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Undo revision 1291644 by ATexan (talk)
The '''line''' is a basic [[geometry|geometric]] [[shape]]. It is such an intuitive concept that most mathematicians think it escapes a formal, general definition. Other such intuitive concepts are the [[point]], the [[plane]], the [[set]], [[symmetry]], and [[infinity]].
The '''line''' is a basic [[geometry|geometric]] [[shape]]. It is a straight, [[breadth]]less figure that extends to [[infinity]] in both directions and consists of infinitely many points strung all in a row with no spaces or breaks.
A line that can be drawn on paper is not actually a line, but a representation of a line since it does not extend infinitely in both directions. Two [[point]]s determine a (unique) line which goes through them. A line can be broken down into finite [[line segment]]s.
A line that can be drawn on paper is not actually a line, but a representation of a line. Two points determine a line. A line can be broken down into finite [[line segment]]s.
Common mathematical representations of a line include:
Common mathematical representations of a line include:
In 2 dimensions in the [[Cartesian plane]]:
In two dimensions in the [[Cartesian plane]]:
*Standard Form: ax + by + c = 0
*Standard Form: ax + by + c = 0
*Slope-Intercept Form: y = mx + b (where m is the [[slope]] of the line, b is the [[y-intercept]])
*Slope-Intercept Form: y = mx + b (where m is the [[slope]] of the line, b is the [[y-intercept]])
*Point-Slope Form: (y - y<sub>0</sub>) = m(x - x<sub>0</sub>) (where m is the slope and (x<sub>0</sub>, y<sub>0</sub>) is a point on the line)
*Point-Slope Form: (y - y<sub>0</sub>) = m(x - x<sub>0</sub>) (where m is the slope and (x<sub>0</sub>, y<sub>0</sub>) is a point on the line)
*Two Points Form: <math>(y - y_1) = \frac{y_2-y_1}{x_2-x_1}(x - x_1)</math> (where <math>(x_1,y_1)</math> and <math>(x_2,y_2)</math> are two different points on the line)
In n dimensions in [[Cartesian coordinates]]:
In n dimensions in [[Cartesian coordinates]]:
*Parametrized Vector Form: r(t) = <x<sub>0</sub>, y<sub>0</sub>,...> + t<x, y,...>
*Parametrized Vector Form: r(t) = <x<sub>0</sub>, y<sub>0</sub>,...> + t<x, y,...>
==Philosophical nature of the line==
The line is such an intuitive concept that most mathematicians think it escapes a formal, rigorous, and general definition. The definition above is unsatisfactory because adjectives such as "straight" are usually defined in terms of resembling lines. Other such intuitive concepts without rigorous definitions are the [[point]], the [[plane]], the [[set]], [[symmetry]], and [[infinity]].
Attempts at providing a rigorous, abstract definition usually face philosophical [[paradox]]es, such as [[Euclid]]'s contention that lines are objects with "breadthless length." In other words, he asserted that a line in infinitely long, but has no width. Lines can also be considered as points strung together, or as curves with infinite [[radius]] of [[curvature]], yet both these definitions have philosophical flaws as well.
[[Category:Geometry]]
[[Category:Geometry]]