect to x, denoted \partial f / \partial x, is the limit \lim_{h \to 0} [f(x+h, y) - f(x, y)] / h, treating y as constant, while \partial f / \partial y is defined analogously by varying y.^[67] These quantities form the components of the gradient vector \nabla f = (\partial f / \partial x, \partial f / \partial y), which points in the direction of the function's steepest increase at a point and whose magnitude |\nabla f| quantifies that rate.^[69] Notably, \nabla f is perpendicular to the level sets of f, as the dot product of \nabla f with any tangent vector to a level curve is zero.^[68] The directional derivative of f in the direction of a unit vector \mathbf{u} = (u_1, u_2) is given by D_{\mathbf{u}} f = \nabla f \cdot \mathbf{u}, representing the instantaneous rate of change along that direction; this leverages the dot product to project the gradient onto \mathbf{u}.^[70] The maximum directional derivative occurs when \mathbf{u} aligns with \nabla f, corresponding to the steepest ascent, with value |\nabla f|, while the minimum (steepest descent) is -|\nabla f| in the opposite direction. A classic example is the distance function to a fixed point (a, b), f(x, y) = \sqrt{(x - a)^2 + (y - b)^2}, whose gradient is \nabla f = \frac{(x - a, y - b)}{f(x, y)}, the unit vector from (a, b) to (x, y), illustrating how the gradient normalizes direction for radial increase.^[71] Another key case involves harmonic functions, which satisfy Laplace's equation \Delta f = \partial^2 f / \partial x^2 + \partial^2 f / \partial y^2 = 0, where the Laplacian \Delta f is the divergence of \nabla f; these functions, such as the real or imaginary parts of holomorphic functions, model steady-state phenomena like electrostatic potentials in the plane.^[72]

Integrals and theorems

In the Euclidean plane, line integrals provide a means to compute quantities along curves, essential for applications in physics such as work done by a force field. For a scalar function f(x, y) and a smooth curve C parametrized by \mathbf{r}(t) = (x(t), y(t)) for t \in [a, b], the line integral \int_C f \, ds is defined as \int_a^b f(x(t), y(t)) \|\mathbf{r}'(t)\| \, dt, where \|\mathbf{r}'(t)\| is the speed of the parametrization, representing the integral of f with respect to arc length.^[73] For a vector field \mathbf{F}(x, y) = (P(x, y), Q(x, y)), the line integral \int_C \mathbf{F} \cdot d\mathbf{r} is given by \int_a^b \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t) \, dt = \int_a^b [P(x(t), y(t)) x'(t) + Q(x(t), y(t)) y'(t)] \, dt, which measures the circulation or flux along the path.^[74] Double integrals extend this to areas, allowing computation of volumes under surfaces or masses of regions in the plane. Over a bounded region D in \mathbb{R}^2, the double integral \iint_D f(x, y) \, dA represents the signed volume beneath the graph of f, evaluated via iterated integrals such as \int_a^b \int_{g(x)}^{h(x)} f(x, y) \, dy \, dx for type I regions.^[75] To simplify evaluation over irregular D, a change of variables x = x(u, v), y = y(u, v) transforms the integral to \iint_{D'} f(x(u, v), y(u, v)) |J| \, du \, dv, where D' is the image region and J = \det \begin{pmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{pmatrix} is the absolute value of the Jacobian determinant, accounting for the scaling of area elements under the transformation.^[76] The fundamental theorem of line integrals connects path integrals to scalar potentials for conservative fields. If \mathbf{F} = \nabla f is the gradient of a scalar potential f (i.e., conservative, with \frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}), then for any piecewise smooth curve C from point P to Q, \int_C \mathbf{F} \cdot d\mathbf{r} = f(Q) - f(P), independent of the path taken, provided the domain is simply connected.^[77] This theorem generalizes the one-dimensional fundamental theorem of calculus to the plane, enabling efficient computation without explicit parametrization.^[78] Green's theorem relates line integrals around closed curves to double integrals over enclosed regions, a cornerstone for planar vector analysis. For a positively oriented, piecewise smooth, simple closed curve C bounding region D, with continuously differentiable P and Q,

\oint_C (P \, dx + Q \, dy) = \iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA,

this equates circulation to the integral of the curl, facilitating conversions between boundary and interior computations.^[79] Originally derived in George Green's 1828 essay on electricity and magnetism, the theorem applies to diverse fields like fluid dynamics.^[80] A key application computes areas: for a closed curve C, the area of D is \frac{1}{2} \oint_C (x \, dy - y \, dx) = \iint_D dA, using P = -y, Q = x.^[81]

