Line (geometry) | EPFL Graph Search

Are you an EPFL student looking for a semester project?

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist embedded in two, three, or higher dimensional spaces. The word line may also refer to a line segment in everyday life that has two points to denote its ends (endpoints). A line can be referred to by two points that lie on it (e.g. ) or by a single letter (e.g. ). Euclid described a line as a "breadthless length" that "lies evenly with respect to the points on itself"; he introduced several postulates as basic unprovable properties from which he constructed all of geometry. 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. When geometry was first formalised by Euclid in the Elements, he defined a general line (now called a curve) to be "breadthless length" with a straight line being a line "which lies evenly with the points on itself". These definitions serve little purpose since they use terms that are not themselves defined. In fact, Euclid himself did not use these definitions in this work and probably included them just to make it clear to the reader what was being discussed. In modern geometry, a line is simply taken as an undefined object with properties given by axioms, but it is sometimes defined as a set of points obeying a linear relationship when some other fundamental concept is left undefined. In an axiomatic formulation of Euclidean geometry, such as that of Hilbert (Euclid's original axioms contained various flaws that have been corrected by modern mathematicians), a line is stated to have certain properties that relate it to other lines and points. For example, for any two distinct points, there is a unique line containing them, and any two distinct lines intersect at most at one point. In two dimensions (i.e., the Euclidean plane), two lines that do not intersect are called parallel.

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (2)

This thesis presents a methodology for the design optimization of hydraulic runner blades. The originality of the methodology comes from the geometric definition of the blade shapes, which uses parame

EPFL2006

IMHEF1994

Related concepts (185)

In analytic geometry, using the common convention that the horizontal axis represents a variable x and the vertical axis represents a variable y, a y-intercept or vertical intercept is a point where the graph of a function or relation intersects the y-axis of the coordinate system. As such, these points satisfy x = 0. If the curve in question is given as the y-coordinate of the y-intercept is found by calculating Functions which are undefined at x = 0 have no y-intercept.

In mathematics, logic, philosophy, and formal systems, a primitive notion is a concept that is not defined in terms of previously-defined concepts. It is often motivated informally, usually by an appeal to intuition and everyday experience. In an axiomatic theory, relations between primitive notions are restricted by axioms. Some authors refer to the latter as "defining" primitive notions by one or more axioms, but this can be misleading. Formal theories cannot dispense with primitive notions, under pain of infinite regress (per the regress problem).

In geometry, a secant is a line that intersects a curve at a minimum of two distinct points. The word secant comes from the Latin word secare, meaning to cut. In the case of a circle, a secant intersects the circle at exactly two points. A chord is the line segment determined by the two points, that is, the interval on the secant whose ends are the two points. A straight line can intersect a circle at zero, one, or two points. A line with intersections at two points is called a secant line, at one point a tangent line and at no points an exterior line.

Related courses (88)

MATH-467: Probabilistic methods in combinatorics

We develop a sophisticated framework for solving problems in discrete mathematics through the use of randomness (i.e., coin flipping). This includes constructing mathematical structures with unexpecte

MATH-261: Discrete optimization

This course is an introduction to linear and discrete optimization. Warning: This is a mathematics course! While much of the course will be algorithmic in nature, you will still need to be able to p

MATH-111(e): Linear Algebra

L'objectif du cours est d'introduire les notions de base de l'algèbre linéaire et ses applications.

Related lectures (804)

Related MOOCs (13)

Introduction to Geographic Information Systems (part 1)

Organisé en deux parties, ce cours présente les bases théoriques et pratiques des systèmes d’information géographique, ne nécessitant pas de connaissances préalables en informatique. En suivant cette

Geographical Information Systems 1

Algebra (part 1)

Un MOOC francophone d'algèbre linéaire accessible à tous, enseigné de manière rigoureuse et ne nécessitant aucun prérequis.

Open Office Configuration & Workspace

Explores open office setup, shared drives, digital modeling tools, and collaborative platforms in architectural projects.

Analytical Geometry: Lines Study

Covers the analytical study of lines in a plane, focusing on equations and geometric interpretations.

Geometry: Euclidean Elements & Vitruvius

Explores Euclid's first proposition, ancient symmetria, and Vitruvius' architectural figures.