Line

A line can be described as an ideal zero-width, infinitely long, perfectly straight curve (the term curve in mathematics includes "straight curves") containing an infinite number of points. In Euclidean geometry, exactly one line can be found that passes through any two points. more...

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The line provides the shortest connection between the points.

In two dimensions, two different lines can either be parallel, meaning they never meet, or may intersect at one and only one point. In three or more dimensions, lines may also be skew, meaning they don't meet, but also don't define a plane. Two distinct planes intersect in at most one line. Three or more points that lie on the same line are called collinear.

Examples

Lines in a Cartesian plane can be described algebraically by linear equations and linear functions. In two dimensions, the characteristic equation is often given by the slope-intercept form:

where:

m is the slope of the line.

b is the y-intercept of the line.

x is the independent variable of the function y.

In three dimensions, a line is described by parametric equations:

where:

x, y, and z are all functions of the independent variable t.

x₀, y₀, and z₀ are the initial values of each respective variable.

a, b, and c are related to the slope of the line, such that the vector (a, b, c) is a parallel to the line.

Formal definitions

This intuitive concept of a line can be formalized in various ways. If geometry is developed axiomatically (as in Euclid's Elements and later in David Hilbert's Foundations of Geometry), then lines are not defined at all, but characterized axiomatically by their properties. While Euclid did define a line as "length without breadth", he did not use this rather obscure definition in his later development.

In Euclidean space Rⁿ (and analogously in all other vector spaces), we define a line L as a subset of the form

where a and b are given vectors in Rⁿ with b non-zero. The vector b describes the direction of the line, and a is a point on the line. Different choices of a and b can yield the same line.

Read more at Wikipedia.org

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