Line

The line is a basic 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 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.

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 segments.

Common mathematical representations of a line include:

In 2 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)

In n dimensions in Cartesian coordinates:

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