Line (geometry)

linestraight linelinesraycollinearrayshalf-linestraightlinearstraight lines
The notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature) with negligible width and depth.wikipedia
516 Related Articles

Curvature

curvednegative curvatureextrinsic curvature
The notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature) with negligible width and depth.
Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.

Geodesic

geodesicsgeodesic flowgeodesic equation
Thus in differential geometry a line may be interpreted as a geodesic (shortest path between points), while in some projective geometries a line is a 2-dimensional vector space (all linear combinations of two independent vectors).
It is a generalization of the notion of a "straight line" to a more general setting.

Euclidean geometry

plane geometryEuclideanEuclidean plane geometry
Euclid described a line as "breadthless length" which "lies equally with respect to the points on itself"; he introduced several postulates as basic unprovable properties from which he constructed all of geometry, which is now called Euclidean geometry to avoid confusion with other geometries which have been introduced since the end of the 19th century (such as non-Euclidean, projective and affine geometry). In Euclidean geometry, the Euclidean distance d(a,b) between two points a and b may be used to express the collinearity between three points by:
Modern school textbooks often define separate figures called lines (infinite), rays (semi-infinite), and line segments (of finite length).

Dimension

dimensionsdimensionalone-dimensional
In two dimensions, i.e., the Euclidean plane, two lines which do not intersect are called parallel.
Thus a line has a dimension of one because only one coordinate is needed to specify a point on itfor example, the point at 5 on a number line.

Axiom

axiomspostulateaxiomatic
Euclid described a line as "breadthless length" which "lies equally with respect to the points on itself"; he introduced several postulates as basic unprovable properties from which he constructed all of geometry, which is now called Euclidean geometry to avoid confusion with other geometries which have been introduced since the end of the 19th century (such as non-Euclidean, projective and affine geometry). In an axiomatic formulation of Euclidean geometry, such as that of Hilbert (Euclid's original axioms contained various flaws which have been corrected by modern mathematicians), a line is stated to have certain properties which relate it to other lines and points.

Projective geometry

projectiveprojective geometriesProjection
Thus in differential geometry a line may be interpreted as a geodesic (shortest path between points), while in some projective geometries a line is a 2-dimensional vector space (all linear combinations of two independent vectors). Euclid described a line as "breadthless length" which "lies equally with respect to the points on itself"; he introduced several postulates as basic unprovable properties from which he constructed all of geometry, which is now called Euclidean geometry to avoid confusion with other geometries which have been introduced since the end of the 19th century (such as non-Euclidean, projective and affine geometry).
In two dimensions it begins with the study of configurations of points and lines.

Parallel (geometry)

parallelparallel linesparallelism
In two dimensions, i.e., the Euclidean plane, two lines which do not intersect are called parallel. In the context of determining parallelism in Euclidean geometry, a transversal is a line that intersects two other lines that may or not be parallel to each other.
In geometry, parallel lines are lines in a plane which do not meet; that is, two lines in a plane that do not intersect or touch each other at any point are said to be parallel.

Plane (geometry)

planeplanarplanes
In two dimensions, i.e., the Euclidean plane, two lines which do not intersect are called parallel. In higher dimensions, two lines that do not intersect are parallel if they are contained in a plane, or skew if they are not. Three points usually determine a plane, but in the case of three collinear points this does not happen.
A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space.

Analytic geometry

analytical geometryCartesian geometrycoordinate geometry
For instance, in analytic geometry, a line in the plane is often defined as the set of points whose coordinates satisfy a given linear equation, but in a more abstract setting, such as incidence geometry, a line may be an independent object, distinct from the set of points which lie on it.
Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and squares, often in two and sometimes in three dimensions.

Skew lines

skewskew angleskew flats
In higher dimensions, two lines that do not intersect are parallel if they are contained in a plane, or skew if they are not.
In three-dimensional geometry, skew lines are two lines that do not intersect and are not parallel.

Arrangement of lines

arrangementarrangements of linesline arrangement
Any collection of finitely many lines partitions the plane into convex polygons (possibly unbounded); this partition is known as an arrangement of lines.
In geometry an arrangement of lines is the partition of the plane formed by a collection of lines.

