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  • ️Weisstein, Eric W.
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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."

Line

A line is uniquely determined by two points, and the line passing through points A and B is denoted <->; AB. Similarly, the length of the finite line segment terminating at these points may be denoted AB^_. 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 lying in different planes that do not intersect one another are said to be skew lines.

LineIntercepts

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

x/a+y/b=1.

(1)

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

ax+by=c.

(2)

The line through (x_1,y_1) with slope m is given by the point-slope form

y-y_1=m(x-x_1).

(3)

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

y=mx+b.

(4)

The line through (x_1,y_1) and (x_2,y_2) is given by the two-point form

y-y_1=(y_2-y_1)/(x_2-x_1)(x-x_1).

(5)

A parametric form is given by

Other forms are

a(x-x_1)+b(y-y_1)=0

(8)

ax+by+c=0

(9)

|x y 1; x_1 y_1 1; x_2 y_2 1|=0.

(10)

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

ax+by=0

(11)

is given by

t[-b; a],

(12)

where t in R. Similarly, vectors of the form

t[a; b]

(13)

are perpendicular to the line.

Three points lie on a line if

|x_1 y_1 1; x_2 y_2 1; x_3 y_3 1|=0.

(14)

The angle between lines

is

tantheta=(A_1B_2-A_2B_1)/(A_1A_2+B_1B_2).

(17)

The line joining points with trilinear coordinates alpha_1:beta_1:gamma_1 and alpha_2:beta_2:gamma_2 is the set of point alpha:beta:gamma satisfying

|alpha beta gamma; alpha_1 beta_1 gamma_1; alpha_2 beta_2 gamma_2|=0

(18)

(beta_1gamma_2-gamma_1beta_2)alpha+(gamma_1alpha_2-alpha_1gamma_2)beta+(alpha_1beta_2-beta_1alpha_2)gamma=0.

(19)

The line through P_1 in the direction (a_1,b_1,c_1) and the line through P_2 in direction (a_2,b_2,c_2) intersect iff

|x_2-x_1 y_2-y_1 z_2-z_1; a_1 b_1 c_1; a_2 b_2 c_2|=0.

(20)

The line through a point alpha^':beta^':gamma^' parallel to

lalpha+mbeta+ngamma=0

(21)

is

|alpha beta gamma; alpha^' beta^' gamma^'; bn-cm cl-an am-bl|=0.

(22)

The lines

are parallel if

a(mn^'-nm^')+b(nl^'-ln^')+c(lm^'-ml^')=0

(25)

for all (a,b,c), and perpendicular if

(ll^'+mm^'+nn^')-(mn^'+m^'n)cosA-(nl^'+n^'l)cosB-(lm^'+l^'m)cosC=0

(26)

for all (a,b,c) (Sommerville 1961, Kimberling 1998, p. 29).

The line through a point alpha^':beta^':gamma^' perpendicular to (◇) is given by

|alpha beta gamma; alpha^' beta^' gamma^'; l-mcosC-ncosB m-ncosA-lcosC n-lcosB-mcosA|=0.

(27)

In three-dimensional space, the line passing through the point (x_0,y_0,z_0) and parallel to the nonzero vector v=(a,b,c) has parametric equations

written concisely as

x=x_0+vt.

(31)

Similarly, the line in three dimensions passing through (x_1,y_1) and (x_2,y_2) has parametric vector equation

x=x_1+(x_2-x_1)t,

(32)

where this parametrization corresponds to x(t=0)=x_1 and x(t=1)=x_2.

See also

Asymptote, Branch Cut, Brocard Line, Cayley Lines, Collinear, Concur, Critical Line, Desargues' Theorem, Erdős-Anning Theorem, Euler Line, Flow Line, Gergonne Line, Imaginary Line, 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, Point, Point-Line Distance--2-Dimensional, Point-Line Distance--3-Dimensional, Plane, Plücker Lines, 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: 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. 156, 105-114, Mar. 1956.MacTutor History of Mathematics Archive. "Straight Line." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Straight.html.Sommerville, D. M. Y. Analytical Conics, 3rd ed. London: G. Bell and Sons, p. 186, 1961.Spanier, J. and Oldham, K. B. "The Linear Function bx+c and Its Reciprocal." Ch. 7 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 53-62, 1987.

Cite this as:

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

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