Euclid defined a line as a ``breadthless length,'' and a straight line as a line which ``lies evenly with the
points on itself'' (Kline 1956, Dunham 1990). Lines are intrinsically 1dimensional objects, but may be embedded in higher dimensional
Spaces. An infinite line passing through points and is denoted
. A Line Segment terminating at these points is denoted .
A line is sometimes called a Straight Line or, more archaically, a Right Line (Casey 1893), to emphasize that
it has no curves anywhere along its length.
Consider first lines in a 2D Plane. The line with xIntercept and
yIntercept is given by
the intercept form

(1) 
The line through with Slope is given by the pointslope form

(2) 
The line with intercept and slope is given by the slopeintercept form

(3) 
The line through and is given by the two point form

(4) 
Other forms are

(5) 

(6) 

(7) 
A line in 2D can also be represented as a Vector. The Vector along the line

(8) 
is given by

(9) 
where . Similarly, Vectors of the form

(10) 
are Perpendicular to the line. Three points lie on a line if

(11) 
The Angle between lines
is

(14) 
The line joining points with Trilinear Coordinates
and
is the set of point
satisfying

(15) 

(16) 
Three lines Concur if their Trilinear Coordinates satisfy
in which case the point is

(20) 
or if the Coefficients of the lines
satisfy

(24) 
Two lines Concur if their Trilinear Coordinates satisfy

(25) 
The line through is the direction and the line through in direction
intersect Iff

(26) 
The line through a point
Parallel to

(27) 
is

(28) 
The lines
are Parallel if

(31) 
for all , and Perpendicular if



(32) 
for all (Sommerville 1924). The line through a point
Perpendicular to
(32) is given by

(33) 
In 3D Space, the line passing through the point and Parallel to the Nonzero Vector

(34) 
has parametric equations

(35) 
See also Asymptote, Brocard Line, Collinear, Concur, Critical Line, Desargues'
Theorem, ErdösAnning Theorem, Line Segment, Ordinary Line, Pencil,
Point, PointLine Distance2D, PointLine Distance3D, Plane, Range (Line Segment), Ray, Solomon's Seal Lines, Steiner Set,
Steiner's Theorem, Sylvester's Line Problem
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. 3095, 1893.
Dunham, W. Journey Through Genius: The Great Theorems of Mathematics. New York: Wiley, p. 32, 1990.
Kline, M. ``The Straight Line.'' Sci. Amer. 156, 105114, Mar. 1956.
MacTutor History of Mathematics Archive. ``Straight Line.''
http://wwwgroups.dcs.stand.ac.uk/~history/Curves/Straight.html.
Sommerville, D. M. Y. Analytical Conics. London: G. Bell, p. 186, 1924.
Spanier, J. and Oldham, K. B. ``The Linear Function and Its Reciprocal.''
Ch. 7 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 5362, 1987.
© 19969 Eric W. Weisstein
19990525