Line

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The line is a basic geometric shape. It is a straight, breadthless 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 points determine a (unique) line which goes through them. A line can be broken down into finite line segments.

Common mathematical representations of a line include:

In two dimensions in the Cartesian plane:

  • 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)
  • Point-Slope Form: (y - y0) = m(x - x0) (where m is the slope and (x0, y0) is a point on the line)
  • Two Points Form: (y - y_1) = \frac{y_2-y_1}{x_2-x_1}(x - x_1) (where (x1,y1) and (x2,y2) are two different points on the line)

In n dimensions in Cartesian coordinates:

  • Parametrized Vector Form: r(t) = <x0, y0,...> + 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 paradoxes, 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.

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