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A line is a straight one-dimensional figure having no thickness and extending infinitely in both directions. A line is sometimes called a straight line or, more archaically, a right line (Casey 1893), to emphasize that it has no "wiggles" anywhere along its length. While lines are intrinsically one-dimensional objects, they may be embedded in higher dimensional spaces.

Harary (1994) called an edge of a graph a "line."

A line is uniquely determined by two distinct points, and the line passing through points and is denoted . Similarly, the length of the finite line segment terminating at these points may be denoted . A line may also be denoted with a single lower-case letter (Jurgensen et al. 1963, p. 22).

Euclid defined a line as a "breadthless length," and a straight line as a line that "lies evenly with the points on itself" (Kline 1956, Dunham 1990).

Consider first lines in a two-dimensional plane. Two lines lying in the same plane that do not intersect one another are said to be parallel lines. Two lines that do not lie in any common plane are said to be skew lines.

The line with nonzero x-intercept and nonzero y-intercept is given by the intercept form

It is not uncommon for lines in intercept form to be rewritten in what's known as standard form:

The line through with slope is given by the point-slope form

The line with y-intercept and slope is given by the slope-intercept form

The line through the distinct points and is given by the two-point form

This form requires ; if , the line is the vertical line . Parametric equations, with , are given by

For and not both zero, other forms are

A line in two dimensions can also be represented as a vector. The vector along the line

is given by

where . Similarly, vectors of the form

are perpendicular to the line.

A line in the plane that does not pass through the origin is uniquely specified by the vector from the origin to the nearest point on the line. Its points satisfy

For with , this vector is

Three points lie on a line if

The angle between lines

has directed angle modulo satisfying

when . The smaller angle between the lines is obtained by taking the corresponding absolute value and choosing .

Let and be the side lengths and opposite angles of a fixed reference triangle. The line joining distinct points with trilinear coordinates and is the set of points satisfying

Two nonparallel lines, one through in the direction and the other through in direction , intersect iff

The line through a point parallel to

is

The lines

are parallel if

for these reference-triangle side lengths, and perpendicular if

for the reference-triangle angles (Sommerville 1961; Kimberling 1998, p. 29).

The line through a point perpendicular to (23) is given by

In three-dimensional space, the line passing through the point and parallel to the nonzero vector has parametric equations

written concisely as

Similarly, the line in three dimensions passing through the distinct points and has the parametric equations summarized by the vector equation

where this parameterization corresponds to and .

See also

2-Dimensional Point-Line Distance, 3-Dimensional Point-Line Distance, Asymptote, Branch Cut, Brocard Line, Cayley Lines, Collinear, Concur, Critical Line, Desargues' Theorem, Erdős-Anning Theorem, Euler Line, Flow Line, Gergonne Line, Imaginary Line, Incidence Structure, Isogonal Line, Isotropic Line, Lemoine Axis, Line-Line Intersection, Line-Plane Intersection, Line Segment, Line Segment Range, Ordinary Line, Pascal Lines, Pedal Line, Pencil, Philo Line, Plane, Plücker Lines, Point, Polar, Radical Line, Ray, Real Line, Secant Line, Simson Line, Skew Lines, Soddy Line, Solomon's Seal Lines, Standard Form, Steiner Set, Steiner's Theorem, Sylvester's Line Problem, Symmedian, Tangent Line, Transversal Line, Trilinear Line, World Line Explore this topic in the MathWorld classroom

Portions of this entry contributed by Christopher Stover

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References

Casey, J. "The Right Line." Ch. 2 in A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl. Dublin, Ireland: Hodges, Figgis, & Co., pp. 30-95, 1893.Dunham, W. Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, p. 32, 1990.Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994.Jurgensen, R. C.; Donnelly, A. J.; and Dolciani, M. P. Modern Geometry: Structure and Method. Boston, MA: Houghton-Mifflin, p. 22, 1963.Kern, W. F. and Bland, J. R. "Lines and Planes in Space." §4 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 9-12, 1948.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Kline, M. "The Straight Line." Sci. Amer. 194, 104-114, 1956.MacTutor History of Mathematics Archive. "Straight Line." https://mathshistory.st-andrews.ac.uk/Curves/Straight/.Sommerville, D. M. Y. Analytical Conics, 3rd ed. London, England: G. Bell and Sons, p. 186, 1961.Spanier, J. and Oldham, K. B. "The Linear Function and Its Reciprocal." Ch. 7 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 53-62, 1987.

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Line

Cite this as:

Weisstein, Eric W., with contributions by Christopher Stover. "Line." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Line.html

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