Editing Line (geometry) (section)

== Definition ==
{{Main|Line coordinates}}

=== Linear equation ===
{{Main|Linear equation}}
[[File:Linear Function Graph.svg|alt=y = –x + 5 (going down) and y = 0.5x + 2 (rising up slower)|thumb|Line graphs of linear equations on the Cartesian plane ]]
Lines in a Cartesian plane or, more generally, in [[affine coordinates]], are characterized by linear equations. More precisely, every line <math>L</math> (including vertical lines) is the set of all points whose [[Cartesian coordinates|coordinates]] (''x'', ''y'') satisfy a linear equation; that is,
<math display="block">L = \{(x,y)\mid ax+by=c\}, </math>
where ''a'', ''b'' and ''c'' are fixed [[real number]]s (called [[coefficient]]s) such that ''a'' and ''b'' are not both zero. Using this form, vertical lines correspond to equations with ''b'' = 0.

One can further suppose either {{math|1=''c'' = 1}} or {{math|1=''c'' = 0}}, by dividing everything by {{mvar|c}} if it is not zero.

There are many variant ways to write the equation of a line which can all be converted from one to another by algebraic manipulation. The above form is sometimes called the ''standard form''. If the constant term is put on the left, the equation becomes
<math display="block">ax + by - c = 0,</math>
and this is sometimes called the ''general form'' of the equation. However, this terminology is not universally accepted, and many authors do not distinguish these two forms.

These forms are generally named by the type of information (data) about the line that is needed to write down the form. Some of the important data of a line is its slope, [[root of a function|x-intercept]], known points on the line and y-intercept.

The equation of the line passing through two different points <math>P_0( x_0, y_0 )</math> and <math>P_1(x_1, y_1)</math> may be written as
<math display="block">(y - y_0)(x_1 - x_0) = (y_1 - y_0)(x - x_0).</math>
If {{math|''x''<sub>0</sub> ≠ ''x''<sub>1</sub>}}, this equation may be rewritten as
<math display="block">y=(x-x_0)\,\frac{y_1-y_0}{x_1-x_0}+y_0</math>
or
<math display="block">y=x\,\frac{y_1-y_0}{x_1-x_0}+\frac{x_1y_0-x_0y_1}{x_1-x_0}\,.</math>In [[Plane (mathematics)|two dimensions]], the equation for non-vertical lines is often given in the ''[[slope–intercept form]]'':

<math display="block"> y = mx + b </math>
where:
* ''m'' is the [[slope]] or [[slope|gradient]] of the line.
* ''b'' is the [[y-intercept]] of the line.
* ''x'' is the [[independent variable]] of the function {{math|1=''y'' = ''f''(''x'')}}.

The slope of the line through points <math>A(x_a, y_a)</math> and <math>B(x_b, y_b)</math>, when <math>x_a \neq x_b</math>, is given by <math>m = (y_b - y_a)/(x_b - x_a)</math> and the equation of this line can be written <math>y = m (x - x_a) + y_a</math>.

As a note, lines in three dimensions may also be described as the simultaneous solutions of two [[linear equation]]s
<math display="block"> a_1 x + b_1 y + c_1 z - d_1 = 0 </math>
<math display="block"> a_2 x + b_2 y + c_2 z - d_2 = 0 </math>
such that <math> (a_1,b_1,c_1)</math> and <math> (a_2,b_2,c_2)</math> are not proportional (the relations <math> a_1 = t a_2, b_1 = t b_2, c_1 = t c_2 </math> imply <math>t = 0</math>). This follows since in three dimensions a single linear equation typically describes a [[plane (geometry)|plane]] and a line is what is common to two distinct intersecting planes.

=== Parametric equation ===
{{Further|Parametric equation}}
Parametric equations are also used to specify lines, particularly in those in [[Three-dimensional space|three dimensions]] or more because in more than two dimensions lines ''cannot'' be described by a single linear equation.

In three dimensions lines are frequently described by parametric equations:
<math display="block">\begin{align}
x &= x_0 + at \\
y &= y_0 + bt \\
z &= z_0 + ct
\end{align}</math>
where:
* ''x'', ''y'', and ''z'' are all functions of the independent variable ''t'' which ranges over the real numbers.
* (''x''<sub>0</sub>, ''y''<sub>0</sub>, ''z''<sub>0</sub>) is any point on the line.
* ''a'', ''b'', and ''c'' are related to the slope of the line, such that the direction [[vector (geometric)|vector]] (''a'', ''b'', ''c'') is parallel to the line.

Parametric equations for lines in higher dimensions are similar in that they are based on the specification of one point on the line and a direction vector.

=== Hesse normal form ===
{{main|Hesse normal form}}
[[File:Hesse normalenform.svg|thumb|Distance from the origin O to the line E calculated with the Hesse normal form. Normal vector in red, line in green, point O shown in blue.]]
The ''normal form'' (also called the ''Hesse normal form'',<ref>{{citation|title=Plane Analytic Geometry: With Introductory Chapters on the Differential Calculus|first=Maxime|last=Bôcher|publisher=H. Holt|year=1915|author-link=Maxime Bôcher| page=44| url=https://books.google.com/books?id=bYkLAAAAYAAJ&pg=PA44|url-status=live|archive-url=https://web.archive.org/web/20160513124511/https://books.google.com/books?id=bYkLAAAAYAAJ&pg=PA44|archive-date=2016-05-13}}</ref> after the German mathematician [[Otto Hesse|Ludwig Otto Hesse]]), is based on the ''[[normal (geometry)|normal]] segment'' for a given line, which is defined to be the line segment drawn from the [[origin (mathematics)|origin]] perpendicular to the line. This segment joins the origin with the closest point on the line to the origin. The normal form of the equation of a straight line on the plane is given by:
<math display="block"> x \cos \varphi + y \sin \varphi - p = 0 ,</math>
where <math>\varphi</math> is the angle of inclination of the normal segment (the oriented angle from the unit vector of the {{math|''x''}}-axis to this segment), and {{math|''p''}} is the (positive) length of the normal segment. The normal form can be derived from the standard form <math>ax + by = c</math> by dividing all of the coefficients by
<math display="block">\sqrt{a^2 + b^2}.</math>
and also multiplying through by <math>-1</math> if <math>c <0.</math>

Unlike the slope-intercept and intercept forms, this form can represent any line but also requires only two finite parameters, <math>\varphi</math> and {{math|''p''}}, to be specified. If {{math|''p'' > 0}}, then <math>\varphi</math> is uniquely defined modulo {{math|2''π''}}. On the other hand, if the line is through the origin ({{math|1=''c'' = ''p'' = 0}}), one drops the {{math|''c''/{{abs|''c''}}}} term to compute <math>\sin\varphi</math> and <math>\cos\varphi</math>, and it follows that <math>\varphi</math> is only defined modulo {{pi}}.