Hilbert's axioms

Grundlagen der GeometrieHilbertaxiomatization of geometry
In an axiomatic formulation of Euclidean geometry, such as that of Hilbert (Euclid's original axioms contained various flaws which have been corrected by modern mathematicians), a line is stated to have certain properties which relate it to other lines and points.

Slope

gradientslopesgradients
In mathematics, the slope or gradient of a line is a number that describes both the direction and the steepness of the line.

Normal (geometry)

normalnormal vectorsurface normal
The normal form (also called the Hesse normal form, after the German mathematician Ludwig Otto Hesse), is based on the normal segment for a given line, which is defined to be the line segment drawn from the origin perpendicular to the line.
In geometry, a normal is an object such as a line or vector that is perpendicular to a given object.

Point (geometry)

pointpointslocation
In an axiomatic formulation of Euclidean geometry, such as that of Hilbert (Euclid's original axioms contained various flaws which have been corrected by modern mathematicians), a line is stated to have certain properties which relate it to other lines and points.
This is usually represented by a set of points; As an example, a line is an infinite set of points of the form, where

Euclid's Elements

ElementsEuclid's ''ElementsEuclid
The "definition" of line in Euclid's Elements falls into this category.
In Book I, Euclid lists five postulates, the fifth of which stipulates ''If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.''

Polar coordinate system

polar coordinatespolarpolar coordinate
In polar coordinates on the Euclidean plane the slope-intercept form of the equation of a line is expressed as:
The reference point (analogous to the origin of a Cartesian coordinate system) is called the pole, and the ray from the pole in the reference direction is the polar axis.

Three-dimensional space

three-dimensional3Dthree dimensions
In three-dimensional space, a first degree equation in the variables x, y, and z defines a plane, so two such equations, provided the planes they give rise to are not parallel, define a line which is the intersection of the planes.
Two distinct points always determine a (straight) line.

Euclidean space

EuclideanspaceEuclidean vector space
In more general Euclidean space, R n (and analogously in every other affine space), the line L passing through two different points a and b (considered as vectors) is the subset
Their great innovation was to prove all properties of the space as theorems by starting from a few fundamental properties, called postulates, which either were considered as evident (for example, there is exactly one straight line passing through two points), or seemed impossible to prove (parallel postulate).

Euclidean distance

Euclidean metricEuclideandistance
In Euclidean geometry, the Euclidean distance d(a,b) between two points a and b may be used to express the collinearity between three points by:
In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" straight-line distance between two points in Euclidean space.

General position

general linear positiongenericallyin general position
Three points usually determine a plane, but in the case of three collinear points this does not happen.
Similarly, three generic points in the plane are not collinear; if three points are collinear (even stronger, if two coincide), this is a degenerate case.

Parabola

parabolicparabolic curveparabolic arc
One description of a parabola involves a point (the focus) and a line (the directrix).

Transversal (geometry)

transversalcorresponding anglestransversals
In the context of determining parallelism in Euclidean geometry, a transversal is a line that intersects two other lines that may or not be parallel to each other.
In geometry, a transversal is a line that passes through two lines in the same plane at two distinct points.

Linear equation

linearlinear equationsslope-intercept form
For instance, in analytic geometry, a line in the plane is often defined as the set of points whose coordinates satisfy a given linear equation, but in a more abstract setting, such as incidence geometry, a line may be an independent object, distinct from the set of points which lie on it. In three-dimensional space, a first degree equation in the variables x, y, and z defines a plane, so two such equations, provided the planes they give rise to are not parallel, define a line which is the intersection of the planes.
The solutions of a linear equation form a line in the Euclidean plane, and, conversely, every line can be viewed as the set of all solutions of a linear equation in two variables.

Perpendicular

perpendicularlyPerpendicularitynormal
Perpendicular lines are lines that intersect at right angles.
In elementary geometry, the property of being perpendicular (perpendicularity) is the relationship between two lines which meet at a right angle (90 degrees).