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 $A$ and $B$ is denoted ${\leftrightarrow\atop{\displaystyle AB}}$. A Line Segment terminating at these points is denoted $\overline{AB}$. 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 $a$ and y-Intercept $b$ is given by the intercept form

$${ x\over a} + { y\over b} = 1.$$ (1)

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

$$y-y_1 = m(x-x_1).$$ (2)

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

$$y = mx+b.$$ (3)

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

$$y-y_1 = { y_2-y_1\over x_2-x_1} (x-x_1).$$ (4)

Other forms are

$$a(x-x_1)+b(y-y_1)=0$$ (5)

$$ax+by+c=0$$ (6)

$$\left\vert\matrix{x & y & 1\cr x_1 & y_1 & 1\cr x_2 & y_2 & 1\cr}\right\vert=0.$$ (7)

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

$$ax+by = 0$$ (8)

is given by

$$t\left[{\matrix{-b\cr a\cr}}\right],$$ (9)

where $t \in \Bbb{R}$. Similarly, Vectors of the form

$$t\left[{\matrix{a\cr b\cr}}\right]$$ (10)

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

$$\left\vert\matrix{x_1 & y_1 & 1\cr x_2 & y_2 & 1\cr x_3 & y_3 & 1\cr}\right\vert=0.$$ (11)

The Angle between lines

$$A_1x+B_1y+C_1 = 0$$ (12)

$$A_2x+B_2y+C_2 = 0$$ (13)

is

$$\tan\theta={A_1B_2-A_2B_1\over A_1A_2+B_1B_2}.$$ (14)

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

$$\left\vert{\matrix{\alpha & \beta & \gamma\cr \alpha_1 & \be... ... \gamma_1\cr \alpha_2 & \beta_2 & \gamma_2\cr}}\right\vert = 0$$ (15)

$$(\beta_1\gamma_2-\gamma_1\beta_2)\alpha+(\gamma_1\alpha_2-\alpha_1\gamma_2)\beta+(\alpha_1\beta_2-\beta_1\alpha_2)\gamma=0.$$ (16)

Three lines Concur if their Trilinear Coordinates satisfy

$$l_1\alpha+m_1\beta+n_1\gamma = 0$$ (17)

$$l_2\alpha+m_2\beta+n_2\gamma = 0$$ (18)

$$l_3\alpha+m_3\beta+n_3\gamma = 0,$$ (19)

in which case the point is

$$m_2n_3-n_2m_3:n_2l_3-l_2n_3:l_2m_3-m_2l_3,$$ (20)

or if the Coefficients of the lines

$$A_1x+B_1y+C_1 = 0$$ (21)

$$A_2x+B_2y+C_2 = 0$$ (22)

$$A_3x+B_3y+C_3 = 0$$ (23)

satisfy

$$\left\vert\matrix{A_1 & B_1 & C_1\cr A_2 & B_2 & C_2\cr A_3 & B_3 & C_3\cr}\right\vert=0.$$ (24)

Two lines Concur if their Trilinear Coordinates satisfy

$$\left\vert{\matrix{l_1 & m_1 & n_1\cr l_2 & m_2 & n_2\cr l_3 & m_3 & n_3\cr}}\right\vert=0.$$ (25)

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

$$\left\vert\matrix{x_2-x_1 & y_2-y_1 & z_2-z_1\cr a_1 & b_1 & c_1\cr a_2 & b_2 & c_2\cr}\right\vert=0.$$ (26)

The line through a point $\alpha':\beta':\gamma'$ Parallel to

$$l\alpha+m\beta+n\gamma=0$$ (27)

is

$$\left\vert\matrix{ \alpha & \beta & \gamma\cr \alpha' & \beta' & \gamma'\cr bn-cm & cl-an & am-bl}\right\vert=0.$$ (28)

The lines

$$l\alpha+m\beta+n\gamma = 0$$ (29)

$$l'\alpha+m'\beta+n'\gamma = 0$$ (30)

are Parallel if

$$a(mn'-nm')+b(nl'-ln')+c(lm'-ml')=0$$ (31)

for all $(a, b, c)$, and Perpendicular if

$2abc(ll'+mm'+nn')-(mn'+m'm)\cos A$
$-(nl'+n'l)\cos B-(lm'+l'm)\cos C=0\qquad$	(32)

for all $(a, b, c)$ (Sommerville 1924). The line through a point $\alpha':\beta':\gamma'$ Perpendicular to (32) is given by

$$\left\vert{\matrix{ \alpha & \beta & \gamma\cr \alpha' & \be... ...n\cos B & \hfill -l\cos C & \hfill -m\cos A\cr}}\right\vert=0.$$ (33)

In 3-D Space, the line passing through the point $(x_0,y_0,z_0)$ and Parallel to the Nonzero Vector

$${\bf v}=\left[{\matrix{a\cr b\cr c\cr}}\right]$$ (34)

has parametric equations

$$\cases{ x = x_0+at\cr y = y_0+bt\cr z = z_0+ct.\cr}$$ (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 $bx+c$ and Its Reciprocal.'' Ch. 7 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 53-62, 1987.