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
![\begin{displaymath}
{ x\over a} + { y\over b} = 1.
\end{displaymath}](l2_170.gif) |
(1) |
The line through
with Slope
is given by the point-slope form
![\begin{displaymath}
y-y_1 = m(x-x_1).
\end{displaymath}](l2_173.gif) |
(2) |
The line with
-intercept
and slope
is given by the slope-intercept form
![\begin{displaymath}
y = mx+b.
\end{displaymath}](l2_174.gif) |
(3) |
The line through
and
is given by the two point form
![\begin{displaymath}
y-y_1 = { y_2-y_1\over x_2-x_1} (x-x_1).
\end{displaymath}](l2_176.gif) |
(4) |
Other forms are
![\begin{displaymath}
a(x-x_1)+b(y-y_1)=0
\end{displaymath}](l2_177.gif) |
(5) |
![\begin{displaymath}
ax+by+c=0
\end{displaymath}](l2_178.gif) |
(6) |
![\begin{displaymath}
\left\vert\matrix{x & y & 1\cr x_1 & y_1 & 1\cr x_2 & y_2 & 1\cr}\right\vert=0.
\end{displaymath}](l2_179.gif) |
(7) |
A line in 2-D can also be represented as a Vector. The Vector along the line
![\begin{displaymath}
ax+by = 0
\end{displaymath}](l2_180.gif) |
(8) |
is given by
![\begin{displaymath}
t\left[{\matrix{-b\cr a\cr}}\right],
\end{displaymath}](l2_181.gif) |
(9) |
where
. Similarly, Vectors of the form
![\begin{displaymath}
t\left[{\matrix{a\cr b\cr}}\right]
\end{displaymath}](l2_183.gif) |
(10) |
are Perpendicular to the line. Three points lie on a line if
![\begin{displaymath}
\left\vert\matrix{x_1 & y_1 & 1\cr x_2 & y_2 & 1\cr x_3 & y_3 & 1\cr}\right\vert=0.
\end{displaymath}](l2_184.gif) |
(11) |
The Angle between lines
is
![\begin{displaymath}
\tan\theta={A_1B_2-A_2B_1\over A_1A_2+B_1B_2}.
\end{displaymath}](l2_188.gif) |
(14) |
The line joining points with Trilinear Coordinates
and
is the set of point
satisfying
![\begin{displaymath}
\left\vert{\matrix{\alpha & \beta & \gamma\cr \alpha_1 & \be...
... \gamma_1\cr \alpha_2 & \beta_2 & \gamma_2\cr}}\right\vert = 0
\end{displaymath}](l2_192.gif) |
(15) |
![\begin{displaymath}
(\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.
\end{displaymath}](l2_193.gif) |
(16) |
Three lines Concur if their Trilinear Coordinates satisfy
in which case the point is
![\begin{displaymath}
m_2n_3-n_2m_3:n_2l_3-l_2n_3:l_2m_3-m_2l_3,
\end{displaymath}](l2_198.gif) |
(20) |
or if the Coefficients of the lines
satisfy
![\begin{displaymath}
\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.
\end{displaymath}](l2_200.gif) |
(24) |
Two lines Concur if their Trilinear Coordinates satisfy
![\begin{displaymath}
\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.
\end{displaymath}](l2_201.gif) |
(25) |
The line through
is the direction
and the line through
in direction
intersect Iff
![\begin{displaymath}
\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.
\end{displaymath}](l2_206.gif) |
(26) |
The line through a point
Parallel to
![\begin{displaymath}
l\alpha+m\beta+n\gamma=0
\end{displaymath}](l2_208.gif) |
(27) |
is
![\begin{displaymath}
\left\vert\matrix{
\alpha & \beta & \gamma\cr
\alpha' & \beta' & \gamma'\cr
bn-cm & cl-an & am-bl}\right\vert=0.
\end{displaymath}](l2_209.gif) |
(28) |
The lines
are Parallel if
![\begin{displaymath}
a(mn'-nm')+b(nl'-ln')+c(lm'-ml')=0
\end{displaymath}](l2_212.gif) |
(31) |
for all
, and Perpendicular if
|
|
|
(32) |
for all
(Sommerville 1924). The line through a point
Perpendicular to
(32) is given by
![\begin{displaymath}
\left\vert{\matrix{
\alpha & \beta & \gamma\cr
\alpha' & \be...
...n\cos B & \hfill -l\cos C & \hfill -m\cos A\cr}}\right\vert=0.
\end{displaymath}](l2_216.gif) |
(33) |
In 3-D Space, the line passing through the point
and Parallel to the Nonzero Vector
![\begin{displaymath}
{\bf v}=\left[{\matrix{a\cr b\cr c\cr}}\right]
\end{displaymath}](l2_218.gif) |
(34) |
has parametric equations
![\begin{displaymath}
\cases{
x = x_0+at\cr
y = y_0+bt\cr
z = z_0+ct.\cr}
\end{displaymath}](l2_219.gif) |
(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.
© 1996-9 Eric W. Weisstein
1999-05-25