## Line

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 1-dimensional 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 2-D Plane. The line with x-Intercept and y-Intercept is given by the intercept form

 (1)

The line through with Slope is given by the point-slope form
 (2)

The line with -intercept and slope is given by the slope-intercept form
 (3)

The line through and is given by the two point form
 (4)

Other forms are
 (5)

 (6)

 (7)

A line in 2-D 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
 (12) (13)

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
 (17) (18) (19)

in which case the point is
 (20)

or if the Coefficients of the lines
 (21) (22) (23)

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
 (29) (30)

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 3-D 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, Erdös-Anning Theorem, Line Segment, Ordinary Line, Pencil, Point, Point-Line Distance--2-D, Point-Line Distance--3-D, 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. 30-95, 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, 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. 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. 53-62, 1987.