Topological and graph-theoretic views

Topological properties

The Euclidean plane, denoted \mathbb{R}^2, is a topological manifold of dimension 2. It is Hausdorff, meaning that for any two distinct points, there exist disjoint open neighborhoods separating them.^[82] Additionally, \mathbb{R}^2 is second-countable, possessing a countable basis for its topology, which ensures that the space is separable and allows for manageable coverings in proofs of topological properties.^[82] As a locally Euclidean space, every point in \mathbb{R}^2 has an open neighborhood homeomorphic to an open subset of \mathbb{R}^2 itself, typically via coordinate charts.^[82] The Euclidean plane is path-connected, as any two points can be joined by a continuous path, such as a straight line segment. It is also simply connected, meaning it is path-connected and has a trivial fundamental group, \pi_1(\mathbb{R}^2) = \{e\}, where loops based at any point can be continuously contracted to a point within the space.^[83] This triviality of the fundamental group distinguishes \mathbb{R}^2 from spaces with "holes," such as the punctured plane \mathbb{R}^2 \setminus \{0\}, whose fundamental group is isomorphic to \mathbb{Z}, generated by loops winding around the origin.^[83] A key consequence is the Jordan curve theorem, which states that every simple closed curve in \mathbb{R}^2 separates the plane into two distinct connected components: a bounded interior region and an unbounded exterior region, with the curve itself forming the boundary between them.^[84] Subsets of the Euclidean plane exhibit compactness properties characterized by the Heine-Borel theorem: a subset of \mathbb{R}^2 is compact if and only if it is closed and bounded.^[85] This equivalence holds because \mathbb{R}^2 is a metric space where closed sets contain all limit points and bounded sets fit within a ball of finite radius, ensuring that every open cover of a closed and bounded set has a finite subcover.^[85] The topology of the Euclidean plane is preserved under homeomorphisms, which are continuous bijections with continuous inverses. Such maps maintain the plane's key invariants, including its simply connectedness and lack of holes, as evidenced by the non-homeomorphism between \mathbb{R}^2 and the punctured plane, due to differing fundamental groups.^[86] This invariance underscores the plane's topological uniqueness among 2-dimensional manifolds without punctures or boundaries.^[86]

Planar graphs and embeddings

A planar graph is a graph that can be embedded in the Euclidean plane such that no two edges cross except possibly at vertices.^[87] This embedding divides the plane into faces, including an unbounded outer face.^[88] For a connected planar graph with V vertices, E edges, and F faces (counting the outer face), Euler's formula states that V - E + F = 2.^[55] This relation holds for any maximal planar embedding, where adding any edge would require a crossing or violate planarity.^[89] Embeddings of planar graphs can use curved edges or straight-line segments. Fáry's theorem asserts that every simple planar graph admits a straight-line embedding in the plane without crossings, preserving the combinatorial structure of a given planar embedding.^[90] Such straight-line drawings position vertices at distinct points in the plane and connect them with line segments that do not intersect except at endpoints.^[91] A subclass of planar graphs is the outerplanar graphs, which can be embedded such that all vertices lie on the boundary of the outer face, with internal edges not crossing.^[92] Outerplanar graphs satisfy a stricter bound from Euler's formula: for a connected outerplanar graph with V \geq 2, E \leq 2V - 3.^[93] Kuratowski's theorem provides a characterization of planarity: a finite graph is planar if and only if it contains no subdivision of the complete graph K_5 (five vertices all connected) or the complete bipartite graph K_{3,3} (two sets of three vertices, each connected to all in the other set).^[94] A subdivision replaces edges with paths, preserving the graph's topological structure.^[95] The graph K_{3,3} exemplifies non-planarity and arises in the utility graph problem, where three houses and three utilities cannot be connected pairwise without crossings in the plane.^[96] Similarly, K_5 cannot be embedded without crossings.^[97] Planar graphs have significant applications, notably in map coloring. The four color theorem states that any planar graph is 4-colorable, meaning its vertices can be colored with at most four colors such that no adjacent vertices share the same color; this was proved in 1976 using a computer-assisted discharge method on unavoidable configurations.^[98]^[99] This theorem directly implies that four colors suffice to color any map in the plane so that adjacent regions differ in color.^[95]

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Sur le problème des courbes gauches en Topologie. Casimir Kuratowski · Fundamenta Mathematicae (1930) ... pdf.png Full (PDF). How to cite. top. MLA; BibTeX; RIS.
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Every planar map is four colorable - Project Euclid
Every planar map is four colorable. K. Appel, W. Haken. DOWNLOAD PDF + SAVE TO MY LIBRARY. Bull. Amer. Math. Soc. 82(5): 711-712 (September 1976).
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[PDF] an elementary proof that the utilities puzzle is impossible - Lomont.org
It is equivalent to the graph K3,3 being nonplanar. Here I present a simple proof that can convince even most non- math majors. 1. The Utilities Puzzle.
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[PDF] Kuratowski's Theorem - UChicago Math
Aug 28, 2017 · Abstract. This paper introduces basic concepts and theorems in graph the- ory, with a focus on planar graphs. On the foundation of the basics, ...
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EVERY PLANAR MAP IS FOUR COLORABLE PART I - Project Euclid
Appel wishes tothank his teacher, Roger Lyndon, for teaching him how to think about mathematics. 429. Page 2. 430. K. APPEL AND W. HAKEN.
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EVERY PLANAR MAP IS FOUR COLORABLE - Project Euclid
EVERY PLANAR MAP IS FOUR COLORABLE. PART I1: REDUCIBILITY. BY. K. APPEL, W. HAKEN, AND J. KOCH. 1. Introduction. In Part I of this paper, a discharging ...