Line

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

(1)

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

(2)

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

(3)

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

(4)

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

(5)

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

			(6)
			(7)

For and not both zero, other forms are

(8)

(9)

(10)

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

(11)

is given by

(12)

where . Similarly, vectors of the form

(13)

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

(14)

For with , this vector is

(15)

Three points lie on a line if

(16)

The angle between lines

			(17)
			(18)

has directed angle modulo satisfying

(19)

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

(20)

(21)

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

(22)

The line through a point parallel to

(23)

is

(24)

The lines

			(25)
			(26)

are parallel if

(27)

for these reference-triangle side lengths, and perpendicular if

(28)

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

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

(29)

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

			(30)
			(31)
			(32)

written concisely as

(33)

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

(34)

where this parameterization corresponds to and .