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Line 1:
Line 1: − A line was described by [[Euclid]] as having "breadthless length." This means that a line in infinitely long, but has no width. Lines can be considered as curves with infinite radius of curvature. 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.+ + + + + − + − + − +
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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]].
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.
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 2 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)
In n dimensions:
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,...>
[[Category:Geometry]]
[[Category:Geometry]]