Editing Line (geometry) (section)

=== 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}}.