Editing Line (geometry) (section)

===Polar coordinates===
[[File:Parametres polaires droite.svg|alt=see caption|thumb|A line on polar coordinates without passing though the origin, with the general parametric equation written above]]
In a [[Cartesian plane]], [[polar coordinates]] {{math|(''r'', ''θ'')}} are related to [[Cartesian coordinates]] by the parametric equations:<ref>{{Cite book |last1=Torrence |first1=Bruce F. |title=The Student's Introduction to MATHEMATICA: A Handbook for Precalculus, Calculus, and Linear Algebra |last2=Torrence |first2=Eve A. |date=29 Jan 2009 |publisher=[[Cambridge University Press]] |isbn=9781139473736 |pages=314}}</ref><math display="block">x=r\cos\theta, \quad y=r\sin\theta.</math>

In polar coordinates, the equation of a line not passing through the [[origin (mathematics)|origin]]—the point with coordinates {{math|(0, 0)}}—can be written
<math display="block">r = \frac p {\cos (\theta-\varphi)},</math>
with {{math|''r'' > 0}} and <math>\varphi-\pi/2 < \theta < \varphi + \pi/2.</math> Here, {{mvar|p}} is the (positive) length of the [[line segment]] perpendicular to the line and delimited by the origin and the line, and <math>\varphi</math> is the (oriented) angle from the {{mvar|x}}-axis to this segment.

It may be useful to express the equation in terms of the angle <math>\alpha=\varphi+\pi/2</math> between the {{mvar|x}}-axis and the line. In this case, the equation becomes
<math display="block">r=\frac p {\sin (\theta-\alpha)},</math>
with {{math|''r'' > 0}} and <math>0 < \theta < \alpha + \pi.</math>

These equations can be derived from the [[#Normal form|normal form]] of the line equation by setting <math>x = r \cos\theta,</math> and <math>y = r \sin\theta,</math> and then applying the [[angle difference identities|angle difference identity]] for sine or cosine.

These equations can also be proven [[geometry|geometrically]] by applying [[trigonometric functions#Right-angled triangle definitions|right triangle definitions]] of sine and cosine to the [[right triangle]] that has a point of the line and the origin as vertices, and the line and its perpendicular through the origin as sides.

The previous forms do not apply for a line passing through the origin, but a simpler formula can be written: the polar coordinates <math>(r, \theta)</math> of the points of a line passing through the origin and making an angle of <math>\alpha</math> with the {{mvar|x}}-axis, are the pairs <math>(r, \theta)</math> such that 
<math display="block">r\ge 0,\qquad \text{and} \quad \theta=\alpha \quad\text{or}\quad \theta=\alpha +\pi.</math>